Chromotopytag:chromotopy.org,2010:9ea8819ec0e23cbaa10796803d04383b2014-05-09T19:39:00ZEric Petersonhttp://chromotopy.orgtag:chromotopy.org,2014-05-01:1398967790On L-functions and algebraic K-theory2014-05-02T15:22:54Z2014-05-01T18:09:50Z<p>Hi all! In this post, I’ll attempt to make some progress towards demystifying the relationship between L-functions and algebraic K-theory, with reference to a very simple example. I’ll say something about L-functions a little lower down, but the only thing I’ll say about K-theory in general is that it’s a way to get a spectrum from, say, a scheme.</p>
<p>The deep connection between these two objects is supposed to be exemplified by the following very approximate and possibly incorrect statement.</p>
<p><strong>Ur-conjecture.</strong> [Lichtenbaum, Bloch-Kato, Beilinson, Scholbach…] Let <script type="math/tex">X</script> be a geometric object, probably a scheme of a particular kind, and let <script type="math/tex">\zeta_X</script> be the zeta function associated to <script type="math/tex">X</script>. Then for <script type="math/tex">n</script> a sufficiently large positive integer,</p>
<script type="math/tex; mode=display">
\zeta_X(n) = \frac {|K_{2n}(X)|_\text{tors}} {|K_{2n - 1}(X)|_\text{tors}} R(n)
</script>
<p>where <script type="math/tex">R(n)</script> is some factor. Obviously this is completely meaningless until you know what <script type="math/tex">R(n)</script> is, but people do have expressions for it (a relevant buzzword here is “Borel regulator”), about which I’ll say no more. The point is that there are formulae for values of L-functions at integers (and I’m sorry, but I’ll use “L-function” and “zeta function” interchangeably) involving torsion orders of K-groups.</p>
<p>A conjecture of this form has been proven for <script type="math/tex">X</script> the ring of integers of a totally real abelian field by piling Iwasawa theory and other manoeuverings onto the cases of the Quillen-Lichtenbaum conjecture proven by Voevodsky. It’s a humongous proof. In general, these conjectures are wide open and should be regarded with reverence.</p>
<h2 id="l-functions">L-functions</h2>
<p>Before I really get going, though, I’m going to say a word about L-functions, which by convention are meromorphic functions on <script type="math/tex">\mathbb{C}</script> with certain properties. I’m going to list these salient properties in bold, and illustrate them using the historically first L-function, which is of course the Riemann zeta <script type="math/tex">\zeta(s)</script>. We’ll later find that each of these properties has a purely homotopy-theoretic analogue on the K-theory spectrum.</p>
<p><strong>Dirichlet series.</strong> To an L-function <script type="math/tex">L</script> is associated a Dirichlet series expansion</p>
<script type="math/tex; mode=display">
\sum_{n \geq 0} a_n n^{-s}.
</script>
<p>with <script type="math/tex">a_n \in \mathbb{C}</script>. <script type="math/tex">\zeta(s)</script> has <script type="math/tex">a_n = 1</script> for all <script type="math/tex">n</script>.</p>
<p><strong>Abscissa of convergence.</strong> There is some <script type="math/tex">\sigma \in \mathbb{R}</script>, called the abscissa of convergence, such that the Dirichlet series converges to <script type="math/tex">L(s)</script> for <script type="math/tex">\mathbf{Re}(s) > \sigma</script>. The abscissa of convergence of <script type="math/tex">\zeta</script> is 1.</p>
<p><strong>Euler product expansion.</strong> There is an infinite product, typically over closed points of a scheme, of nice simple functions that also converges to <script type="math/tex">L(s)</script> on some respectable proportion of <script type="math/tex">\mathbb{C}</script>. The Euler product expansion of <script type="math/tex">\zeta</script> is</p>
<script type="math/tex; mode=display">
\prod_{p \text{ prime}} \frac 1 {1 - p^s}.
</script>
<p><strong>Analytic continuation.</strong> This is implicit in the fact that I said an L-function was a meromorphic function on <script type="math/tex">\mathbb{C}</script> in the first place, but if you take the more practical view that an L-function is defined by its Dirichlet series or Euler product, there’s some content here, and analytic continuation statements are usually hard to prove.</p>
<p><strong>Functional equation.</strong> An L-function has a (possibly twisted) reflective symmetry: there is some real number <script type="math/tex">\tau</script>, frequently equal to the abscissa of convergence, such that</p>
<script type="math/tex; mode=display">
L(s) = A(s) L(\tau - s).
</script>
<p>For the Riemann zeta, we have (lifted from Wikipedia) <script type="math/tex">\tau = 1</script> and </p>
<script type="math/tex; mode=display">
A(s) = 2^s \pi^{s-1} \sin \left (\frac {\pi s} 2 \right) \Gamma(1 - s)
</script>
<p>This just comes from the Archimedean L-factor (we’ll say nothing about that here) but there can be other things too. Analytic continuation and functional equation are usually proved together.</p>
<p>You probably expect to be able to formulate a Riemann hypothesis for a general L-function, too, but we won’t get into that here.</p>
<h2 id="the-simplest-l-function">The simplest L-function</h2>
<p>I said the Riemann zeta was the historically first L-function, but it’s not the simplest. The simplest is the zeta function of a finite field <script type="math/tex">\mathbb{F}_q</script>, which is a single Euler factor:</p>
<script type="math/tex; mode=display">
\Xi_q(s) = \frac 1 {1 - q^s}.
</script>
<p>The number theorists probably use different notation for this, but I am a carefree rogue who does as he wills.</p>
<p><script type="math/tex">\Xi_q</script> is manifestly meromorphic; it is its own Euler product; its Dirichlet series is given by expanding the above expression as a geometric series, with abscissa of convergence <script type="math/tex">0</script>; and its functional equation is</p>
<script type="math/tex; mode=display">
\Xi_q(-s) = -q^s \Xi_q(s).
</script>
<p>This is the only example of an L-function which one can comfortably manipulate with one’s hands without substantial analytic ingenuity. It’s this basic example that we’ll spend much of the rest of this blog post studying.</p>
<h2 id="the-k-theory-of-a-finite-field">The K-theory of a finite field</h2>
<p>In a 1972 paper, Quillen proved one of my all-time favourite theorems by describing the homotopy type of the K-theory spectrum of a finite field.</p>
<p><strong>Theorem.</strong> [Quillen] The connected component <script type="math/tex">K(\mathbb{F}_q)_0</script> of the <script type="math/tex">0</script>-space of the K-theory spectrum of <script type="math/tex">\mathbb{F}_q</script> fits into a fibre sequence</p>
<script type="math/tex; mode=display">
K(\mathbb{F}_q)_0 \to BU \overset{\psi^q - 1} \to BU.
</script>
<p>where <script type="math/tex">\psi^q</script> is the <script type="math/tex">q</script>th Adams operation.</p>
<p>This is the only example of a (global) K-theory spectrum which one can comfortably manipulate with one’s hands without substantial topological ingenuity. By Bott periodicity, we know the homotopy groups of BU:</p>
<script type="math/tex; mode=display">% <![CDATA[
\pi_n BU = \begin{cases} \mathbb{Z} & n = 2m > 0 \text{ is even;} \\
0 & \text{otherwise.} \end{cases}
%]]></script>
<p>We also know that <script type="math/tex">\psi^q</script> acts by multiplication by <script type="math/tex">q^m</script> on <script type="math/tex">\pi_{2m} BU</script>. That means we can read off the positive-degree homotopy groups of <script type="math/tex">K(\mathbb{F}_q)</script>:</p>
<script type="math/tex; mode=display">% <![CDATA[
K_n (\mathbb{F}_q) = \begin{cases} \mathbb{Z}/(q^m - 1) & n = 2m - 1 \text{ is odd;} \\
0 & \text{otherwise.} \end{cases}
%]]></script>
<p>Motivated by the ur-conjecture stated above, let’s define a function <script type="math/tex">f_q : \mathbb{N} \to \mathbb{C}</script> by</p>
<script type="math/tex; mode=display">
f_q(n) = \frac {|K_{2n}(\mathbb{F}_q)|} {|K_{2n - 1}(\mathbb{F}_q)|}.
</script>
<p>Suppose, as if possessed by a ghost, you wrote this function down without ever having heard of an L-function. You would still notice that</p>
<script type="math/tex; mode=display">
f_q(n) = \frac 1 {q^n - 1}
</script>
<p>is the restriction to <script type="math/tex">\mathbb{N}</script> of an obvious complex-analytic function: <script type="math/tex">-\Xi_q</script>. This is the easiest case of the conjecture.</p>
<p>Now what of the values of <script type="math/tex">\Xi_q</script> at negative integers? We might hope that they have something to do with the negative homotopy groups of <script type="math/tex">K(\mathbb{F}_q)</script>, which is, after all, a spectrum. But as things stand, these negative homotopy groups are zero. This rift is signaling that the arithmetic incarnation of the bare K-theory spectrum is not the L-function; rather, it’s the Dirichlet series of the L-function, or the part of the L-function to the right of the abscissa of convergence. The topological object that we should really associate to the full L-function is the K(1)-localisation of the K-theory spectrum. To do this, we need to fix an ambient prime <script type="math/tex">p</script>, and let’s make sure it’s different from the characteristic of <script type="math/tex">\mathbb{F}_q</script>.</p>
<h2 id="the-k1-localisation-of-the-k-theory-of-a-finite-field">The K(1)-localisation of the K-theory of a finite field</h2>
<p>I’m going to self-consciously write out all <script type="math/tex">L_{K(1)}</script>s in full. It’ll take up space, but hopefully it’ll aid clarity of notation.</p>
<p>Here’s a slight variant of Quillen’s calculation; if you like, it arises by applying the Bousfield-Kuhn functor to the statement of Quillen’s theorem above.</p>
<p><strong>Proposition.</strong> The K(1)-localised K-theory spectrum <script type="math/tex">L_{K(1)} K(\mathbb{F}_q)</script> fits into a fibre sequence</p>
<script type="math/tex; mode=display">
L_{K(1)} K(\mathbb{F}_q) \to KU^\wedge_p \overset{\psi^q -1} \to KU^\wedge_p.
</script>
<p>Thus the homotopy groups of <script type="math/tex">L_{K(1)} K(\mathbb{F}_q)</script>, positive and negative, are given by</p>
<script type="math/tex; mode=display">% <![CDATA[
\pi_n L_{K(1)} K(\mathbb{F}_q) = \begin{cases} \mathbb{Z}_p / (q^m - 1) & n = 2m -1 \text{ odd}\\
\mathbb{Z}_p & n = 0 \\
0 & \text{otherwise.}
\end{cases}
%]]></script>
<p>Until further notice, when we talk about the “value” of an L-function at a point, we’ll mean up to multiplication by a rational number which is a p-adic unit. With this caveat, we now have the identity</p>
<script type="math/tex; mode=display">
\Xi_q(n) = \frac {|\pi_{2n} L_{K(1)} K(\mathbb{F}_q)|} {|\pi_{2n - 1} L_{K(1)} K(\mathbb{F}_q)|}
</script>
<p>for all <script type="math/tex">n \in \mathbb{Z}</script>, except at <script type="math/tex">n = 0</script>, where things are untidy because <script type="math/tex">\Xi_q</script> has a pole and both homotopy groups in the formula are infinite, and a regulator gets involved. That’s nice.</p>
<p>We need to make a couple more observations about the homotopy groups of the localised K-theory spectrum.</p>
<p>First, the natural map</p>
<script type="math/tex; mode=display">
K(\mathbb{F}_q)^\wedge_p \to L_{K(1)} K(\mathbb{F}_q)
</script>
<p>is an equivalence in degrees greater than <script type="math/tex">0</script>. (It’s also an equivalence in degree <script type="math/tex">0</script>, but I’d like to propose that we regard that as an accident.) This <script type="math/tex">0</script> is the very same <script type="math/tex">0</script> that occurs as the abscissa of convergence of <script type="math/tex">\Xi_q</script>, and it also shows up as the étale cohomological dimension of <script type="math/tex">\mathbb{F}_q</script> minus 1 - a number I’d like to refer to as the Quillen-Lichtenbaum constant of <script type="math/tex">\mathbb{F}_q</script>.</p>
<p>Second, the functional equation for <script type="math/tex">\Xi_q(n)</script> is reflected in the homotopy groups of <script type="math/tex">L_{K(1)} K(\mathbb{F}_q)</script> as follows. For any integer <script type="math/tex">m</script>, the <script type="math/tex">p</script>-adic valuation of <script type="math/tex">q^m - 1</script> is the same as that of <script type="math/tex">q^{-m} - 1</script>. Thus for any odd integer <script type="math/tex">n</script>, we have an isomorphism</p>
<script type="math/tex; mode=display">
\pi_n L_{K(1)} K(\mathbb{F}_q) \cong \pi_{-2 - n} L_{K(1)} K(\mathbb{F}_q).
</script>
<p>Can this apparent duality be given a topological explanation?</p>
<h2 id="gross-hopkins-and-spanier-whitehead-duality">Gross-Hopkins and Spanier-Whitehead duality</h2>
<p>We’re going to do this by invoking a total of four or five different dualities, one of which is deep and will be totally blackboxed, so get ready.</p>
<p>Let’s start tackling this by rearranging the above isomorphism slightly. Both of the groups there arise as quotients of <script type="math/tex">\mathbb{Z}_p</script>, so they come with a choice of generator: the image of <script type="math/tex">1</script>. That means that either group can be identified with its Pontryagin dual, so let’s do it on the left:</p>
<script type="math/tex; mode=display">
(\pi_n L_{K(1)} K(\mathbb{F}_q))^\vee \cong \pi_{-2 - n} L_{K(1)} K(\mathbb{F}_q).
</script>
<p>There, now it looks more like a sensible duality. And the Pontryagin dual in there is a strong hint that Brown-Comenetz duality is involved. Briefly, the Brown-Comenetz dualising spectrum is a spectrum <script type="math/tex">I</script> defined by the natural isomorphism</p>
<script type="math/tex; mode=display">
\pi_n \mathbf{Map}(E, I) \cong (\pi_{-n} E)^\vee
</script>
<p>for spectra <script type="math/tex">E</script>. To save pixels, we’ll write <script type="math/tex">IE</script> for <script type="math/tex">\mathbf{Map}(E, I)</script>. Now our isomorphism reads</p>
<script type="math/tex; mode=display">
\pi_{-n} I(L_{K(1)} K(\mathbb{F}_q)) \cong \pi_{-2-n} L_{K(1)} K(\mathbb{F}_q).
</script>
<p>But height 1 Gross-Hopkins duality states that for a K(1)-local spectrum E,</p>
<script type="math/tex; mode=display">
IME \cong \mathbf{Map}(E, L_{K(1)} \mathbb{S}^{-2}).
</script>
<p>Here <script type="math/tex">M</script> is the first monochromatic slice functor, which for a K(1)-local spectrum is just the fibre of the map to its rationalisation. <script type="math/tex">M</script> isn’t doing anything too destructive here, so let’s omit it for now and figure out its role in a moment. We’ve arrived at</p>
<script type="math/tex; mode=display">
\pi_{-2-n} \mathbf{Map}(L_{K(1)} K(\mathbb{F}_q), L_{K(1)} \mathbb{S}) \cong \pi_{-2-n} L_{K(1)} K(\mathbb{F}_q).
</script>
<p>at least for odd <script type="math/tex">n \neq -1</script>. So we’d be home if <script type="math/tex">K(\mathbb{F}_q)</script> were to be K(1)-locally Spanier-Whitehead self-dual. And it is! I learned this in Dustin Clausen’s thesis defence, but one can prove it by showing that</p>
<script type="math/tex; mode=display">
KU^\wedge_p \wedge KU^\wedge_p \overset{\mu} \to KU^\wedge_p \overset{J} \to L_{K(1)} \mathbb{S}^{-1}
</script>
<p>exhibits <script type="math/tex">KU^\wedge_p</script> as K(1)-locally self-dual up to a shift of 1, compatibly with Adams operations, and plugging this into the fiber sequence for <script type="math/tex">L_{K(1)}K(\mathbb{F}_q)</script>. And thus the duality is explained.</p>
<p>Let’s circle back for a moment and say a quick word about the role of the monochromatic slice functor <script type="math/tex">M</script>. Since <script type="math/tex">L_{K(1)} K(\mathbb{F}_q)</script> doesn’t have many torsion-free homotopy groups, the only effect of <script type="math/tex">M</script> is to convert the <script type="math/tex">\mathbb{Z}_p</script>s in degrees <script type="math/tex">0</script> and <script type="math/tex">-1</script> into <script type="math/tex">\mathbb{Q}_p/ \mathbb{Z}_p</script>s in degrees <script type="math/tex">-1</script> and <script type="math/tex">-2</script>. But that was exactly what was needed to get the original duality statement to work in those degrees, once we put in the Pontryagin dual. Thus the duality is not merely explained but embellished with a pristine little bow.</p>
<h2 id="the-dictionary-and-some-questions">The dictionary, and some questions</h2>
<p>As promised, we’ve given homotopy-theoretic interpretations of (almost) all the fundamental analytic properties of L-functions. Here’s a table to summarise:</p>
<table>
<thead>
<tr>
<th style="text-align: center">Analysis</th>
<th style="text-align: center"> </th>
<th style="text-align: center">Topology</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center"> </td>
<td style="text-align: center"> </td>
<td style="text-align: center"> </td>
</tr>
<tr>
<td style="text-align: center">L-function</td>
<td style="text-align: center"> </td>
<td style="text-align: center">K(1)-localised K-theory spectrum</td>
</tr>
<tr>
<td style="text-align: center">Dirichlet series</td>
<td style="text-align: center"> </td>
<td style="text-align: center">Bare K-theory spectrum</td>
</tr>
<tr>
<td style="text-align: center">Abscissa of convergence</td>
<td style="text-align: center"> </td>
<td style="text-align: center">Quillen-Lichtenbaum constant</td>
</tr>
<tr>
<td style="text-align: center">Euler product</td>
<td style="text-align: center"> </td>
<td style="text-align: center">?? (But see below)</td>
</tr>
<tr>
<td style="text-align: center">Analytic continuation</td>
<td style="text-align: center"> </td>
<td style="text-align: center">K(1)-localisation</td>
</tr>
<tr>
<td style="text-align: center">Functional equation</td>
<td style="text-align: center"> </td>
<td style="text-align: center">Gross-Hopkins + Spanier-Whitehead duality</td>
</tr>
</tbody>
</table>
<p>(I apologise for the poor formatting - I’m not used to Markdown, and the latex support doesn’t seem to run to table environments).</p>
<p>Some questions I don’t know how to answer at this point:</p>
<ul>
<li>Let X be a Dedekind domain, L its field of fractions, and <script type="math/tex">k(x)</script> the residue field at a closed point <script type="math/tex">x \in X</script>. Then there’s a cofiber sequence</li>
</ul>
<script type="math/tex; mode=display">
\bigvee_{x \in X} K(k(x)) \to K(X) \to K(L)
</script>
<p>The left-hand arrow looks an awful lot like it’s trying to express K(X) as an Euler product over x of K(k(x)), but I don’t know how to make that precise. K(L) is a nuisance.</p>
<ul>
<li>
<p>Now let X be a smooth proper variety over <script type="math/tex">\mathbb{F}_q</script>. There’s a spectral sequence with <script type="math/tex">E_2</script> term given by étale cohomology of X with suitably Tate-twisted <script type="math/tex">\mathbb{Z}_p</script> coefficients which abuts to the K(1)-local K-theory of X. Can we use this to relate the formula for the zeta function of X in terms of the characteristic polynomial of Frobenius on étale cohomology to a K-theoretic description of the special values?</p>
</li>
<li>
<p>What’s the analogue of <script type="math/tex">K(\mathbb{F}_q)</script> for an archimedean L-factor? This might be related to work of Connes and Consani on cyclic homology with adelic coefficients.</p>
</li>
</ul>
<h2 id="p-adic-l-functions">p-adic L-functions</h2>
<p>This section is tangential to the rest of the post, but I think it suggests some interesting avenues of investigation. Up until now we’ve been discussing L-functions defined on <script type="math/tex">\mathbb{C}</script>, but L-functions defined on <script type="math/tex">\mathbb{Z}_p</script> are also of interest to number theorists. We’ll construct the p-adic analogue of an Euler factor, and see that it throws up some unexpected topology.</p>
<p>In this section, we’ll assume that <script type="math/tex">p</script> is odd. Things are probably substantially more awkward when <script type="math/tex">p = 2</script>.</p>
<p>Let’s return (up to sign) to our function <script type="math/tex">f_q</script>, this time regarded as a function <script type="math/tex"> \mathbb{N} \to \mathbb{Q}_p</script>:</p>
<script type="math/tex; mode=display">
f_q(n) = \frac 1 {1 - q^n}.
</script>
<p>We’d like to extend <script type="math/tex">f_q</script> to a continuous function <script type="math/tex">E_q: \mathbb{Z}_p \to \mathbb{Q}_p</script>. p-adic continuity sounds weak compared to complex analyticity, but it’s actually kind of hard to be p-adically continuous, and since <script type="math/tex">\mathbb{N}</script> is dense in <script type="math/tex">\mathbb{Z}_p</script>, we’re going to have at most one choice of extension.</p>
<p>In fact, we don’t have any choices, a fact which won’t surprise anyone who’s studied p-adic L-functions before. The proof of this is a byproduct of the construction of the real p-adic L-function. Let</p>
<script type="math/tex; mode=display">
S = \{n \in \mathbb{N} \, | \, q^n \equiv 1 \mod p.\}
</script>
<p>Observe that <script type="math/tex">S</script> is already dense in <script type="math/tex">\mathbb{Z}_p</script>, so if we can find a continuous function <script type="math/tex">\mathbb{Z}_p \to \mathbb{Q}_p</script> that agrees with <script type="math/tex">f_q</script> on <script type="math/tex">S</script>, that’s the best we can do, whether the two functions agree on the whole of <script type="math/tex">\mathbb{N}</script> or not.</p>
<p>Here’s the crux: There’s a continuous function </p>
<script type="math/tex; mode=display">
\log : \mathbb{Z}_p \to p\mathbb{Z}_p
</script>
<p>known as the Iwasawa logarithm, and a continuous function </p>
<script type="math/tex; mode=display">
\exp: p\mathbb{Z}_p \to \mathbb{Z}_p
</script>
<p>known as the p-adic exponential. They have most of the properties you’d expect of a pair of functions called <script type="math/tex">\exp</script> and <script type="math/tex">\log</script>, except that the identity</p>
<script type="math/tex; mode=display">
\exp (\log (x)) = x
</script>
<p>only holds when <script type="math/tex">x</script> is congruent to <script type="math/tex">1</script> mod <script type="math/tex">p</script>. In general,</p>
<script type="math/tex; mode=display">
\exp (\log(x)) = p^r \omega x
</script>
<p>where the integer <script type="math/tex">r</script> and the <script type="math/tex">(p-1)</script>st root of unity <script type="math/tex">\omega</script> are uniquely determined by the requirement that <script type="math/tex">p^r \omega x \equiv 1 \mod p</script>.</p>
<p>So we can follow our hearts and define</p>
<script type="math/tex; mode=display">
E_q(s) = \frac 1 {1 - \exp(s \log q)}
</script>
<p>but we have to live with the fact that <script type="math/tex">E_q(n) = f_q(n)</script> only when <script type="math/tex">n \in S</script>. For general <script type="math/tex">n \in \mathbb{N}</script>,</p>
<script type="math/tex; mode=display">
E_q(n) = \frac 1 {1 - (\omega q)^n}
</script>
<p>for a suitable <script type="math/tex">(p-1)</script>st root of unity <script type="math/tex">\omega</script>.</p>
<p>Fine. Where’s the topology? What spectrum is <script type="math/tex">E_q</script> telling us about? At integers outside <script type="math/tex">S</script>, the values of <script type="math/tex">E_q</script> don’t match up with the homotopy groups of <script type="math/tex">L_{K(1)} K(\mathbb{F}_q)</script> anymore. Instead,</p>
<script type="math/tex; mode=display">
E_q(n) = \frac {|\pi_{2n} M|}{|\pi_{2n - 1} M|}
</script>
<p>for all integers <script type="math/tex">n \neq 0</script>, where M is a K(1)-local spectrum defined as the fiber of</p>
<p><script type="math/tex">\psi^{\omega q} - 1 : KU^\wedge_p \to KU^\wedge_p</script>.</p>
<p>So, a couple more questions:</p>
<ul>
<li>
<p>What’s special about M? Why is it any better than <script type="math/tex">L_{K(1)}K(\mathbb{F}_q)</script>? What other spectra are “p-adically continuous” in this sense?</p>
</li>
<li>
<p>Constructing p-adic L-functions more serious than <script type="math/tex">E_q</script> usually requires some fairly heavy p-adic analysis. Can we partly bypass this by proving p-adic continuity results for the Picard-graded homotopy groups of certain K(1)-local spectra and using the orders of these homotopy groups? I believe some such continuity results already exist, in the folklore if not in the literature.</p>
</li>
</ul>
<p>Thanks for reading this many words! Let me finish up by remarking that some of the stuff in this post carries through to K-theory spectra of rings of integers, and using work of Dwyer and Mitchell as a bridge, one can obtain results on the special values of Kubota-Leopoldt-type p-adic zeta functions. If I can think of enough interesting stuff to say about that, it might be the subject of a future post.</p>
<p>I think the Chromotopy platform doesn’t currently support blog comments, but if you have any comments or questons, please email me at sglasman at math dot mit dot edu.</p>
<hr />
<p>Update: Dustin Clausen emailed me with some helpful comments about the Spanier-Whitehead self-duality of <script type="math/tex">L_{K(1)} K(\mathbb{F}_q)</script>, resulting in some alterations in the text.</p>
tag:chromotopy.org,2014-03-13:1394732648The K-Theory of Endomorphisms2014-05-01T18:10:47Z2014-03-13T17:44:08Z<p>For the purposes of this post, I am going to assume that you are all familiar with the basics of Algebraic <script type="math/tex">K</script>-theory. If you’re not, just treat it as a black box, a gadget which takes in one of :</p>
<ul>
<li>a (simplicial) ring (spectrum)</li>
<li>an augmented (simplicial) bimodule over a ring</li>
<li>an exact category</li>
<li>a Waldhausen category</li>
<li>a small, stable <script type="math/tex">\infty</script>-category</li>
</ul>
<p>and spits out a spectrum <script type="math/tex">K(\mathcal{C})</script>, such that <script type="math/tex">K(\mathcal{C})</script> does the job of ”storing all Euler characteristics” or “additive invariants”.</p>
<p>An important example is when <script type="math/tex">\mathcal{C}</script> is the category of finitely generated projective modules over a ring <script type="math/tex">R</script>, and then we call this <script type="math/tex">K(R)</script>. It is a generally-accepted fact that the <script type="math/tex">K</script>-theory of rings is generically a very difficult and often useful thing to compute (knowledge of the <script type="math/tex">K</script>-groups of <script type="math/tex">\mathbb{Z}</script>, for example, would be quite valuable to many). In an ideal world, we would be able to understand the functor</p>
<script type="math/tex; mode=display">
A \mapsto \tilde{K}(A) = hofib(K(A) \rightarrow K(R))
</script>
<p>on the category of algebras augmented over a given ring <script type="math/tex">R</script>, but this ends up being fairly intractable. One might hope that it would be easier to look at what happens when restricted to free augmented algebras, that is looking at the functor on (flat) <script type="math/tex">R</script>-bimodules</p>
<script type="math/tex; mode=display">
M \mapsto \tilde{K}(T_R (M)),
</script>
<p>where <script type="math/tex">T_R (M)</script> is the tensor algebra on <script type="math/tex">M</script>. This is pretty tough, but it turns out that we can do it. Let’s take a baby step first, and try to understand the K-theory of the “linearized” tensor algebra. The 1st Goodwillie derivative of the identity functor on augmented <script type="math/tex">R</script>-algebras is given by</p>
<script type="math/tex; mode=display">
P_1 (id) (A) = A / I^2 ,
</script>
<p>where <script type="math/tex">I</script> is the augmentation ideal of <script type="math/tex">A</script>. In the case where <script type="math/tex">A = T_R (M)</script> for some <script type="math/tex">R</script>-bimodule <script type="math/tex">M</script>, then the Goodwillie derivative <script type="math/tex">P_1 (id)(T_R (M)) = R \oplus M</script>, where <script type="math/tex">R \oplus M</script> is the the square-zero extension of <script type="math/tex">R</script> by <script type="math/tex">M</script>, the ring given by demanding that <script type="math/tex">M^2 = 0</script>. In studying the <script type="math/tex">K</script>-theory of square-zero extensions, then, we are also studying the <script type="math/tex">K</script>-theory of the linearization of the tensor algebra functor (note how many simplifications we have already done!).</p>
<p>A different strategy for dealing with this computational difficulty is to try and understand how the <script type="math/tex">K</script>-theory of a ring changes as we “perturb” the ring. To do this, we look at “parametrized <script type="math/tex">K</script>-theory” (or the “<script type="math/tex">K</script>-theory of parametrized endomorphisms”):</p>
<p><strong>Definition.</strong> Let <script type="math/tex">R</script> be a ring, and let <script type="math/tex">M</script> be an <script type="math/tex">R</script>-bimodule. We define the <em>parametrized</em> <script type="math/tex">K</script>-<em>theory of</em> <script type="math/tex">R</script> with coefficients in <script type="math/tex">M</script>, <script type="math/tex">K(R;M)</script>, to be the <script type="math/tex">K</script>-theory of the exact category of pairs <script type="math/tex">(P,f)</script> where <script type="math/tex">P</script> is a finitely-generated projective <script type="math/tex">R</script>-module and <script type="math/tex">f: P \rightarrow P \otimes_{R} M</script> is a map of <script type="math/tex">R</script>-modules.</p>
<p>We think about <script type="math/tex">K(R;M)</script> as being the <script type="math/tex">K</script>-theory of endomorphisms with coefficients that are allowed to be in <script type="math/tex">M</script>. If our picture of a finitely-generated projective <script type="math/tex">R</script>-module <script type="math/tex">P</script> is as living as a summand of a rank <script type="math/tex">n</script> free <script type="math/tex">R</script>-module, then an element of the above exact category is an <script type="math/tex">n \times n</script> matrix with entries in <script type="math/tex">M</script> that commutes with the projection map <script type="math/tex">R^n \rightarrow R^n</script> defining <script type="math/tex">P</script>.</p>
<p>Why should we look towards endomorphisms as perturbations? Well, the picture is supposed to be the following: Let <script type="math/tex">R</script> be a ring and <script type="math/tex">M</script> an <script type="math/tex">R</script>-bimodule. This is the same data as a sheaf <script type="math/tex">\mathcal{F}_M</script> over <script type="math/tex">\operatorname{Spec}(R)</script>, and we would like to think of a deformation of this over, say, <script type="math/tex">R[t]/t^2</script> as a flat sheaf <script type="math/tex">\mathcal{F}_{M'}</script> over <script type="math/tex">R[t]/t^2</script> (a module) that restricts to <script type="math/tex">\mathcal{F}_M</script> over <script type="math/tex">\operatorname{Spec}(R)</script>.</p>
<p align="center"><a href="images/Screen-Shot-2014-03-13-at-2.21.02-PM.png"><img class="alignnone size-medium wp-image-1475 aligncenter" alt="Screen Shot 2014-03-13 at 2.21.02 PM" src="images/Screen-Shot-2014-03-13-at-2.21.02-PM-300x86.png" width="300" height="86" /></a></p>
<p>which corresponds to an element of <script type="math/tex">\operatorname{Ext}^1_R (M,M)</script>, or a derived endomorphism. The idea is that an extension of <script type="math/tex">M</script> corresponds to a deformation of <script type="math/tex">M</script>, which is a reasonable perspective.</p>
<p>The above definition is not immediately seen to be relevant to our original stated desire to study perturbations, but in the investigations of Dundas and McCarthy of stable <script type="math/tex">K</script>-theory, the <script type="math/tex">K</script>-theory of endomorphisms naturally comes up. In this paper they prove the following theorem:</p>
<p><strong>Theorem.</strong> For <script type="math/tex">R</script> a ring and <script type="math/tex">M</script> a discrete <script type="math/tex">R</script>-bimodule)</p>
<script type="math/tex; mode=display">
\tilde{K}(R;B_\bullet M) \simeq \text{hofib} \left\{ K(R \oplus M) \rightarrow K(R) \right\} = \tilde{K}(R \oplus M),
</script>
<p>where <script type="math/tex">R \oplus M</script> is again the the square-zero extension of <script type="math/tex">R</script> by <script type="math/tex">M</script>.</p>
<p>We think of <script type="math/tex">R \oplus M</script> as a perturbation of <script type="math/tex">R</script> by <script type="math/tex">M</script>, as the elements that we are adding on (those coming from the direct summand <script type="math/tex">M</script>) are “so small’’ that they multiply to zero. This result relates the “perturbation’’ approach to understanding <script type="math/tex">K</script>-theory to our earlier potential approach of understanding the free objects in the category of augmented <script type="math/tex">R</script>-algebras.</p>
<p>That’s great, but we’d like to actually have some idea of what these things are. To get a hint at what we should be looking for, we go back to what was classically studied by Almkvist et al.</p>
<h3 id="theory-of-endomorphisms-the-classical-story"><script type="math/tex">K</script>-Theory of Endomorphisms: The Classical Story</h3>
<p><strong>Definition.</strong> Let <script type="math/tex">R</script> be a ring and consider the category <script type="math/tex">End(R)</script>, whose objects are pairs <script type="math/tex">(P,f)</script> with <script type="math/tex">P</script> a finitely-generated projective <script type="math/tex">R</script>-module and <script type="math/tex">f:P \rightarrow P</script> an endomorphism. The morphisms in this category are commutative diagrams of the appropriate type.</p>
<p>Now, we might ask what possible “additive invariants’’ there are on this category, having in mind a few examples. The key (and, as it turns out, universal) one is the following:</p>
<p><strong>Example.</strong> [The Characteristic Polynomial]
Let <script type="math/tex">(P,f) \in End(R)</script>, then the <em>characteristic polynomial</em> of <script type="math/tex">f</script> is given by</p>
<script type="math/tex; mode=display">
\lambda_t (f) = \sum_{i \geq 0} Tr(\Lambda^i f)t^i.
</script>
<p>This can also be obtained in the usual way as a determinant.</p>
<p>The important property of the characteristic polynomial is that if we have a commutative diagram in <script type="math/tex">End(R)</script></p>
<p style="text-align: center"><a href="images/Screen-Shot-2014-03-13-at-2.20.50-PM.png"><img class="alignnone size-medium wp-image-1474" alt="Screen Shot 2014-03-13 at 2.20.50 PM" src="images/Screen-Shot-2014-03-13-at-2.20.50-PM-300x85.png" width="300" height="85" /></a></p>
<p>with exact rows, then</p>
<script type="math/tex; mode=display">
\lambda_t (f) = \lambda_t (f') \cdot \lambda_t (f''),
</script>
<p>meaning that the characteristic polynomial takes short exact sequences of endomorphisms to products (which are sums in the abelian group where the characteristic polynomials lie).</p>
<p>We now know that a search for additive invariants is a desire to compute <script type="math/tex">K</script>-theory, and so we define:</p>
<p><strong>Definition.</strong> <script type="math/tex">K_0 (End(R))</script> is defined to be the free abelian group on isomorphism classes of objects in <script type="math/tex">End(R)</script> modulo the subgroup generated by the relations <script type="math/tex">[(P,f)] = [(P',f')] + [(P'',f'')]</script> if there is a commutative diagram</p>
<p align="center"><a href="images/Screen-Shot-2014-03-13-at-2.20.46-PM.png"><img class="alignnone size-medium wp-image-1473 aligncenter" alt="Screen Shot 2014-03-13 at 2.20.46 PM" src="images/Screen-Shot-2014-03-13-at-2.20.46-PM-300x85.png" width="300" height="85" /></a></p>
<p>with the rows exact. There is a natural splitting</p>
<script type="math/tex; mode=display">
K_0 (End(R)) \simeq K_0 (R) \times \tilde{K}_0 (End(R))
</script>
<p>coming from thinking of the category of finitely generated projective <script type="math/tex">R</script>-modules as living in <script type="math/tex">End(R)</script> as the guys with <script type="math/tex">0</script> endomorphisms.</p>
<p>Of course, <script type="math/tex">End(R)</script> is an exact category, and we could define higher <script type="math/tex">K</script>-groups for this as well.</p>
<p><script type="math/tex">K_0 (End(R))</script> is the repository for additive invariants, in that it has the following universal property:</p>
<p><strong>Proposition.</strong> Let <script type="math/tex">F</script> be a map from the commutative monoid of isomorphism classes of objects in <script type="math/tex">End(R)</script> to an abelian group <script type="math/tex">A</script>, such that <script type="math/tex">F</script> splits short exact sequences as above. Then <script type="math/tex">F</script> factors through <script type="math/tex">K_0 (End(R))</script>, or <script type="math/tex">K_0(End(R))</script> is the initial abelian group for which short exact sequences of endomorphisms split.</p>
<p>What does this look like, though? Well, let’s go back to the characteristc polynomial:</p>
<p><strong>Theorem.</strong> The map</p>
<script type="math/tex; mode=display">
c: [(P,f)] \mapsto ([P], \lambda_t (f))
</script>
<p>is an isomorphism</p>
<script type="math/tex; mode=display">
K_0 (End(R)) \simeq K_0 (A) \times \tilde{W}(R),
</script>
<p>where</p>
<script type="math/tex; mode=display">
\tilde{W} (R) = \left \{ \dfrac{1 + a_1 t + \dots + a_n t^n}{1+ b_1t + \dots + b_m t^m} ; \; a_i, b_j \in R \right \}
</script>
<p>is the multiplicative group of fractions with constant term 1.</p>
<p>The moral of this is that the characteristic polynomial encodes all of the additive information about an endomorphism (the trace is a special case of this, of course, being read off by the constant term of the characteristic polynomial). Something interesting is the following:</p>
<p><strong>Proposition.</strong> The inclusion <script type="math/tex">\tilde{W}(R) \rightarrow W(R)</script> exhibits <script type="math/tex">\tilde{K}_0 (R)</script> as a dense <script type="math/tex">\lambda</script>-subring of <script type="math/tex">W(R)</script>, where <script type="math/tex">W(R)</script> are the big Witt vectors of <script type="math/tex">R</script>, modelled as power series with constant term <script type="math/tex">1</script>.</p>
<p>This means that we might as well think of the big Witt vectors as being limits of characteristic polynomials of endomorphisms, which is the starting point for the line of thought that led to the Lindenstrauss and McCarthy results.</p>
<p>It is also important to mention what some of the uses of this equivalence are:</p>
<ul>
<li>Calculations that may be difficult to perform on Witt vectors might be easy if we think of them as coming from characteristic polynomials of endomorphisms;</li>
<li>Operations that exist on Witt vectors get turned into operations on K-theory, and vice-versa:</li>
<li>The <script type="math/tex">n</script>-th ghost map is given by <script type="math/tex">gh_n ([P,f] = tr(f^n))</script>;</li>
<li>The Frobenius map is given by <script type="math/tex">F_n ([P,f]) = [P, f^n]</script>;</li>
<li>The Verschiebung map is given by <script type="math/tex">[V_n ([P,f]) = [P^{\oplus n}, v_n f]</script>, where <script type="math/tex">v_n f</script> is represented by a shift in the first <script type="math/tex">n-1</script> factors and action by <script type="math/tex">f</script> in the last factor. This represents an <script type="math/tex">n</script>-th root of <script type="math/tex">f</script>, in that <script type="math/tex">V_n^n</script> looks like applying <script type="math/tex">f</script> to all of the blocks of <script type="math/tex">P^{\oplus n}</script>.</li>
</ul>
<h3 id="lindenstrauss-mccarthy-and-topological-witt-vectors">Lindenstrauss-McCarthy and Topological Witt Vectors</h3>
<p>The first thing to do to start generalizing the previous construction is to allow for a “parametrization’’, like we said before.</p>
<p><strong>Definition.</strong> Let <script type="math/tex">R</script> be a ring and <script type="math/tex">M</script> an <script type="math/tex">R</script>-bimodule. <script type="math/tex">End(R;M)</script> is the category whose objects are pairs <script type="math/tex">(P,f)</script> with <script type="math/tex">P</script> a finitely generated projective <script type="math/tex">R</script>-module and <script type="math/tex">f: P \rightarrow P \otimes_R M</script> a map of <script type="math/tex">R</script>-modules. As in <script type="math/tex">End(R)</script>, morphisms are commutative diagrams.</p>
<p><script type="math/tex">End(R;M)</script> inherits an exact/Waldhausen structure from considering what’s happening in the “base’’, and so it makes sense to define <em>parametrized</em> <script type="math/tex">K</script><em>-theory</em> as</p>
<script type="math/tex; mode=display">
K(R;M) := K(End(R;M)).
</script>
<p>There is a natural map from <script type="math/tex">End(R;M)</script> to the category of finitely generated projective <script type="math/tex">R</script>-modules that forgets the endomorphism, and so we can reduce and consider</p>
<script type="math/tex; mode=display">
\tilde{K}(R;M) := hofib(K(R;M) \rightarrow K(R)).
</script>
<p>We can also consider <script type="math/tex">M</script> to be simplicial by geometrically realizing, and this assembles to a functor</p>
<script type="math/tex; mode=display">
\tilde{K}(R;-): \left\{\text{simplicial } R\text{-bimodules} \right \}\; \rightarrow \text{Spectra},
</script>
<p>which is a good setting to do Goodwillie calculus.</p>
<p>The remarkable result of Lindenstrauss-McCarthy is the following:</p>
<p><strong>Theorem.</strong> The functors <script type="math/tex">\tilde{K} (R; -), W(R;-)</script> from simplicial <script type="math/tex">R</script>-bimodules to spectra have the same Taylor tower, where <script type="math/tex">W(R;-)</script> is a “topological Witt vectors’’ construction.</p>
<p>Now, this is only remarkable if I tell you what this <script type="math/tex">W(R;-)</script> functor is, so let’s do that. Lindenstrauss and McCarthy describe this using the (notationally cumbersome) language of FSPs instead of spectra or spectral categories, which makes everything a bit hard to read. The idea is a bit simpler, however.</p>
<p><strong>Construction.</strong> Let <script type="math/tex">R</script> be a ring, <script type="math/tex">M</script> a bimodule. We define a bunch of spectra <script type="math/tex">U^n (R;M)</script> with <script type="math/tex">C_n</script> action by letting <script type="math/tex">U^n (R;M)</script> be the derived cyclic tensor product (over <script type="math/tex">R</script>) of <script type="math/tex">n</script> copies of <script type="math/tex">M</script>.</p>
<p>There is an evident <script type="math/tex">C_n</script> action on <script type="math/tex">U^n (R;M)</script> given by permuting the factors, and moreover when <script type="math/tex">n \mid N</script>, we have natural restriction maps between the fixed points</p>
<script type="math/tex; mode=display">
res : U^N (R;M)^{C_N} \rightarrow U^n (R;M)^{C_n}.
</script>
<p>We then define</p>
<script type="math/tex; mode=display">
W(R;M) := holim_{res} U^n (R;M)^{C_n}.
</script>
<p><strong>Example.</strong> <script type="math/tex">U^1 (R;M) \simeq THH(R;M)</script> as normally defined, and if <script type="math/tex">R = M</script> then we have that <script type="math/tex">W(R;M) \simeq TR(R;M)</script>.</p>
<p>This functor <script type="math/tex">W(R;M)</script> is meant to be an attempt to define <script type="math/tex">TR</script> in the absence of the cyclic symmetry that is present when <script type="math/tex">M = R</script>. The remarkable thing is that this actually works! The nomenclature is justified by the following <script type="math/tex">p</script>-typical statement:</p>
<p><strong>Theorem.</strong> (Hesselholt-Madsen) <script type="math/tex">\pi_0 (TR(R;p)) \simeq W_{(p)}(R),</script></p>
<p>where the latter is a suitable version of <script type="math/tex">W(R;M)</script> that takes a homotopy limit over only <script type="math/tex">U^{p^n}(R;M)^{C_{p^n}}</script>.</p>
<p>The virtue of this setup is that we have explicit understanding of what the layers of the Taylor tower of <script type="math/tex">W(R;-)</script> look like, as well as various splitting theorems, e.g. we have the following ``fundamental cofibration sequence’’:</p>
<p><strong>Theorem.</strong> There is a homotopy fiber sequence</p>
<script type="math/tex; mode=display">
U^n (R; M)_{hC_n} \rightarrow W^{n}(R; M) \rightarrow W^{(n-1)}(R; M),
</script>
<p>where <script type="math/tex">W^{n}(R;M)</script> are the functors obtained by truncating the homotopy limit defining <script type="math/tex">W(R;M)</script>.</p>
<p>Moreover, we have that <script type="math/tex">W^{n}(R;M)</script> is the <script type="math/tex">n</script>th polynomial approximation of <script type="math/tex">W (R;M)</script>, so this is the fiber sequence that computes the layers of the Taylor tower.</p>
<h3 id="generalizations">Generalizations</h3>
<p>The problem with the Lindenstrauss-McCarthy story is that it is only proven for discrete rings, but all of the constructions floating around can just as well be made when <script type="math/tex">R</script> is a connective ring spectrum and <script type="math/tex">M</script> is a (simplicial) <script type="math/tex">R</script>-bimodule.</p>
<p>One of the great tricks that one can do is to use the resolution of connective ring spectra by simplicial rings. That is, given a connective ring spectrum <script type="math/tex">R</script>, there is a <script type="math/tex">id</script>-Cartesian cube with initial vertex <script type="math/tex">R</script>, and such that all other vertices are canonically stably equivalent to the Eilenberg-MacLane spectrum of a simplicial ring.</p>
<p>The following fact lets us promote results about simplicial rings to those about connective ring spectra:</p>
<p><strong>Theorem.</strong> If <script type="math/tex">\chi</script> is an <script type="math/tex">(id)</script>-Cartesian <script type="math/tex">k</script>-cube of connective ring spectra, then the cube <script type="math/tex">K(\chi)</script> is <script type="math/tex">(k+1)</script>-Cartesian.</span></p>
<p><strong>Idea for proving generalization:</strong> Show that the functors <script type="math/tex">W(-;-), \tilde{K}(-;-)</script> have a similar property, so that the equivalence given by Lindenstrauss and McCarthy can be promoted to one for connective ring spectra.</p>
<p>This works just fine, and we arrive at:</p>
<p><strong>Theorem.</strong> (P.)<strong> </strong>The Lindenstrauss-McCarthy result holds for connective ring spectra.</p>
<p>Unfortunately, this is mostly just a one-off trick, and we are still looking for a good conceptual understanding of what the Lindenstrauss-McCarthy equivalence really <em>means</em>. It largely goes back to the Dundas-McCarthy theorem mentioned earlier, which can also be proven for connective ring spectra, but fails to admit a generalizable proof. The issue that arises there is that their proof relies heavily on the linear models of <script type="math/tex">K</script>-theory when working over discrete rings, and these simply faily to work when we move to other contexts (their proof involves the addition of maps, a simplicial model of <script type="math/tex">K</script>-theory, etc.).</p>
<p>If we had a good way of dealing with the coherence issues in defining the Dundas-McCarthy maps, then we might be able to generalize the proof and understand what’s going on, but that’s something that has yet to be done.</p>
tag:chromotopy.org,2014-02-26:1393395294Hypothetical abelian varieties from K-theory2014-05-09T19:39:00Z2014-02-26T06:14:54Z<p>I heard an idea tossed around recently that I’d like to share with you all. I worry that it might be a little half-baked, as I’ve only heard about it recently and so maybe haven’t spent enough time sketching out the edges of it. Maybe writing this will help. Throughout, <script type="math/tex">k</script> is a perfect field of positive characteristic <script type="math/tex">p</script>.</p>
<p>Where Lie theorists study Lie algebras, formal algebraic geometers study (covariant) Dieudonné modules. The essential observation is that the sorts of formal Lie groups appearing in algebraic topology are commutative and one-dimensional, meaning that their associated Lie algebras are one-dimensional vector spaces with vanishing brackets, and so it is unreasonable to think that it would carry much interesting information about its parent formal Lie group. To correct for this, one takes the collection of <em>all</em> curves on the formal group (i.e., without reducing to their linear equivalence classes, as one does when building the Lie algebra) and remembers enough structure stemming from the group multiplication that this assignment</p>
<script type="math/tex; mode=display">
D: \left\{\text{formal groups}\right\} \longrightarrow \left\{\begin{array}{c}\text{collections of}\;p\text{-typical curves} \\ + \\ \text{structure} \end{array}\right\}
</script>
<p>becomes an equivalence. Such collections of curves form modules, called “Dieudonné modules”, over a certain ground ring, called the Cartier ring, which is built upon the ambient field <script type="math/tex">k</script>. The three relevant pieces of structure are the actions of homotheties, of the Frobenius, and of the Verschiebung, which were all described <a href="what-are-witt-vectors">way back in this post on Witt vectors</a>. (In the notation of that post, we’re interested just in <script type="math/tex">F_p</script> and <script type="math/tex">V_p</script>.) Altogether, this gives a formula for the Cartier ring:</p>
<script type="math/tex; mode=display">
\mathrm{Cart}(k) = \mathbb{W}(k)\langle F,V \rangle / \left(\begin{array}{c}VF = FV = p, \\ Fa = a^\sigma F, \\ aV = Va^\sigma\end{array}\right).
</script>
<p>Say that a Dieudonné module is <em>formal</em> when it is: finite rank; free as a <script type="math/tex">\mathbb{W}(k)</script>-module; reduced, meaning that it’s <script type="math/tex">V</script>-adically complete; and uniform, meaning that the natural map <script type="math/tex">M / VM \to V^k M / V^{k+1} M</script> is an isomorphism. Then, the Dieudonné functor on finite height formal groups lands in the subcategory of formal Dieudonné modules, and there it restricts to an equivalence. Actually, more is true: the Dieudonné module of a <script type="math/tex">p</script>-divisible group can be made sense of, and Dieudonné modules which are free of finite rank (but without reducedness or uniformity) are equivalent to <script type="math/tex">p</script>-divisible groups.</p>
<p>Dieudonné modules are thrilling because <em>they are just modules</em>, whereas <script type="math/tex">p</script>-divisible groups are these unwieldy ind-systems of finite group schemes. The relative simplicity of the data of a Dieudonné module allows one to compute basic invariants very quickly:</p>
<blockquote>
<p><strong>Theorem:</strong> Take <script type="math/tex">\mathbb{G}</script> to be a <script type="math/tex">p</script>-divisible group and <script type="math/tex">M</script> its Dieudonné module. There is a natural isomorphism of <script type="math/tex">k</script>-vector spaces <script type="math/tex">T_0 \mathbb{G} \cong M / VM</script>. In the case that <script type="math/tex">\mathbb{G}</script> is one-dimensional, then the rank of <script type="math/tex">M</script> as a <script type="math/tex">\mathbb{W}(k)</script>-module agrees with the height <script type="math/tex">d</script> of <script type="math/tex">\mathbb{G}</script>. Moreover, if <script type="math/tex">\gamma</script> is a coordinate on <script type="math/tex">\mathbb{G}</script> (considered as a curve), then <script type="math/tex">\{\gamma, V\gamma, \ldots, V^{d-1} \gamma\}</script> forms a basis for <script type="math/tex">M</script>.</p>
</blockquote>
<p>There is also something like a classification of the simple Dieudonné modules over <script type="math/tex">\bar{\mathbb{F}}_p</script> (where, among other things, the étale component of a p-divisible group carries little data):</p>
<blockquote>
<p><strong>Theorem</strong> (Dieudonné): For <script type="math/tex">m</script> and <script type="math/tex">n</script> coprime and positive, set</p>
</blockquote>
<script type="math/tex; mode=display">
G_{m,n} = \mathrm{Cart}(\bar{\mathbb{F}}_p) / (V^m = F^n).
</script>
<blockquote>
<p>Additionally, allow the pairs <script type="math/tex">(m, n) = (0, 1)</script> to get</p>
</blockquote>
<script type="math/tex; mode=display">
G_{0,1} = D(\mathbb{G}_m[p^\infty])
</script>
<blockquote>
<p>and <script type="math/tex">(m, n) = (1, 0)</script> to get</p>
</blockquote>
<script type="math/tex; mode=display">
G_{1,0} = D(\mathbb{Q} / \mathbb{Z}).
</script>
<blockquote>
<p>(All these modules <script type="math/tex">G_{m,n}</script> have formal dimension <script type="math/tex">n</script> and height <script type="math/tex">(m+n)</script>.) For any simple Dieudonné module <script type="math/tex">M</script>, there is an <em>isogeny</em> <script type="math/tex">M \to G_{m,n}</script>, i.e., a map with finite kernel and cokernel. Moreover, up to isogeny every Dieudonné module is the direct sum of simple objects.</p>
</blockquote>
<p>Every abelian variety <script type="math/tex">A</script> comes with a p-divisible group <script type="math/tex">A[p^\infty]</script> arising from its system of <script type="math/tex">p</script>-power order torsion points. This connection is remarkably strong; for instance, there is the following theorem:</p>
<blockquote>
<p><strong>Theorem</strong> (Serre–Tate): Over a <script type="math/tex">p</script>-adic base, the infinitesimal deformation theories of an abelian variety <script type="math/tex">A</script> and its <script type="math/tex">p</script>-divisible group <script type="math/tex">A[p^\infty]</script> agree naturally.[1]</p>
</blockquote>
<p>For this reason and others, the <script type="math/tex">p</script>-divisible group of an abelian variety carries fairly strong content about the parent variety. On the other hand, it is not immediately clear which <script type="math/tex">p</script>-divisible groups arise in this way. Toward this end, there is a symmetry condition:</p>
<blockquote>
<p><strong>Lemma</strong> (“Riemann–Manin symmetry condition”): As every abelian variety is isogenous to its Poincaré dual and the corresponding (Cartier) duality on <script type="math/tex">p</script>-divisible groups sends (<script type="math/tex">V</script> to <script type="math/tex">F</script> and hence) <script type="math/tex">G_{m,n}</script> to <script type="math/tex">G_{n,m}</script>, these summands must appear in pairs in the isogeny type of the Dieudonné module of <script type="math/tex">A[p^\infty]</script>.</p>
</blockquote>
<p>As a simple example, this gives the usual categorization of elliptic curves: an elliptic curve is <script type="math/tex">1</script>-dimensional, hence has <script type="math/tex">p</script>-divisible group of height <script type="math/tex">2</script>. One possibility, called a supersingular curve, is for the Dieudonné module to be isogenous to <script type="math/tex">G_{1,1}</script>; this is a <script type="math/tex">1</script>-dimensional formal group of height <script type="math/tex">2</script> and it satisfies the symmetry condition. The only other possibility, called an ordinary curve, is for the Dieudonné module to be isogenous to <script type="math/tex">G_{1,0} \oplus G_{0,1}</script>; this is the sum of a <script type="math/tex">1</script>-dimensional formal group of height <script type="math/tex">1</script> with an étale component of height <script type="math/tex">1</script>, and it too satisfies the symmetry condition.</p>
<p>A remarkable theorem is that the converse of the symmetry lemma holds as well:</p>
<blockquote>
<p><strong>Theorem</strong> (Serre, Oort; conjectured by Manin): If a Dieudonné module <script type="math/tex">M</script> satisfies the above symmetry condition, then there exists an abelian variety whose <script type="math/tex">p</script>-divisible group is isogenous to <script type="math/tex">M</script>.</p>
</blockquote>
<p>Both proofs of this theorem are very constructive. Serre’s proof explicitly names abelian hypersurfaces whose <script type="math/tex">p</script>-divisible groups are of the form <script type="math/tex">G_{m,n} \oplus G_{n,m}</script>, for instance.</p>
<p>Now, finally, some input from algebraic topology. The Morava <script type="math/tex">K</script>-theories of Eilenberg–Mac Lane spaces give a collection of formal groups which can be interpreted in the following way:</p>
<blockquote>
<p><strong>Theorem</strong> (Ravenel–Wilson): For <script type="math/tex">% <![CDATA[
1 < m \le n %]]></script>, the <script type="math/tex">p</script>-divisible group associated to <script type="math/tex">K(n)^* K(\mathbb{Q}/\mathbb{Z}, m)</script> is (in a suitable sense) the <script type="math/tex">m</script>th exterior power of the <script type="math/tex">p</script>-divisible group <script type="math/tex">K(n)^* K(\mathbb{Q}/\mathbb{Z}, 1)</script>. It is smooth, has formal dimension <script type="math/tex">\binom{n-1}{m-1}</script>, and has height <script type="math/tex">\binom{n}{m}</script>. (Additionally, it is zero for <script type="math/tex">m > n</script>.)</p>
</blockquote>
<blockquote>
<p><strong>Theorem</strong> (Buchstaber–Lazarev): For <script type="math/tex">% <![CDATA[
1 < m \le n %]]></script>, the same <script type="math/tex">p</script>-divisible group has Dieudonné module isogenous to the product of <script type="math/tex">\frac{1}{n_0} \cdot \binom{n}{m}</script> copies of <script type="math/tex">G_{n_0-m_0,m_0}</script>, where <script type="math/tex">m_0/n_0</script> is the reduced fraction of <script type="math/tex">m/n</script>.</p>
</blockquote>
<p>The conclusion of Buchstaber and Lazarev is that this means that these <script type="math/tex">p</script>-divisible groups almost never have realizations as abelian varieties, since they mostly don’t satisfy the symmetry condition. The only time that they do is something of an accident: when <script type="math/tex">n</script> is even and <script type="math/tex">m = n/2</script>, then the corresponding Dieudonné module is isogenous to that of a large product of copies of a supersingular elliptic curve. However, Ravenel observed that Pascal’s triangle is symmetric:</p>
<blockquote>
<p><strong>Observation</strong> (I heard this from Ravenel, but surely Buchstaber and Lazarev knew of it): The sum of all of the Dieudonné modules</p>
</blockquote>
<script type="math/tex; mode=display">
\bigoplus_{m=0}^{\text{$n$ (or $\infty$)}} D(\operatorname{Spf} K(n)^* K(\mathbb{Q}/\mathbb{Z}, m))
</script>
<blockquote>
<p>satisfies the Riemann-Manin symmetry condition.</p>
</blockquote>
<p>This is an interesting observation. In light of the comments at the end of the Buchstaber–Lazarev paper, one wonders: why privilege <script type="math/tex">m = n/2</script>? But, even more honestly, why privilege <script type="math/tex">m = 1</script> in our study of chromatic homotopy theory? A recurring obstacle to our understanding of higher-height cohomology theories has been the disconnection from the picture of globally defined abelian varieties. Could there be a naturally occurring abelian variety whose <script type="math/tex">p</script>-divisible group realizes the large (<script type="math/tex">2^{n-1}</script>-dimensional!) formal group associated to the <script type="math/tex">K</script>-theoretic Hopf ring of Eilenberg–Mac Lane spaces?</p>
<p>The explicit nature of the solutions to Manin’s conjecture show us that, yes, it is certainly possible to write down large products of hypersurfaces to give a positive answer to this question with the words “naturally occurring” deleted. This alone isn’t very helpful, however, and so to pin down what we might be even talking about there are a number of smaller observations that might help:</p>
<ul>
<li>If such an abelian variety existed, there would be a strange filtration imposed on its <script type="math/tex">p</script>-divisible group arising from the degrees of the Eilenberg–Mac Lane spaces. The Riemann–Manin symmetry condition tells us that these <script type="math/tex">G_{m,n}</script> and <script type="math/tex">G_{n,m}</script> factors must come in pairs, but in almost all situations, one pair appears on one side of the middle-dimension <script type="math/tex">n/2</script> and the other appears on the other side. What could this mean in terms of the hypothetical abelian variety?</li>
<li>Relatedly, this pairing arises from the “<script type="math/tex">\circ</script>-product” structure on the level of Hopf algebras, or as a sort of Hodge-star operation on the level of Dieudonné modules. What sort of structure on the abelian variety would induce such an operation, and in such an orderly fashion?</li>
<li>There should be accessible examples of these hypothetical varieties at low heights. For instance, the one associated to height <script type="math/tex">1</script> via mod-<script type="math/tex">p</script> complex <script type="math/tex">K</script>-theory is (canonically, not merely isogenous to) a sum <script type="math/tex">G_{1,0} \oplus G_{0,1}</script> — i.e., it comes from an ordinary elliptic curve. Can we identify <em>which</em> elliptic curve — and in what sense we can even ask this question? Is there a naturally occurring map from the forms of <script type="math/tex">\mathbb{G}_m</script> to the ordinary locus on the (noncompactified?) moduli of elliptic curves? What about forms of <script type="math/tex">\hat{\mathbb{G}}_m</script>? What if we select a supersingular elliptic curve instead — is there an instructive assignment to abelian varieties whose Dieudonné module has isogeny type <script type="math/tex">G_{1,0} \oplus G_{1,1} \oplus G_{0,1}</script>? (On the face of it, this last bit doesn’t look so helpful, but maybe it is.)</li>
</ul>
<p>[1] - Presumably this can be expressed by saying that the map from a moduli of abelian varieties to a moduli of <script type="math/tex">p</script>-divisible groups is formally étale, but no one seems to say this, so maybe I’m missing something.</p>
<hr />
<p>I’ve been making an effort to learn some arithmetic geometry recently. I started with local class field theory, which was mind-blowing. When I was a first year, someone sat me down and instructed me that I must take a course in complex geometry to become competent — and they were right, and it was a wonderful course, and I’m really glad I got that advice. I have no idea how (especially as I’ve been bumbling about with formal groups for so long!) it slipped past me that local class field theory is another one of these core competencies, and really one of the great achievements of twentieth century mathematics.</p>
<p>I gave a kind of measly talk about this a month ago, when I was trying to stir up interest in a reading group. The notes are a little batty, but they’re fun enough, and you can find them <a href="http://math.berkeley.edu/~ericp/latex/misc/rg-fargues.pdf">here</a>.</p>
<hr />
<p>Speaking of mind-blowing things, last week there was a week-long workshop on perfectoid spaces at MSRI, which I attended between one-third and one-half of. There are video lectures available <a href="http://www.msri.org/workshops/731">on the MSRI website</a>; at the very least everyone should watch Scholze’s introduction, just to get a sense of what all the fuss is about, and then ideally both of Weinstein’s lectures, which were excellent and very much adjacent to the subject of this post — this blog, really.</p>
<p>And, as an uninspired parting remark, we topologists do have access to the pro-system</p>
<script type="math/tex; mode=display">\left( \cdots \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty\right) \ldots</script>
tag:chromotopy.org,2013-11-27:1385574210Some Thoughts on Descent and Descent Data2013-11-27T17:43:30Z2013-11-27T17:43:30Z
<p>So this is just an attempt to make clear some things I’ve been thinking about lately in the areas of descent in algebraic geometry and topology. All of the following is well known and well documented in many places. I’ve particularly learned a great deal from Tyler Lawson about it, as well as the collective knowledge of the nLab, the Stacks Project and MathOverflow. I’ve also spent some time with Jacob Lurie’s various works, including DAG XI and section 6.2 of Higher Algebra. I’m probably also getting quite a bit incorrect, which is due to me and not any of the above mentioned people or references. Please send me an e-mail if there’s anything glaring, and I’ll attempt to correct it or make it rigorous. At the moment this is rather vague stuff…</p>
<p>Everyone’s pretty familiar with the (co)equalizer sheaf condition. That is, if we’ve got a map of rings <script type="math/tex">R \to S</script> and we want to know whether or not an <script type="math/tex">S</script>-module <script type="math/tex">M</script> has a (possibly unique) “representative” among <script type="math/tex">R</script>-modules, we need to make sure that when we tensor <script type="math/tex">M</script> up to being an <script type="math/tex">S\otimes_R S</script>-module along either the left unit <script type="math/tex">\eta_L:S\cong S\otimes_R R\to S\otimes_R S</script> or the right unit <script type="math/tex">\eta_R:S\cong R\otimes_R S\to S\otimes_R S</script>, we get the same <script type="math/tex">S\otimes_R S</script> module.</p>
<p>You might be more familiar with the opposite of this diagram that you get from thinking of such a module as a sheaf on <script type="math/tex">Spec(S)</script>. Obviously we’re leaving out some details here. If I’m working with, say, stacks (valued in groupoids) rather than sheaves, I’ll need to extend this diagram another level, which says that the cocycle condition isn’t satisfied up to equality, but rather to coherent isomorphisms (which then satisfy another condition up to equality!). If we’re working in a sheaf valued in, say, spaces or quasicategories, we have to extend this diagram all the way up (to <script type="math/tex"> \infty</script>).</p>
<p>Now, I want to talk a bit about what we mean by a category of “descent data.” Generally for a map of “rings” (in whichever (discrete or quasi-)category we’re interested in) <script type="math/tex"> \varphi:R\to S</script>, the “descent problem” associated to this map is the question: Given an <script type="math/tex"> S</script>-module <script type="math/tex"> M</script>, when is it the case that there is an <script type="math/tex"> R</script>-module <script type="math/tex"> M_0</script> such that <script type="math/tex"> M\cong M_0\otimes_R S</script>? Again, in the case of modules, this is just the same thing as satisfying the sheaf condition. It’s saying that we’ve got some sheaf over <script type="math/tex"> Spec(S)</script> and we want to “descend” it down to <script type="math/tex"> Spec(R)</script>. Note that one typically leaves unsaid exactly <em>how</em> <script type="math/tex"> S</script> is an <script type="math/tex"> R</script>-module, since it’s assumed to be clear that it’s along the map <script type="math/tex"> \varphi</script>. However, sometimes I’ll use the notation <script type="math/tex"> M_0\otimes_\varphi S</script> to indicate the precise way in which <script type="math/tex"> S</script> is an <script type="math/tex"> R</script> module, and perhaps more importantly, exactly in what way we’re lifting up the module structure on <script type="math/tex"> M_0</script>. Given the previous discussion, you can probably already see why this is going to be important to us.</p>
<p>The category of “descent data” for such a map <script type="math/tex"> \varphi</script> should be, intuitively, things over <script type="math/tex"> S</script> with some kind of information on how to produce a thing over <script type="math/tex"> R</script>. There are many, many ways to formalize this. If you’re used to dealing with “covers” that look like <script type="math/tex"> \mathcal{U}=\{U_i\}_{i\in I}</script>, you’ll be thinking of descent data as something like “matching families” or gluing data. In that case, a descent datum is collection of objects over each element of the cover that agree on pairwise intersections. If by “agree” in the previous sentence we mean “are equal” then that’s all we need for descent datum. However, if by “agree” we mean “are isomorphic” (e.g. if we’re dealing with categories of modules or vector bundles or something rather than sets) then we have to throw in the “cocycle condition” which says that we can glue together these isomorphisms in the right way.</p>
<p>But if our “things over <script type="math/tex"> S</script>” are spaces, then the isomorphisms between the restrictions won’t be equal, but rather isomorphic again. And we’ve got to glue those isomorphisms together in the right way. But there’s a rather nice way to frame this. Let’s take a map of rings <script type="math/tex"> R\to S</script>, and start with an <script type="math/tex"> S</script>-module <script type="math/tex"> M</script>. We can glue <script type="math/tex"> M</script> along double intersections if, as we said above, the two possible ways of tensoring <script type="math/tex"> M</script> up to <script type="math/tex"> S\otimes_R S</script>-modules are isomorphic, and these isomorphisms satisfy a cocycle condition. Now from here, there are three ways to tensor <script type="math/tex"> M\otimes_S(S\otimes_R S)</script> up to an <script type="math/tex"> S\otimes_R S\otimes_R S</script>-module. All of these ways need to be isomorphic, and these isomorphisms need to satisfy a cocycle condition one level up. And so this just keeps going, up and up and up. It turns out that there’s a really nice way to phrase all of this.</p>
<p>First we need the following:</p>
<blockquote>
<p><strong>Definition:</strong> Let <script type="math/tex">A^\bullet</script> be a cosimplicial ring, and <script type="math/tex">M^\bullet</script> be a cosimplicial module over <script type="math/tex"> A</script>. That is, <script type="math/tex"> A^\bullet</script> is a functor <script type="math/tex"> A:\Delta\to CRng</script> and <script type="math/tex"> M^\bullet</script> is a functor <script type="math/tex"> M:\Delta\to CRng\times Mod</script> whose value in <script type="math/tex"> CRng</script> is <script type="math/tex"> A^\bullet</script>. Then <script type="math/tex"> M^\bullet</script> is said to be <em>co-cartesian</em> over <script type="math/tex"> A^\bullet</script> if for every map <script type="math/tex"> \varphi:[n]\to[m]</script> in <script type="math/tex"> \Delta</script>, the map <script type="math/tex"> M(\phi):M^n\to M^m</script> induces an isomorphism <script type="math/tex"> M^n\otimes_{A(\phi)} A^m\cong M^m</script></p>
</blockquote>
<p>Let’s just unwind this definition for a second here. At level <script type="math/tex"> n\geq 0</script> in the cosimplicial module <script type="math/tex"> M</script>, there are a whole bunch of maps <script type="math/tex"> M^n\to M^m</script> for any other <script type="math/tex"> m\geq 0</script>. What this condition is saying is the codomain of these maps is the same thing as just tensoring up to that cosimplicial level <em>along</em> the associated map in <script type="math/tex"> A^\bullet</script>!</p>
<p>If you think back to what we talked about regarding sheaves of sets, you’ll see that at the first two levels of the cosimplicial diagram, this is the same as being a descent datum! Accordingly, we define the descent data for a map of homotopical rings <script type="math/tex"> R\to S</script> to be the cosimplicial modules over the cosimplicial ring <script type="math/tex"> S\to S\otimes_R S\to S\otimes_R S\otimes_R S\to\cdots</script> which are cocartesian, where we replace the isomorphisms in our definition of cocartesian with homotopy equivalences.</p>
<p>At the moment, what really interests me is the fact that this structure allows us to attempt to compute all possible descent data for a given <script type="math/tex"> S</script>-module along a map <script type="math/tex"> \varphi:R\to S</script>, and we do this by way of a Bousfield-Kan spectral sequence. Suppose we’re given an <script type="math/tex"> S</script>-module <script type="math/tex"> M</script>. Then what we’re interested now in computing are co-cartesian modules over the cosimplicial ring <script type="math/tex"> S\to S\otimes_R S\to\cdots</script> whose bottom level is the module <script type="math/tex"> M</script>. The requirement that this cosimplicial module be cocartesian is strong though, and the first bit of help it gives us is to tell us that <script type="math/tex"> M^n\simeq M\otimes_S (S\otimes_R\cdots \otimes_R S)</script>, where there are <script type="math/tex"> n</script> copies of <script type="math/tex"> S</script> on the right hand side.</p>
<p>What’s really cool about this though is that it connects this notion of descent datum with the notion of a descent datum being a comodule over a canonical descent coring. If you’re not familiar with that, in the discrete case it says that for a map of rings <script type="math/tex"> R\to S</script>, the category of descent data is equivalent to the category of <script type="math/tex"> S\otimes_R S</script> comodules, where we regard <script type="math/tex"> S\otimes_R S</script> as a coring, called the <em>canonical descent coring</em>. Sometimes it’s also called the Sweedler coring. It’s not hard to see the the data we’ve given above for being a co-cartesian module ends up being the same thing as being a comodule over <script type="math/tex"> S\otimes_R S</script> in a homotopically coherent way. For instance, the first level of the cosimplicial module tells us that we have a map (well, two maps) <script type="math/tex"> M\to M\otimes_R S\cong M\otimes_S S\otimes_R S</script>, which is a coaction of the canonical coring on <script type="math/tex"> M</script>. The usefulness of the co-cartesian criterion is that we know that the two ways that <script type="math/tex"> M\otimes_S S\otimes_R S</script> can be isomorphic to <script type="math/tex"> M\otimes_R S</script> are equivalent. In other words, descent data with a fixed base <script type="math/tex"> S</script>-module <script type="math/tex"> M</script> are the same thing as <script type="math/tex"> S\otimes_R S</script>-comodules structures on <script type="math/tex"> M</script>. So how can we work out what possible such structures there are?</p>
<p>This is where the BKSS comes in. Notice that such a comodule structure on <script type="math/tex"> M</script> is going to be a system of maps from the constant cosimplicial object on <script type="math/tex"> M</script>, let’s denote it by <script type="math/tex"> \tilde{M}^\bullet</script>, to the cosimplicial object which is just <script type="math/tex"> M</script> tensored (over <script type="math/tex"> S</script>!) with <script type="math/tex"> S\otimes_R S\to S\otimes_R S\otimes_R S\to\cdots</script>. So, to compute the space of descent data on <script type="math/tex"> M</script>, we need to attempt to compute the homotopy of the cosimplicial mapping space/spectrum/object of your favorite model category <script type="math/tex"> Hom(\tilde{M}^\bullet, M\otimes_S (S\otimes_R S)^\bullet)</script>.</p>
<p>I hope to write more soon about how this space we’re computing can be compared to the space of twisted forms for <script type="math/tex"> M</script> along the map <script type="math/tex"> R\to S</script>, and how if that map is a Galois or Hopf-Galois extension, the above computation actually has other interpretations in terms of the Galois group or the Hopf-Galois algebra of the extension.</p>
<p>P.S. - Please comment and let me know if anything above here seems off, or just plain wrong. I’d love to change it if so!</p>
<p> </p>
tag:chromotopy.org,2013-11-16:1384624261Truncations of E-theory2013-11-16T17:51:01Z2013-11-16T17:51:01Z<p>Jack Morava has a very pleasant introduction to some of the rationale behind chromatic homotopy theory and how algebraic geometry gets used to organize topological things, called <a href="http://arxiv.org/abs/0707.3216">Complex Cobordism and Algebraic Topology</a>. It’s short, and it’s worth your time to read whether you’re an expert or a novice. I’d opened it again recently to help prepare for a talk I was to give, and I found a gem of a paragraph which I hadn’t really processed before. I’d like to try to share that with you now.</p>
<p>The story starts the usual way: there is a sense in which the homology groups <script type="math/tex">MU_* X</script> can be thought of as a sheaf over the moduli space of formal group laws, equivariant against the action of the group of formal diffeomorphisms <script type="math/tex">\Lambda</script>, thought of as change-of-coordinate maps. Equivalently, this can be packaged into saying that <script type="math/tex">MU_* X</script> forms a sheaf over the moduli <em>stack</em> of <em>formal groups</em>. After <script type="math/tex">p</script>-localization, the geometric points of this stack admit a nice description: the positive characteristic ones are indexed by “heights” <script type="math/tex">1 \le n \le \infty</script>, along with one stray rational point of “height 0”. There is a construction of a family of formal group laws, called the Honda formal group laws, which produce examples of each of these formal groups of finite height, and one can think of some of the action of <script type="math/tex">\Lambda</script> as being “spent” on moving an arbitrary formal group law into one of these canonical forms. However, some of <script type="math/tex">\Lambda</script> will remain — the part which gives automorphisms of the Honda formal group law — and this part is referred to as the “stabilizer group” <script type="math/tex">\mathbb{S}_n</script>.</p>
<p>E-theory comes in by attempting to restrict the sheaf <script type="math/tex">MU_* X</script> to one of these geometric points. The inclusion of such a point is not typically a flat map (and so one would not expect restriction to produce another homology theory), but this can be corrected by instead considering the inclusion of its formal neighborhood. Then, an E-theory (up to the fuss I was raising in a <a href="topologized-objects">previous post</a>) is exactly given by the restriction of the bordism sheaf to the deformation space of one of these geometric points on the moduli of formal groups.</p>
<p>Good. Now let me tell you fewer generalities and more formulas. The Honda formal group law is cooked up so that its multiplication-by-p endomorphism, called its “<script type="math/tex">p</script>-series”, is given by <script type="math/tex">[p]_n(x) = x^{p^n}</script>. Any formal group law in positive characteristic carries a Frobenius endomorphism <script type="math/tex">S: x \mapsto x^p</script>, and so in the endomorphism ring of the Honda formal group law, there is a relation <script type="math/tex">S^n = p</script>. It’s a theorem of Cartier that if the base field has sufficiently many roots of unity, then this is actually a complete description of the endomorphism ring: it is the maximal order of a division algebra</p>
<script type="math/tex; mode=display">
\mathbf{o}_n = \mathbb{W}(k)\langle S \rangle / \left( \begin{array}{c} Sw = w^\varphi S \\ S^n = p \end{array} \right),
</script>
<p>where the angle brackets denote a free noncommuting associative algebra. The stabilizer group <script type="math/tex">\mathbb{S}_n</script> can then be identified with the compositional units of the endomorphism ring: <script type="math/tex">\mathbb{S}_n = \mathbf{o}_n^\times</script>.</p>
<p>What about the point at infinite height? The notation is actually arranged so that you can sort of take limits in n: the <script type="math/tex">p</script>-series becomes <script type="math/tex">[p]_\infty(x) = 0</script>, and the endomorphism ring becomes</p>
<script type="math/tex; mode=display">
\mathbb{W}(k)\langle\!\langle S \rangle\!\rangle / \left(\begin{array}{c}Sw = w^\varphi S \\ 0 = p\end{array}\right) = k \langle\!\langle S \rangle\!\rangle / (Sw = w^\varphi S).
</script>
<p>Much of the chromatic story is unable to be made explicit, but these two objects are quite familiar: the formal group law whose <script type="math/tex">p</script>-series vanishes is precisely the additive formal group law <script type="math/tex">x +_\infty y = x + y</script>. The endomorphisms of the additive formal group law are indeed exactly the power series concentrated in degrees which are powers of p, matching the description of the limiting endomorphism algebra. In terms of homology theories, the additive formal group law belongs to ordinary homology (as it is the only formal group law it can carry for degree limitations), and I’ve <a href="steenrod-algebra">previously done a questionable job</a> in justifying a connection between this endomorphism algebra and the Steenrod algebra.</p>
<p>Here, finally, is where Jack’s story becomes nonclassical. He says that the Morava E-theory of a finite spectrum gives a representation of the stabilizer group <script type="math/tex">\mathbb{S}_n</script>, and if we take these division algebras <script type="math/tex">\mathbf{o}_n</script> and quotient them by p, then we can pull back these E-theory representations to get a sequence of representations over the group</p>
<script type="math/tex; mode=display">
(\mathbf{o}_n/p)^\times = \left( k\langle S \rangle / \left( \begin{array}{c} Sw = w^\phi S \\ S^n = 0 \end{array} \right) \right)^\times.
</script>
<p>Then, he says, these should “limit” in an appropriate sense to the representation that ordinary mod-<script type="math/tex">p</script> homology produces. This is a weird statement — the stabilizer groups themselves do not fit into any such sequence, and the limit appears to be illegitimate — but in the barest and fuzziest sense, it is sort of reasonable. The Atiyah–Hirzebruch spectral sequence computing the connective Morava <script type="math/tex">K</script>-theory of a finite spectrum from its ordinary mod-<script type="math/tex">p</script> homology will be affected by sparseness, and so for very large <script type="math/tex">n</script>, we wouldn’t expect the differential behavior of the <script type="math/tex">E(n)</script>-AHSS to differ from the <script type="math/tex">E(n+1)</script>-AHSS.</p>
<p>I would posit that something more might be happening.</p>
<p>The height of a formal group law in positive characteristic has a multitude of different definitions. One of them is computed by finding the degree of the bottom nonzero term in the <script type="math/tex">p</script>-series, which makes it plain that the characterization of the Honda formal group law given above is really a formal group law of height n. Another is that there is an integral equation expressing the logarithm associated to any formal group law <script type="math/tex">x +_! y</script>:</p>
<script type="math/tex; mode=display">
\log_!(x) = \int \left( \left.\frac{\partial(x +_! y)}{\partial y} \right|_{y=0} \right)^{-1} dx.
</script>
<p>For finite height formal group laws, this integral equation has no solution — eventually you end up having to integrate a monomial like <script type="math/tex">x^{p^n-1}</script>. Indeed, the degree of the bottommost obstruction to this integral equation is in a degree of precisely that form, and that value <script type="math/tex">n</script> is referred to as the “height”.</p>
<p>I bring this up as an attempt to find geometric justification for this quotient by p that Jack performs. By setting p equal to zero in the endomorphism algebra, we are asserting that the map <script type="math/tex">x \mapsto x^{p^n}</script> has the action of the zero map, which can be minimally justified by setting <script type="math/tex">x^{p^n}</script> itself to zero — i.e., restricting to the <script type="math/tex">(p^n-1)</script>th order neighborhood on the formal group. On this neighborhood, the formal group law possesses a truncated logarithm which compares it to a truncated additive formal group law, and Jack is somehow asserting that as this neighborhood grows to encompass the whole formal object, the bordism sheaves “over the truncated objects” should fit together to recover the bordism sheaf over the additive object. (I’ve left “over the truncated objects” in quotes, because I don’t see how one can make this sane. The bordism sheaf doesn’t live “over the formal group” in any sense I can see.) But, if something like this were true, I would expect that the finiteness hypothesis could be relaxed to allow for bounded-below complexes, or at least suspension spectra. (Maybe it’s also worth pointing out that the E-theory of a <em>space</em> carries power operations, and in particular an action by <script type="math/tex">\psi^p</script>. Possibly some construction like <script type="math/tex">E_n / \psi^p</script> is relevant?) In any case, this is something I would like to know about.</p>
<p>This dishonest geometry also suggests another intriguing connection: there is a fast mnemonic for remembering the standard quotient algebras <script type="math/tex">\mathcal{A}(n)</script> of the mod-2 Steenrod algebra. Since the dual Steenrod algebra itself corepresents automorphisms of the additive formal group law, these should corepresent a certain subset of them — a certain subgroup, even, since these are quotient Hopf algebras. Indeed, <script type="math/tex">\mathcal{A}(n)</script> corepresents precisely the power series which vanish above order <script type="math/tex">2^n</script>, together with further restrictions so that their inverses and compositions remain supported in this range. So, these algebras are another kind of “truncation” of the infinite height stabilizer group — possibly not quite the same, but certainly not far off from what Jack is talking about. It was of interest for a while to try to find spectra <script type="math/tex">x(n)</script> with the property that <script type="math/tex">H\mathbb{F}_2^* x(n) = \mathcal{A} /\!/ \mathcal{A}(n)</script>; such spectra would behave as though they were taking mod-2 cohomology, modified so that its coalgebra of cooperations was <script type="math/tex">\mathcal{A}(n)_*</script> (cf., the Adams spectral sequence method for computing the real <script type="math/tex">K</script>-theory of spaces). This fell out, though, once the Hopf invariant one theorem was proven: it shows that there can be no such spectrum for <script type="math/tex">n > 3</script>. However, the spectra <script type="math/tex">x(0)</script>, <script type="math/tex">x(1)</script>, and <script type="math/tex">x(2)</script> all exist, and there is the following intriguing table:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{array}{|c||c|c|c|}\hline n & x(n) & X(n) & L_{K(n)} X(n) \\ \hline \hline 0 & H\mathbb{Z}_{(2)} & H\mathbb{Z}_{(2)} & E_0 \\ 1 & \mathrm{ko} & \mathrm{KO} & E_1^{hC_2} \\ 2 & \mathrm{tmf} & \mathrm{TMF} & E_2^{hG_{24}} \\ \hline \end{array}
%]]></script>
<p>I would like to think that the rightmost column of this table this isn’t an accident. In light of the above discussion with endomorphism algebras, is there some reason why this table should be true? Is there a natural candidate for subgroups of the stabilizer group which extend the E-theory side of this table, if not the <script type="math/tex">x(n)</script> side? Can partial theorems be proven about the nonexistent <script type="math/tex">x(n)</script> spectra, given these <script type="math/tex">K(n)</script>-local models? Can one see from the perspective of formal groups and some knowledge of the stabilizer groups this famous theorem that <script type="math/tex">x(3)</script> does not exist? It would also be nice to see the information flow the other way — to think of E-theories as repairing some specific failure in the ideology of <script type="math/tex">x(3)</script> that prevents it from existing and which they circumvent.</p>
<p>Hm!</p>
<hr />
<p>This doesn’t really fit anywhere, but Morava has another article, titled <a href="http://link.springer.com/article/10.1007%2FBF01214905">Forms of K-theory</a>, which has another really unique perspective on things. Published in 1989 and written in 1973, he manages to construct <script type="math/tex">p</script>-adic <script type="math/tex">\mathrm{TMF}</script> in a neighborhood of the cuspidal curve, along with a scattering of other really interesting ideas. It’s really cool to see some of the lesser known tributaries that led into chromatic homotopy theory, now that so much has become at least quasi-standard and worn smooth.</p>
<p>In this same preparation spree, I also rediscovered a talk by Mike Hopkins, <a href="http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter08/MikeTalk1v3.pdf">From Spectra to Stacks</a>, where he produces stacks associated to any ring spectrum rather than just those which are complex oriented. He does this by passing them through <script type="math/tex">MU</script> in a certain sense, and what’s interesting is this remark of his that <script type="math/tex">MU</script> isn’t special in this regard; any spectrum satisfying a few nicety properties will do. Morava, in Forms of <script type="math/tex">K</script>-theory, remarks that it’s not clear why complex bordism should have such a rich tie to arithmetic geometry, and in some sense this Hopkins remark explains it: that tie is always there, but complex bordism is the first example of a spectrum <script type="math/tex">MU</script> where the algebra of <script type="math/tex">MU_*</script> and <script type="math/tex">MU_* MU</script> is free enough and ‘spread out’ enough that topology doesn’t have a chance to come in and piss all over the rug. Why there’s a connection between formal groups and <em>topology</em>, however, is no less of a mystery.</p>
<p>Anyway, these are also worth reading.</p>
tag:chromotopy.org,2013-10-20:1382291461Loop spaces and descent2013-10-20T17:51:01Z2013-10-20T17:51:01Z
<p>Hey all. I’ve had this account forever, and figured now is as good a time as any to start posting. These days I’m thinking about derived algebraic geometry, specifically about the relationship between derived (free) loop spaces and crystalline cohomology of (ordinary and derived) schemes in positive characteristic. I’ll post more about that later, but as a warmup I thought I would discuss a derived view on and generalization of the fairly classical theory of descent. The details will be a little sketchy in order to emphasize the fundamental ideas, but hopefully anything I leave out will be some combination of intuitively clear and enlightening exercise. This in mind, here we go.</p>
<p>Suppose <script type="math/tex">f \colon X \to Y</script> is a map of schemes, <script type="math/tex">\mathcal{F}</script> a quasicoherent sheaf on <script type="math/tex">X</script>. Descent in this setting takes the following form.</p>
<blockquote>
<p>Question: When is <script type="math/tex">\mathcal{F}</script> isomorphic to <script type="math/tex">f^*\mathcal{G}</script> for some quasicoherent sheaf <script type="math/tex">\mathcal{G}</script> on <script type="math/tex">Y</script>?</p>
</blockquote>
<p>Let’s first look for some necessary conditions. Suppose we have such a <script type="math/tex">\mathcal{G}</script>, in which case we say that <script type="math/tex">\mathcal{F}</script> satisfies descent. Write <script type="math/tex">p_0, p_1 \colon X \times_Y X \to X</script> for the projections onto the first and second factors, respectively. Then there necessarily exists an isomorphism <script type="math/tex">\alpha \colon p_0^*\mathcal{F} \cong p_1^*\mathcal{F}</script> by commutation of the relevant fiber square and naturality of the pullback functors. Going a step further and examining the threefold fiber product <script type="math/tex">X \times_Y X \times_Y X</script>, we see that <script type="math/tex">\alpha</script> has to satisfy a transitivity criterion, classically known as the cocycle condition. Working out precisely what this condition says is not difficult, but is beyond my energy to typeset, so I’ll leave it as an exercise. In any case, we have found that if <script type="math/tex">\mathcal{F}</script> satisfies descent, it does so via <em>descent data</em> in the form of an <script type="math/tex">\alpha</script> satisfying the cocycle condition. The collection of such pairs <script type="math/tex">(\mathcal{F}, \alpha)</script> forms a category in the evident way, which we call the category of descent data for <script type="math/tex">f</script>. The question now becomes how close the category of descent data is to the category of quasicoherent sheaves on <script type="math/tex">Y</script>. A fairly general and important class of morphisms <script type="math/tex">f</script> are covered in the following theorem due to Grothendieck.</p>
<blockquote>
<p>Theorem: (Grothendieck) If <script type="math/tex">f \colon X \to Y</script> is faithfully flat, i.e. flat and surjective, then the categories of descent data for <script type="math/tex">f</script> and quasicoherent sheaves on <script type="math/tex">Y</script> are equivalent through the above construction.</p>
</blockquote>
<p>A nice, simple example of the theorem arises when <script type="math/tex">X, Y</script> are spectra of fields <script type="math/tex">L, K</script>, respectively, and <script type="math/tex">f</script> is induced by a Galois extension <script type="math/tex">K \hookrightarrow L</script>. Then Grothendieck’s theorem states that the category of vector spaces over <script type="math/tex">K</script> is equivalent to the category of vector spaces over <script type="math/tex">L</script> equipped with descent data <script type="math/tex">\alpha</script> as above. The usual statement of Galois descent, however, holds that this category is equivalent to the category of <script type="math/tex">L</script>-vector spaces with semilinear actions of the Galois group <script type="math/tex">Gal(L/K)</script>. This arises from a different, equivalent formulation of descent data which we will now discuss.</p>
<p>The starting point of this alternative perspective is the observation that the fiber product <script type="math/tex">X \times_Y X</script> forms the morphisms of a groupoid scheme, with the objects given by <script type="math/tex">X</script>. That is, there are the two projection maps <script type="math/tex">p_0, p_1</script> from before, which we will rename <script type="math/tex">s, t</script>, along with a unit morphism <script type="math/tex">\eta \colon X \to X \times_Y X</script> given by the diagonal, and an antipode <script type="math/tex">\tau \colon X \times_Y X \to X \times_Y X</script> given by interchanging the two factors. The subtlest bit of structure is in the composition <script type="math/tex">(X \times_Y X) \times_X (X \times_Y X)</script>. This is given by first identifying this with the iterated fiber product <script type="math/tex">X \times_Y X \times_Y X</script>, applying <script type="math/tex">f</script> to the middle factor so as to map to <script type="math/tex">X \times_Y Y \times_Y X</script>, then identifying this with <script type="math/tex">X \times_Y X</script>. If you’ve ever seen or dealt with Hopf algebroids before, this data is just the nonaffine generalization of the spectrum of such a gadget. In any case, we can now see that a descent datum for <script type="math/tex">\mathcal{F}</script> is the same as an action of this so-called <em>descent groupoid</em>, the cocycle condition giving the relevant associativity. For typesetting reasons, I’ll leave that translation to you, and state that the categories of descent data and quasicoherent sheaves equivariant for the descent groupoid are equivalent. A good example to keep in mind is that of a Galois extension of fields, as above. In this case, the descent groupoid can be identified with <script type="math/tex">Spec(L) \times Gal(L/K)</script>, and the descent groupoid action translates to a semilinear Galois action.</p>
<p>Now let’s switch to some homotopy theory. Suppose the morphism we’re concerned with is the inclusion of a basepoint <script type="math/tex">x \colon pt \to X</script> into some space <script type="math/tex">X</script>, and we’re given a space <script type="math/tex">Y</script> over <script type="math/tex">pt</script> (that is, a space.)</p>
<blockquote>
<p>Question: When is <script type="math/tex">Y</script> weakly equivalent to <script type="math/tex">x^*Z</script> for some space <script type="math/tex">Z</script> over <script type="math/tex">X</script>?</p>
</blockquote>
<p>As stated, the answer to this question is “always,” since we may just take <script type="math/tex">Z = Y \times X</script>. A more interesting question is the following.</p>
<blockquote>
<p>Question: Can we classify all <script type="math/tex">Z</script> over <script type="math/tex">X</script> with the property that <script type="math/tex">Y \simeq x^* Z</script>?</p>
</blockquote>
<p>We can apply the same reasoning as in the case of schemes to start building a guess at the category of descent data. The interesting difference here is that we are asking <em>homotopical</em> questions, and so we ought to replace all categorical concepts we used with higher-categorical, homotopy-coherent notions. For example, we should now consider the <em>homotopy</em> fiber product <script type="math/tex">pt \times_X^R pt</script> with its associated projections <script type="math/tex">p_0, p_1</script> to the point. (Of course, the point is the terminal object in spaces so these maps are equivalent.) It’s a classical homotopy-theoretic observation that this fiber product is weakly equivalent to the based loop space <script type="math/tex">\Omega_x X</script>. Moreover, the appropriate notion of groupoid object here is a sort of <script type="math/tex">\mathbb{E}_1</script>-groupoid, and the discussion from above endows the pair <script type="math/tex">(\Omega_x X, pt)</script> with the structure of such a gadget. A groupoid with a single object is just a group, though, and correspondingly we obtain <script type="math/tex">\Omega_x X</script> as a grouplike <script type="math/tex">\mathbb{E}_1</script>-space. In fact, if you go through the explicit construction carefully, you see that this structure is just the usual loop composition, and the category of descent data is equivalent to the category of spaces with an action of <script type="math/tex">\Omega_x X</script>. (Here we should really be saying “homotopical category” or “<script type="math/tex">\infty</script>-category,” but this just comes with the homotopical territory as it were.) The corresponding descent theorem is now the following, due originally to Dror, Dwyer and Kan.</p>
<blockquote>
<p>Theorem: If <script type="math/tex">X</script> is connected, then the homotopy theories of spaces over <script type="math/tex">X</script> and <script type="math/tex">\Omega_x X</script>-spaces are equivalent.</p>
</blockquote>
<p>This actually jives rather well with Grothendieck’s theorem above. Here we should be thinking of spaces in a “stacky” fashion, and so the “points” are really elements of <script type="math/tex">\pi_0</script>. The connectedness hypothesis ensures that <script type="math/tex">x \colon pt \to X</script> is surjective, and in some sense there is only one point for the “fiber” to vary over, so flatness is guaranteed. A simple example to keep in mind is when <script type="math/tex">X = BG</script> for a discrete group <script type="math/tex">G</script>. Then the map <script type="math/tex">x</script> is homotopy equivalent to the universal cover of <script type="math/tex">X</script>, the based loop space is equivalent to <script type="math/tex">G</script>, and the descent statement is familiar from covering space theory.</p>
<p>To this point we haven’t really mixed the algebraic geometry with the homotopy theory, so let me give a taste of what that looks like by putting a bit of old-school stable homotopy theory in this language. Things will get really sketchy now, but I’ll say more in a later post about how this all shakes out rigorously. Suppose we have a connective <script type="math/tex">\mathbb{E}_\infty</script>-ring spectrum <script type="math/tex">E</script>. Then to first approximation, we should think of <script type="math/tex">Spec(E)</script> as being the ordinary scheme <script type="math/tex">Spec(\pi_0 E)</script> with a collection of quasicoherent sheaves <script type="math/tex">Spec(\pi_n E)</script> providing “higher nilpotents.” Alternatively, we have the topological space <script type="math/tex">\mid Spec(\pi_0 E) \mid</script> with a sheaf of <script type="math/tex">\mathbb{E}_\infty</script>-ring spectra on it given by localizing <script type="math/tex">E</script> appropriately over every open set. Quasicoherent sheaves on <script type="math/tex">Spec(E)</script> are just <script type="math/tex">E</script>-module spectra, and the terminal affine spectral scheme is given by <script type="math/tex">Spec(S)</script>, the Zariski spectrum of the sphere spectrum. Consider the morphism <script type="math/tex">Spec(H\mathbb{F}_p) \to Spec(S)</script> given by the unit <script type="math/tex">S \to H\mathbb{F}_p</script>. Here the right-hand side is the Eilenberg-Mac Lane spectrum of the finite field <script type="math/tex">\mathbb{F}_p</script> as usual. The descent question here takes the following form.</p>
<blockquote>
<p>Question: Given a <script type="math/tex">H\mathbb{F}_p</script>-module spectrum <script type="math/tex">M</script>, when is it weakly equivalent to <script type="math/tex">H\mathbb{F}_p \wedge N</script> for some spectrum <script type="math/tex">N</script>?</p>
</blockquote>
<p>To get started on this, we should identify the derived descent groupoid. Since we’re dealing with affine spectral schemes, the relevant fiber product is just computed as a smash product, and it’s not hard to see that this spectrum is the Hopf-algebra spectrum <script type="math/tex">H\mathbb{F}_p \wedge H\mathbb{F}_p</script> whose homotopy groups form the mod <script type="math/tex">p</script> dual Steenrod algebra. Thus the <script type="math/tex">\infty</script>-category of descent data is equivalent to the <script type="math/tex">\infty</script>-category of comodule spectra over the dual Steenrod algebra spectrum. Convergence of the Adams spectral sequence gives the following descent theorem.</p>
<blockquote>
<p>Theorem: There is an equivalence of homotopy theories between that of comodule spectra over the dual Steenrod algebra spectrum that are compact as <script type="math/tex">H\mathbb{F}_p</script>-module spectra and of compact <script type="math/tex">p</script>-complete spectra.</p>
</blockquote>
<p>This is definitely not the strongest version of this theorem possible, but it includes two hypotheses that show up all over the place in these descent results. The first is finiteness, which is required in many Koszul duality-type contexts such as this (more on that later.) The other is completion, which arises from thinking hard about the surjectivity in the Grothendieck descent theorem above. Very, very roughly, we should only expect to see information about sheaves supported on the formal neighborhood of the image of whatever morphism we’re descending along. In this case, this means spectra concentrated at the formal neighborhood of <script type="math/tex">(p)</script> in <script type="math/tex">Spec(S)</script>, which corresponds to <script type="math/tex">p</script>-complete spectra. This completion business is pretty important, and very much tied up in the crystalline story. My next post will discuss that in more detail, but for now I’m out of breath.</p>
tag:chromotopy.org,2013-05-02:1367517061Topologized objects in algebraic topology2014-04-22T20:45:32Z2013-05-02T17:51:01Z<p>I attended MIT’s Talbot workshop last week, which was super fun and – at times – even productive. I had a couple facts about Morava <script type="math/tex">E</script>-theory come up repeatedly in side-conversations, and I think it’s worth sharing those with the rest of you now.</p>
<p>Here’s the point: continuous Morava <script type="math/tex">E</script>-theory is the only sort of Morava <script type="math/tex">E</script>-theory. Let me explain by taking a moment to recall where it comes from. The central thesis of chromatic homotopy theory is that the stable category is tightly bound to the moduli stack of <script type="math/tex">1</script>-dimensional, commutative, smooth formal groups via the homology theory <script type="math/tex">MU_*</script> of complex bordism. Generally, to any ring spectrum <script type="math/tex">E</script> we can associate a simplicial scheme given by the following fancy construction:</p>
<script type="math/tex; mode=display">
\mathrm{Spec}\,\pi_* E \begin{array}{c}\leftarrow \\ \rightarrow \\ \leftarrow\end{array} \mathrm{Spec}\,\pi_* \left(\begin{array}{c}E \\ \wedge \\ E\end{array}\right) \begin{array}{c}\leftarrow \\ \rightarrow \\ \leftarrow \\ \rightarrow \\ \leftarrow \end{array} \mathrm{Spec}\,\pi_* \left(\begin{array}{c}E \\ \wedge \\ E \\ \wedge \\ E\end{array}\right) \cdots.
</script>
<p>The assignment <script type="math/tex">X \mapsto E_* X</script> can be thought of as sending <script type="math/tex">X</script> to a quasicoherent sheaf over this simplicial scheme. In the case of <script type="math/tex">MU</script>, the associated simplicial scheme is equivalent to the moduli of formal groups, and various niceness results in algebraic topology say that this assignment is not <em>too</em> lossy — in many ways, the stable category behaves like <script type="math/tex">\mathrm{QCoh}(\mathcal{M}_{\mathbf{fg}})</script>.</p>
<p>Number theorists know quite a bit about formal groups, and we can shamelessly piggyback on their hard work to learn things about the stable category. For instance, after localizing at a prime, there are countably many geometric points on the moduli of formal groups, belonging to the Honda formal groups and enumerated by “height”. In stable homotopy theory, these geometric points are reflected by the following theorem:</p>
<blockquote>
<p>(One of the Hopkins-Smith periodicity theorems:) There is a sequence of homology theories called Morava <script type="math/tex">K</script>-theories, which are complex oriented with formal group given by these Honda formal groups. Moreover, these give “exhaustive” list of field spectra, in the sense that every homology theory with natural Künneth isomorphisms is some kind of field extension of a Morava <script type="math/tex">K</script>-theory.</p>
</blockquote>
<p>The next place to look for instruction from the number theorists after the points of <script type="math/tex">\mathcal{M}_{\mathbf{fg}}</script> is in its local geometry. This analysis was carried out by Lubin and Tate, who found the following result:</p>
<blockquote>
<p>(Lubin-Tate:) Every geometric point on <script type="math/tex">\mathcal{M}_{\mathbf{fg}}</script> is smooth. The infinitesimal deformation space of the Honda formal group with height <script type="math/tex">n</script> has dimension <script type="math/tex">(n-1)</script>.</p>
</blockquote>
<p>This, too, is reflected in stable homotopy theory: associated to any formal group <script type="math/tex">G</script> over a perfect field there is a spectrum <script type="math/tex">E_n</script>, called Morava <script type="math/tex">E</script>-theory, whose coefficient ring is given by the infinitesimal deformation space of <script type="math/tex">G</script>.</p>
<p>Morava <script type="math/tex">K</script>-theories have lots of nice properties, largely owing to their Künneth isomorphisms, but Morava <script type="math/tex">E</script>-theories also have lots of nice properties from the perspective of structured ring spectra, largely owing to the fact that they are mixed- rather than positive-characteristic. Of course, the <script type="math/tex">K</script>- and <script type="math/tex">E</script>-theory of a particular formal group are closely related to one another, and so a lot of algebraic topology gets done by using whichever is appropriate and transferring results back and forth as necessary.</p>
<p>This all sounds great, but there’s a wrinkle: everything in the above story works perfectly when studying cohomology theories, but there are <em>two</em> sorts of Morava <script type="math/tex">E</script>-homology people consider: ordinary and continuous. These are defined respectively by the following two formulas:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{array}{cc}E_* X = \pi_* E \wedge X, & E^\vee_* X = \pi_* L_K (E \wedge X)\end{array}.
%]]></script>
<p>The left-hand formula is exactly what you’d expect, so why are we studying the right-hand formula? After all, it’s not even properly a homology theory — localization is not expected to preserve infinite wedges!</p>
<p>The literal meaning “homology theory” aside, the right-hand formula is in fact the one we want. To start to see why, let’s recall the following formula for <script type="math/tex">K</script>-localization: if <script type="math/tex">X</script> is an <script type="math/tex">E</script>-local spectrum (and <script type="math/tex">X \wedge E</script> is always so local), then we have the formula</p>
<script type="math/tex; mode=display">
L_K X = \lim_I M^0(I) \wedge X,
</script>
<p>where <script type="math/tex">M^0(I)</script> is a finite spectrum with the property <script type="math/tex">BP_* M^0(I) = BP_* / (p^{I_0}, v_1^{I_1}, \ldots, v_{n-1}^{I_{n-1}})</script>. It is, again, a consequence of the periodicity theorems that such spectra exist for <script type="math/tex">I \gg 0</script>, along with the expected quotient maps used to build the inverse system.</p>
<p>This formula indicates what’s really going on: the infinitesimal deformation space of a point is always defined as a formal scheme, i.e., as the formal colimit of a directed system of finite schemes, each of which captures a “nilpotent of degree <script type="math/tex">m</script>” piece of the deformation space, with <script type="math/tex">m</script> growing large. In the case of a smooth point, the deformation space is given by a power series ring, and this colimit amounts to remembering the adic topology on this ring. This is actually a huge deal — the formal spectrum of a power series ring (i.e., the spectrum with this topology taken into account) is what you’d expect: a single point, with structure sheaf at the point given by the power series ring. On the other hand, just the spectrum of a power series ring is terrifyingly large and repugnant — it has all sorts of prime ideals that you wouldn’t expect, on account of the existence of transcendental power series.</p>
<p>In turn, that’s what the extra <script type="math/tex">K</script>-localization is doing: it’s recalling that Lubin-Tate space (and hence Morava <script type="math/tex">E</script>-theory) is given as an ind-scheme (resp., as a pro-spectrum), and certainly we shouldn’t forget this crucial fact about our set-up. This extra step has all sorts of niceness consequences in algebraic topology; for instance, we have the following:</p>
<blockquote>
<p>(Hovey-Strickland:) When <script type="math/tex">X</script> is a spectrum such that <script type="math/tex">E^* X</script> is even-concentrated, then <script type="math/tex">K^* X</script> is given by reduction at the maximal ideal: <script type="math/tex">K^* X = E^* X / I_n</script>. Conversely, when <script type="math/tex">K^* X</script> is even-concentrated, then <script type="math/tex">E^* X</script> is pro-free with this same reduction property.</p>
</blockquote>
<p>There is an analogue of this property for continuous <script type="math/tex">E</script>-homology <script type="math/tex">E^\vee_*</script> but <em>not</em> for <script type="math/tex">E_*</script>. In fact, the failure of this for the noncontinuous version is dramatic: its Bousfield class is given by a wedge of Morava <script type="math/tex">K</script>-theories, with heights ranging from <script type="math/tex">0</script> up to <script type="math/tex">n</script>. You can sort of see this coming — without continuity to control the image of the power series generators, you can imagine sending one of them to an invertible element in some target ring, thereby decreasing the height of the pulled-back formal group law. On the other hand, the Hovey-Strickland theorem says that the <em>cohomological</em> Bousfield classes of <script type="math/tex">K</script>- and <script type="math/tex">E</script>-theory coincide, as you would reasonably expect. That there is a Hovey-Strickland theorem for <script type="math/tex">K</script>-homology and continuous Morava <script type="math/tex">E</script>-homology says, among other things, that we’re specifically repairing this defect in the homological Bousfield class of <script type="math/tex">E</script>-theory.</p>
<p>OK, deep breath.</p>
<p>I think these are neat facts, coming from being careful about the insertion of formal geometry into algebraic topology. However, there’s a different fact that this perspective doesn’t quite jive with and which I would like to understand better: fracture squares. Chromatic fracture is a big deal in algebraic topology; its noble goal is to reconstruct higher chromatic spectra from chromatic layers and gluing data. It takes the form of the following pullback square:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{array}{ccc} L_{E(n)} X & \to & L_{K(n)} X \\ \downarrow & & \downarrow \\ L_{E(n-1)} X & \to & L_{E(n-1)} L_{K(n)} X,\end{array}
%]]></script>
<p>implicitly noting that the Bousfield classes of <script type="math/tex">E</script>- and <script type="math/tex">K</script>-theory don’t depend upon the choice of formal group, but just of its height <script type="math/tex">n</script>. If you like, you can inductively expand this out entirely in terms of <script type="math/tex">K</script>-theoretic localizations, and you’ll find you’re taking a limit over a big punctured cube whose vertices are given by all the <script type="math/tex">K</script>-theoretic localizations applied in sequence, with variations on which localizations you skip. Moreover, this construction and theorem crucial — it’s how we chromatic homotopy theorists do everything, from understanding finite spectra to constructing <script type="math/tex">\operatorname{TMF}</script>.</p>
<p>That’s why it’s a puzzle that this doesn’t seem to play nicely with formal geometry. Suppose that <script type="math/tex">X</script> is an <script type="math/tex">E</script>-theory; then one of the corners in this fracture cube takes the form <script type="math/tex">L_{K(t)} E_n</script>. The homotopy of this spectrum is computable, given noncanonically in choice of generators by:</p>
<script type="math/tex; mode=display">
\pi_0 L_{K(t)} E_n = W(k)[\![u_1, \ldots, u_{n-1}]\!][u_t^{-1}]^\wedge_{(p, v_1, \ldots, u_{t-1})}.
</script>
<p>What the hell is this thing? The first thing to say is that this doesn’t have an interpretation in terms of classical formal geometry: we’re inverting something in the maximal ideal (which is not that frightening; consider building <script type="math/tex">\mathbb{Q}_p</script> from <script type="math/tex">\mathbb{Z}_p</script>) and then recompleting against a nonclosed ideal (which <em>is</em> frightening; it means the completion map is not continuous in the respective adic topologies). You might also worry that I claimed that this presentation is noncanonical in choice of generators, then inverted a particular element — but a surprising fact is that the recompletion erases this choice of inversion. Selecting different generators will give isomorphic rings after passing to the recompletion. This indicates that we are, in some sense, performing a geometric operation.</p>
<p>But what operation and what geometry?</p>
<p>I’ve spoken a little bit to Nat Stapleton about this, who studies (among other things) “transchromatic character maps,” which use in an essential way this height-jumping feature of this strange localization of <script type="math/tex">E</script>-theory. We have some ideas (more accurately: he has some ideas) about what moduli problem this ring presents, but our phrasing of this is seriously crippled by the lack of an algebro-geometric framework in which to work. I’m at a loss as to what to do — we’ve spoken to a few algebraic geometers about this, and the most promising thing they’ve suggested has been Huber’s theory of adic spaces, but even this doesn’t quite accomplish the task at hand.</p>
<p>Who knows.</p>
<p>To wrap it up, topologized objects appear elsewhere in stable homotopy theory as well. For instance, work of Ando-Morava-Sadofsky and of Ando-Morava use pro-systems of Thom spectra to do some pretty cool transchromatic things. The Tate construction can also be thought of in such a way, as can parts of Lin’s theorem / the Segal conjecture. So, these sorts things are all around, and given their collective mysterious nature, I feel that we really don’t have a good way of thinking about them yet.</p>
tag:chromotopy.org,2013-04-19:1366393861Stanley symmetric functions2013-04-19T17:51:01Z2013-04-19T17:51:01Z
<p>Besides the <a href="reduced-words">original definition</a> in terms of reduced words, the Stanley symmetric functions <script type="math/tex">F_w</script> can be computed in at least three ways, which seem mysteriously different enough to me that it’s worth talking about all of them.</p>
<h3 id="combinatorics">Combinatorics</h3>
<p>It turns out that in the Schur expansion <script type="math/tex">F_w = \sum_{\lambda} a_{\lambda w} s_{\lambda}</script>, the coefficients <script type="math/tex">a_{\lambda w}</script> are nonnegative integers, so it’s natural to ask for a combinatorial interpretation for them. Edelman and Greene gave a nice answer to this question. The <em>column word</em> of a semistandard tableau <script type="math/tex">T</script> is the word gotten by reading up the first column, then the second column, and so on. Say <script type="math/tex">EG(w)</script> is the set of semistandard tableaux whose column words are reduced words for <script type="math/tex">w</script>. Then <script type="math/tex">a_{\lambda w}</script> is the number of tableaux in <script type="math/tex">EG(w)</script> with shape <script type="math/tex">\lambda</script>.</p>
<p>For example, <script type="math/tex">Red(2143) = \{13, 31\}</script>, and each of these is the column word of exactly one tableau: <script type="math/tex">% <![CDATA[
\begin{array}{cc} 1 & 3 \end{array} %]]></script> and <script type="math/tex">\begin{array}{c} 1\\ 3 \end{array}</script> respectively, so <script type="math/tex">F_{2143} = s_{11} + s_{2}</script>. <script type="math/tex">Red(321) = \{121, 212\}</script>, but only the second of these can be the column word of a semistandard tableau, namely <script type="math/tex">% <![CDATA[
\begin{array}{cc} 1 & 2\\ 2 & \end{array} %]]></script>, so <script type="math/tex">F_{321} = s_{21}</script>.</p>
<p>In fact Edelman-Greene do better than this. <a href="reduced-words">Previously</a> I mentioned that the equality <script type="math/tex">\mid Red(w)\mid = \sum_{\lambda} a_{w\lambda} f^{\lambda}</script> falls out of the definition of <script type="math/tex">F_w</script>, where <script type="math/tex">f^{\lambda}</script> is the number of standard tableaux of shape <script type="math/tex">\lambda</script>. Given the interpretation of <script type="math/tex">a_{w\lambda}</script>, this suggests there should be a bijection</p>
<script type="math/tex; mode=display">
Red(w) \leftrightarrow \{(P, Q) : P \in EG(w), \text{shape}(P) = \text{shape}(Q), Q \text{ standard} \}.
</script>
<p>Edelman-Greene provide just such a bijection. If you know the <a href="http://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted_correspondence">Robinson-Schensted(-Knuth) correspondence</a>, this should look familiar—Edelman-Greene’s bijection is very similar, and includes Robinson-Schensted as the special case <script type="math/tex">w = 2143\cdots (2n)(2n-1)</script>.</p>
<p>This also solves the problem of sampling uniformly from <script type="math/tex">Red(w)</script>, as long as we can compute <script type="math/tex">EG(w)</script>. For example, take the special case <script type="math/tex">w = n(n-1)\cdots 1</script>, where <script type="math/tex">EG(w)</script> has just one element <script type="math/tex">P_0</script> (because <script type="math/tex">F_w = s_{(n-1, n-2, \ldots, 1)}</script>). It turns out to be easy to uniformly choose a standard tableau <script type="math/tex">Q</script> of a particular shape (using the “hook walk” algorithm), so choose <script type="math/tex">Q</script> with the same shape as <script type="math/tex">P_0</script> and apply the Edelman-Greene bijection to the pair <script type="math/tex">(P_0, Q)</script> to obtain a reduced word.</p>
<h3 id="representation-theory">Representation theory</h3>
<p>The Edelman-Greene bijection shows that the coefficients <script type="math/tex">a_{w\lambda}</script> are nonnegative integers. Since the Schur functions <script type="math/tex">s_{\lambda}</script> are irreducible characters for <script type="math/tex">GL(E)</script> (if <script type="math/tex">\text{dim}(E)</script> is large enough), this means <script type="math/tex">F_w</script> is the character of a <script type="math/tex">GL(E)</script>-module. This module turns out to have a nice description not relying on knowing its decomposition into irreducibles.</p>
<p>Here’s how to get the module <script type="math/tex">E^{\lambda}</script> whose character is <script type="math/tex">s_{\lambda}</script> (called a <em>Schur module</em>). Start with the left <script type="math/tex">GL(E)</script>-module <script type="math/tex">E^{\otimes n}</script>, where <script type="math/tex">n = \mid\lambda\mid</script>. This is also a right <script type="math/tex">S_n</script>-module, permutations acting by permuting tensor factors. Construct the <a href="http://en.wikipedia.org/wiki/Young_symmetrizer">Young symmetrizer</a> <script type="math/tex">c_{\lambda}</script>, a member of the group algebra <script type="math/tex">\mathbb{C}[S_n]</script>. Then the Schur module <script type="math/tex">E^{\lambda}</script> is <script type="math/tex">E^{\otimes n}c_{\lambda}</script>.</p>
<p>The construction of the Young symmetrizer <script type="math/tex">c_{\lambda}</script> uses the Ferrers diagram of <script type="math/tex">\lambda</script>, but only actually relies on having a set of boxes to put numbers in, not that the boxes are arranged in any particular way. That is, any finite set <script type="math/tex">D \subset \mathbb{N}^2</script> has a Young symmetrizer <script type="math/tex">c_D</script> using the same definition, and so there’s a <script type="math/tex">GL(E)</script>-module <script type="math/tex">E^D = E^{\otimes n}c_D</script>.</p>
<p>The Stanley symmetric function <script type="math/tex">F_w</script> turns out to be the character of one of these guys. The <em>inversion set</em> of a permutation <script type="math/tex">w</script> is <script type="math/tex">% <![CDATA[
I(w) = \{(i,j) : i < j, w(i) > w(j)\} %]]></script>. Then <script type="math/tex">F_w</script> is the character of <script type="math/tex">E^{I(w^{-1})}</script>. [I’m not sure who to credit this to… Kraśkiewicz and Pragacz? Let’s say yes, Poles unite!!]</p>
<p>For example,</p>
<script type="math/tex; mode=display">% <![CDATA[
I(321) = \{(1,2),(1,3),(2,3)\} = \begin{array}{ccc} \cdot & \square & \square \\ & \cdot & \square \\ & & \cdot \end{array}
%]]></script>
<p>is more or less the Ferrers diagram of <script type="math/tex">(2,1)</script>, so we recover again that <script type="math/tex">F_{321} = s_{21}</script>. The equality <script type="math/tex">F_{2143} = s_{11} + s_{2}</script> from this point of view ends up being the classic decomposition of 2-tensors into symmetric and antisymmetric parts, <script type="math/tex">E \otimes E \simeq Sym^2(E) \oplus \bigwedge^2(E)</script>.</p>
<h3 id="geometry">Geometry</h3>
<p>Schubert calculus, in modern terms, amounts to doing calculations in the cohomology ring of Grassmannians or flag varieties (or more generally, a semisimple algebraic group mod a parabolic subgroup). Take <script type="math/tex">B^+, B^-</script> the subgroups of upper and lower triangular matrices, respectively, in <script type="math/tex">GL_n</script>. A <em>(complete) flag</em> in <script type="math/tex">\mathbb{C}^n</script> is a sequence of subspaces <script type="math/tex">0 = V_0 \subseteq V_1 \subseteq \cdots \subseteq V_n = \mathbb{C}^{n}</script> with <script type="math/tex">\text{dim}(V_i) = i</script>. The collection of flags in <script type="math/tex">\mathbb{C}^n</script> is naturally in bijection with the <em>flag variety</em> <script type="math/tex">Fl(n) = GL_n / B^+</script>, a projective variety.</p>
<p>The <a href="http://en.wikipedia.org/wiki/LU_decomposition">LU decomposition</a> (or LPU) gives the <em>Bruhat decomposition</em> of <script type="math/tex">GL_n</script>: the disjoint union <script type="math/tex">GL_n = \bigcup_{w \in S_n} B^- w B^+</script>, interpreting <script type="math/tex">w</script> as a permutation matrix. This descends to a decomposition <script type="math/tex">Fl(n) = \bigcup_{w \in S_n} B^- w B^+ / B^+</script>. Each <script type="math/tex">B^- w B^+ / B^+</script> is a <em>Schubert cell</em> <script type="math/tex">X_w^o</script>, homeomorphic to some <script type="math/tex">\mathbb{C}^k</script>; the <em>Schubert variety</em> <script type="math/tex">X_w</script> is the closure of <script type="math/tex">X_w^o</script>. The Schubert cells form a CW decomposition of <script type="math/tex">Fl(n)</script>, and have even real dimensions, so the classes <script type="math/tex">[X_w]</script> Poincaré dual to the Schubert varieties form a <script type="math/tex">\mathbb{Z}</script>-basis of <script type="math/tex">H^*(Fl(n), \mathbb{Z})</script>.</p>
<p>As for the ring structure, by realizing <script type="math/tex">Fl(n)</script> as the total space of the top of a tower of projective bundles over <script type="math/tex">\mathbb{P}^1</script>, one can compute that <script type="math/tex">H^*(Fl(n))</script> is a certain quotient of <script type="math/tex">\mathbb{Z}[x_1, \ldots, x_n]</script>. Lots of classical enumerative geometry questions can be phrased in terms of counting intersection points between various Schubert varieties, so one should try to find polynomials representing the Schubert classes <script type="math/tex">[X_w]</script> in this picture of <script type="math/tex">H^*(Fl(n))</script>.</p>
<p>Lascoux and Schützenberger found a nice set of such polynomials, the <em>Schubert polynomials</em> <script type="math/tex">\mathfrak{S}_w</script>. They have the nice property of being stable under the inclusions <script type="math/tex">Fl(n) \to Fl(n+1)</script>, meaning that for <script type="math/tex">w \in S_n</script>, <script type="math/tex">\mathfrak{S}_w</script> represents <script type="math/tex">[X_w]</script> in <script type="math/tex">H^*(Fl(N))</script> whenever <script type="math/tex">N \geq n</script>, if we view <script type="math/tex">w</script> as a permutation of <script type="math/tex">[N]</script> fixing <script type="math/tex">n+1, \ldots, N</script>.</p>
<p>Schubert polynomials enjoy another kind of stability. Write <script type="math/tex">1^m \times w</script> for the permutation <script type="math/tex">1\cdots m (w(1)+m)(w(2)+m) \cdots</script>. It turns out that as <script type="math/tex">m \to \infty</script>, <script type="math/tex">\mathfrak{S}_{1^m \times w}</script> converges to a power series in <script type="math/tex">x_1, x_2, \ldots</script>. This power series is exactly <script type="math/tex">F_{w^{-1}}</script>!</p>
tag:chromotopy.org,2013-04-04:1365097861Determinantal K-theory2013-04-04T17:51:01Z2013-04-04T17:51:01Z<p>Craig Westerland and I have been going around giving talks about something we stumbled across in K(n)-local homotopy theory and can’t yet really explain. It’s a really neat construction, and I think it’s mind-blowing that it works as well as it does, and I can’t hope to tell everyone about it in a talk — so instead I’ll blog about it a little. There are some serious words coming up, so hold on to your hats.</p>
<p>This story picks up where <a href="spectral-cotangent-space">the previous post about spectral tangent spaces</a> left off: there’s some machine which swallows spectra-with-diagonals whose cohomology looks like that of <script type="math/tex">\mathbb{C}\mathrm{P}^\infty</script> and produces elements of the K(n)-local Picard group. In fact, I can say how this machine works: suppose you’re given a K-algebra A and a scheme-theoretic point <script type="math/tex">x: A \to K</script>. The kernel of this map is the ideal I, which is thought of as functions vanishing at x. This ideal receives a multiplication map <script type="math/tex">I \otimes_K I \to I</script> restricting the one for A, and the quotient (or cokernel) is the definition of the cotangent space: <script type="math/tex">T^*_x A = I / I \otimes_K I</script>. We can say all these words for spectra too, but the theory works out better if we use coalgebras rather than algebras, essentially because the Spanier-Whitehead duals of infinite objects are complicated. So: we have a pointed coalgebra spectrum <script type="math/tex">\eta: \mathbb{S} \to C</script>. This has a cofiber M, which is a C-comodule spectrum. It also supports a projection of the diagonal <script type="math/tex">M \to M \square_C M</script>, the fiber of which we define to be the tangent space <script type="math/tex">T_\eta C</script> at <script type="math/tex">\eta</script>. The effort in this construction comes from deciding what the cotensor product <script type="math/tex">M \square_C M</script> should mean, and how to control it once the decision is made — but it turns out that this is accomplishable.</p>
<p>So, like I said in the previous post, there are two obvious choices of spectra you can feed into this: <script type="math/tex">\Sigma^\infty_+ \mathbb{C}\mathrm{P}^\infty</script> and <script type="math/tex">\Sigma^\infty_+ K(\mathbb{Z}, n+1)</script>. In the first case, the spectrum you get out is the 2-sphere, and in the latter case, the spectrum you get out is a nonstandard invertible spectrum called the “determinantal sphere,” which is meant to describe its image as a line bundle over Lubin-Tate space:</p>
<script type="math/tex; mode=display">
\mathcal{E}_n(T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1)) \cong \Omega^{n-1}_{LT_n / \mathbb{Z}_p}.
</script>
<p>It’s not super important what this actually means — mostly it means that for <script type="math/tex">n > 1</script>, it simply cannot be a standard sphere.</p>
<p>So, that catches us up with the previous post. Here’s the first remarkable fact: this process is iterable. The definition of cotangent space studies 1-jets of functions, but we can consider pieces of m-jets instead, i.e., the quotients <script type="math/tex">I^{\otimes_K m} / I^{\otimes_K m+1}</script>, or even the spectral fibers <script type="math/tex">A_m = \mathrm{fib}\;M^{\square_C m} \to M^{\square_C m+1}</script>. It turns out that there is a K(n)-local equivalence <script type="math/tex">A_m = (A_1)^{\wedge m}</script> describing these further quotients as smash powers of the first one. What’s interesting about this is most visible in the case <script type="math/tex">C = \mathbb{C}\mathrm{P}^\infty</script>: these spectra <script type="math/tex">M^{\square_C m}</script> form a sort of cofiltration of C whose filtration quotients are given by <script type="math/tex">A_m = (A_1)^{\wedge m} = \mathbb{S}^{2m}</script>. What that must mean is that we’re picking up the cellular filtration of <script type="math/tex">\mathbb{C}\mathrm{P}^\infty</script>. Of course, this machine works just as well with <script type="math/tex">C = K(\mathbb{Z}, n+1)</script>, and once again I encourage you to think of this jet decomposition as a kind of cellular filtration of C — but this time “cell” is relaxed to mean a disk attached along an arbitrary element of the K(n)-local Picard group, rather than merely a standard sphere. And this is really interesting — the classical cellular structure of <script type="math/tex">K(\mathbb{Z}, n+1)</script> is dismally complicated, and that of its K(n)-localization is even worse, but if you enlarge your viewpoint a little bit things turn out to simplify dramatically.</p>
<p>Having made this comparison between these two spectra, we may as well keep going. The inclusion of the 2-skeleton into <script type="math/tex">\mathbb{C}\mathrm{P}^\infty</script> has another name: it is the Bott class in homotopy <script type="math/tex">\beta: \mathbb{C}\mathrm{P}^1 \to \mathbb{C}\mathrm{P}^\infty</script>. Using the jet decomposition of <script type="math/tex">K(\mathbb{Z}, n+1)</script>, we get a “Bott class” there too:</p>
<script type="math/tex; mode=display">
\beta: T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1) \to \Sigma^\infty_+ K(\mathbb{Z}, n+1),
</script>
<p>which is just the inclusion of the fiber <script type="math/tex">T_\eta C \to M</script> from before. What good is identifying the Bott element? — well, Snaith’s theorem says that you can use it to build complex K-theory:</p>
<blockquote>
<p>(Snaith:) As ring spectra, <script type="math/tex">\Sigma^\infty_+ \mathbb{C}\mathrm{P}^\infty[\beta^{-1}] \simeq KU.</script></p>
</blockquote>
<p>Of course, we can do this too: we define a spectrum <script type="math/tex">R_n = \Sigma^\infty_+ K(\mathbb{Z}, n+1)[\beta^{-1}]</script>, which I strongly recommend thinking of as a K-theory spectrum — so strongly that I’m going to call it “determinantal K-theory”. I’ll try to convince you that this is a good name in a moment. For now, let’s justify the name by noting that when <script type="math/tex">n = 1</script> it is precisely complex K-theory.</p>
<p>Craig Westerland has a remarkable theorem placing this object in classical homotopy theory:</p>
<blockquote>
<p>(Westerland:) There is an equivalence <script type="math/tex">R_n \simeq E_n^{hS\mathbb{S}_n}</script> of ring spectra, where <script type="math/tex">S\mathbb{S}_n</script> is the subgroup of “special” elements of the Morava stabilizer group. These are the elements in the kernel of the determinant map <script type="math/tex">\mathbb{S}_n \to \mathbb{Z}_p^\times</script> which arises from considering <script type="math/tex">\mathbb{S}_n</script> as acting on a certain division algebra as its units.</p>
</blockquote>
<p>This is neat enough even if you don’t know about classical Picard computations — for instance, this says that <script type="math/tex">R_n</script> receives the structure of an <script type="math/tex">E_\infty</script> ring spectrum — but the story turns out to be better than that. Since <script type="math/tex">S\mathbb{S}_n</script> is such a large subgroup, there are very few automorphisms of <script type="math/tex">R_n</script> left. Specifically, taking <script type="math/tex">\gamma</script> to be a topological generator of <script type="math/tex">\mathbb{Z}_p^\times</script>, there is a fiber sequence</p>
<script type="math/tex; mode=display">
(L_{K(n)} \mathbb{S}^0 =) E_n^{h\mathbb{S}_n} \to E_n^{hS\mathbb{S}_n} \xrightarrow{\psi^\gamma - 1} E_n^{hS\mathbb{S}_n},
</script>
<p>identifying <script type="math/tex">R_n</script> as “half” of the K(n)-local sphere in this precise sense.</p>
<p>… And then you refuse to stop there. The space BU is constructed from the spectrum KU by the formula <script type="math/tex">BU = \Omega^\infty KU \langle 1 \rangle</script>, and so we too can build an analogue <script type="math/tex">W_n = \Omega^\infty R_n \langle 1 \rangle</script>. This turns out to have a natural map <script type="math/tex">J_n: W_n \to BGL_1\mathbb{S}^0</script>, generalizing the classical J-homomorphism when n is set to 1. It even detects a family of elements identical in shape to the image of J, but in the homotopy groups graded along powers of the determinantal sphere. It <em>also</em> yields a Thom spectrum <script type="math/tex">X_n = \mathrm{Thom}(J_n)</script>, in analogy to MU, and there is then a theory of orientations. The spectrum <script type="math/tex">R_n</script> <em>also</em> has the same universality property as complex K-theory: it is initial among ring spectra which are multiplicatively oriented, where “orientation” is taken relative to this new replacement for MU. And the list goes on.</p>
<p>To me, this is really impressive and exciting — but it’s also young, which means there are loads of things unanswered and pending. I’ll point out two big ones, but there are plenty of others too if you go looking.</p>
<ol>
<li>There is no mention of geometry in this discussion. In particular, there is no known analogue of the spaces <script type="math/tex">BU(m)</script> for <script type="math/tex">m \neq 1, \infty</script> — Snaith’s theorem lets us make the jump from knowing about “line bundles” to knowing about virtual vector bundles with no intermediate geometric step. This is cool, but it’s also sort of bad news — it cuts us off from the enormous sector of mathematics that has developed around classical K-theory, and it also makes us rather blind about what to do next. Curiously, it’s rather easy to construct analogues of the spaces <script type="math/tex">\Omega SU(m)</script> — but doing so gives no indication of any underlying geometry.</li>
<li>There’s a second filtration that we may have accidentally confused with the skeletal filtration. Namely, there are equivalences <script type="math/tex">\mathbb{C}\mathrm{P}^\infty \simeq BU(1)</script> and <script type="math/tex">\mathbb{R}\mathrm{P}^\infty \simeq BO(1)</script>, and the first filtration quotients of the bar construction for these two spaces are <script type="math/tex">S^2 \simeq \Sigma U(1)</script> and <script type="math/tex">S^1 \simeq \Sigma O(1)</script> respectively. So, we might further guess that <script type="math/tex">K(\mathbb{Z}, n+1)</script> can be written as <script type="math/tex">BG(1)</script> in the K(n)-local stable category in a nonstandard way — it may be possible to pick <script type="math/tex">G(1) = \Sigma^{-1} T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1)</script> and produce an appropriate <script type="math/tex">A_\infty</script> multiplication on it, i.e., there may be interesting “elements of Hopf invariant one” that live on nonstandard spheres. That would certainly be a push toward becoming geometrically informed about <script type="math/tex">R_n</script>.</li>
</ol>
<p>In any case, there seems to be an elephant out there, and we seem to have gotten a pretty good hold of its tail. It will be really cool to see how this all sorts out in the distant future.</p>
tag:chromotopy.org,2013-04-01:1364838661The (very nice) Bousfield Lattice of Harmonic Spectra2013-04-01T17:51:01Z2013-04-01T17:51:01Z
<p>So this is my first post here on Chromotopy (thanks to Matt Pancia for making me aware of this blog, and Eric Peterson for letting exercise my typing fingers!). Just thought I might mention a little doodle about Bousfield lattices that occurred to me recently. It’s nothing too complicated, and it’s kind of cute, so I’ll include the proof. This is something that I realized while I was futzing around with general Bousfield lattices, notions of Stone duality (i.e. categorical anti-equivalences between certain species of lattice and certain species of topological space). Ultimately, I suspect it follows from one or two of the many beautiful theorems hidden away in papers like “Axiomatic Stable Homotopy Theory” and other stuff by Hovey, Strickland and Palmieri.</p>
<p>Here’s the point: in Mark Hovey and Neil Strickland’s beautiful (and incredibly dense) paper “Morava K-theories and Localisation” they show that the Bousfield lattice of the <script type="math/tex">E(n)</script>-local category of spectra is precisely the finite Boolean algebra generated by <script type="math/tex">n</script> objects, specifically the Bousfield classes of Morava K-theories, <script type="math/tex">\langle K(n)\rangle</script>. The natural next question, for me at least, is what the Bousfield lattice of the harmonic category (i.e. the <script type="math/tex">p</script>-local stable homotopy category localized at the infinite wedge of Morava K-theories <script type="math/tex">\bigvee_n K(n)</script>) looks like (let’s denote this category by <script type="math/tex">\mathcal{H}</script>). And it turns out, it’s precisely what one would expect! That is, it’s the colimit of the Bousfield classes of the <script type="math/tex">E(n)</script>-local categories, i.e. the infinite Boolean algebra generated by a countable number of minimal elements.</p>
<p>How do we know this? Well, what we really want to do is determine the <em>harmonic</em> Bousfield class of a harmonic spectrum <script type="math/tex">X</script>, which I’ll denote by <script type="math/tex">\langle X\rangle_H</script>. What does that really mean? It means that <script type="math/tex">\langle X\rangle_H=\{ Y\in\mathcal{H}:Y\wedge X\simeq \ast\}</script>. If you’re quick (which I am not), you certainly already see how to do this. If you’re like me, it might take you a few days, so since it’s not that big of a deal, you should just keep reading.</p>
<p>Let <script type="math/tex">supp(X)=\{K(n):X\wedge K(n) \not\simeq \ast\}</script> and <script type="math/tex">cosupp(X)=\{K(n):X\wedge K(n)\simeq \ast\}</script> . What we’re going to do is show that <script type="math/tex">\langle X\rangle_H=\bigvee_{K(n)\in supp(X)} \langle K(n)\rangle</script>. Suppose that <script type="math/tex">Y\in\langle X\rangle_H</script>. So, <script type="math/tex">Y\wedge X\simeq\ast</script>, and imortantly, <script type="math/tex">Y\wedge X\wedge K(n)\simeq\ast</script>. In other words, <script type="math/tex">Y\in\langle X\wedge K(n)\rangle_H=\langle K(n)\rangle_H</script>. Where this last equality follows from the fact that <script type="math/tex">K(n)\wedge X=\vee\Sigma^{d_i} K(n)</script> for some collection of <script type="math/tex">d_i</script>. Hence, <script type="math/tex">Y\in\langle \bigvee_{supp(X)}K(n)\rangle=\bigvee_{supp(X)}\langle K(n)\rangle</script>.</p>
<p>Now, suppose that <script type="math/tex">Y\in\langle \bigvee_{supp(X)}K(n)\rangle</script>. Then we want to show that <script type="math/tex">Y\wedge X\simeq\ast</script>. So, for every <script type="math/tex">K(n)\in supp(X)</script>, it’s clear that <script type="math/tex">Y\wedge X\wedge K(n)\simeq\ast</script>. But for every <script type="math/tex">K(m)\in cosupp(X)</script>, we have that <script type="math/tex">Y\wedge X\wedge K(m)\simeq\ast</script> also! So, since <script type="math/tex">X\wedge Y</script> is harmonic, but <script type="math/tex">X\wedge Y\wedge K(n)\simeq\ast</script> for every <script type="math/tex">n</script>, it’s also contractible!</p>
<p>One remark: I’ve neatly brushed under the rug here that colocalizing subcategories (e.g. harmonic spectra) may not be closed under smash product (I don’t think). So, I really should have written <script type="math/tex">\wedge_H</script> or something every time I wrote <script type="math/tex">\wedge</script>, but it works out fine if you just assume that after I took the smash product, I localized at the harmonic category again.</p>
<p>Anyway, the reason I think this is interesting is that this lattice has a really nice Zariski spectrum, and I first started looking at it hoping that it might approximate the Bousfield lattice of <script type="math/tex">p</script>-local spectra in some nice way. But as far as I can tell, it’s just too simple (you never get something for nothing, I suppose). I was hoping to make something of some ideas of Jack Morava regarding sheaves of spectra over the Bousfield lattice, but it may be that Bousfield lattices are just too coarse to do this kind of thing, or rather, tensor-triangulated categories are too coarse. We really need to rewrite the HPS paper I referenced above in the language of infinity categories and derived algebraic geometry.</p>
<p>It’s also interesting to note that the little proof above implies that the telescope conjecture holds inside the category of harmonic spectra. Obviously this doesn’t really say anything about the telescope conjecture outside this category. It’s also known, for instance, that the telescope conjecture holds inside the category of <script type="math/tex">BP</script>-local spectra. However, the Bousfield lattice in that category is not known. It’s structure is the content of a conjecture from Doug Ravenel’s well known ‘84 paper that is still open!</p>
<p>I hope to write more later about Stone duality for lattices/locales and Bousfield lattices in general.</p>
<p>:)</p>