Chromotopy

Determinantal K-theory

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 hnew on to your hats.

This story picks up where the previous post about spectral tangent spaces left off: there’s some machine which swallows spectra-with-diagonals whose cohomology looks like that of $\mathbb{C}\mathrm{P}^\infty$ 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 $x: A \to K$. The kernel of this map is the ideal I, which is thought of as functions vanishing at x. This ideal receives a multiplication map $I \otimes_K I \to I$ restricting the one for A, and the quotient (or cokernel) is the definition of the cotangent space: $T^*_x A = I / I \otimes_K I$. 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 $\eta: \mathbb{S} \to C$. This has a cofiber M, which is a C-comodule spectrum. It also supports a projection of the diagonal $M \to M \square_C M$, the fiber of which we define to be the tangent space $T_\eta C$ at $\eta$. The effort in this construction comes from deciding what the cotensor product $M \square_C M$ should mean, and how to control it once the decision is made — but it turns out that this is accomplishable.

So, like I said in the previous post, there are two obvious choices of spectra you can feed into this: $\Sigma^\infty_+ \mathbb{C}\mathrm{P}^\infty$ and $\Sigma^\infty_+ K(\mathbb{Z}, n+1)$. 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:

$$\mathcal{E}_n(T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1)) \cong \Omega^{n-1}_{LT_n / \mathbb{Z}_p}.$$

It’s not super important what this actually means — mostly it means that for $n > 1$, it simply cannot be a standard sphere.

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 $I^{\otimes_K m} / I^{\otimes_K m+1}$, or even the spectral fibers $A_m = \mathrm{fib}\;M^{\square_C m} \to M^{\square_C m+1}$. It turns out that there is a K(n)-local equivalence $A_m = (A_1)^{\wedge m}$ describing these further quotients as smash powers of the first one. What’s interesting about this is most visible in the case $C = \mathbb{C}\mathrm{P}^\infty$: these spectra $M^{\square_C m}$ form a sort of cofiltration of C whose filtration quotients are given by $A_m = (A_1)^{\wedge m} = \mathbb{S}^{2m}$. What that must mean is that we’re picking up the cellular filtration of $\mathbb{C}\mathrm{P}^\infty$. Of course, this machine works just as well with $C = K(\mathbb{Z}, n+1)$, 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 $K(\mathbb{Z}, n+1)$ 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.

Having made this comparison between these two spectra, we may as well keep going. The inclusion of the 2-skeleton into $\mathbb{C}\mathrm{P}^\infty$ has another name: it is the Bott class in homotopy $\beta: \mathbb{C}\mathrm{P}^1 \to \mathbb{C}\mathrm{P}^\infty$. Using the jet decomposition of $K(\mathbb{Z}, n+1)$, we get a “Bott class” there too:

$$\beta: T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1) \to \Sigma^\infty_+ K(\mathbb{Z}, n+1),$$

which is just the inclusion of the fiber $T_\eta C \to M$ from before. What good is identifying the Bott element? — well, Snaith’s theorem says that you can use it to build complex K-theory:

(Snaith:) As ring spectra, $\Sigma^\infty_+ \mathbb{C}\mathrm{P}^\infty[\beta^{-1}] \simeq KU.$

Of course, we can do this too: we define a spectrum $R_n = \Sigma^\infty_+ K(\mathbb{Z}, n+1)[\beta^{-1}]$, 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 $n = 1$ it is precisely complex K-theory.

Craig Westerland has a remarkable theorem placing this object in classical homotopy theory:

(Westerland:) There is an equivalence $R_n \simeq E_n^{hS\mathbb{S}_n}$ of ring spectra, where $S\mathbb{S}_n$ is the subgroup of “special” elements of the Morava stabilizer group. These are the elements in the kernel of the determinant map $\mathbb{S}_n \to \mathbb{Z}_p^\times$ which arises from considering $\mathbb{S}_n$ as acting on a certain division algebra as its units.

This is neat enough even if you don’t know about classical Picard computations — for instance, this says that $R_n$ receives the structure of an $E_\infty$ ring spectrum — but the story turns out to be better than that. Since $S\mathbb{S}_n$ is such a large subgroup, there are very few automorphisms of $R_n$ left. Specifically, taking $\gamma$ to be a topological generator of $\mathbb{Z}_p^\times$, there is a fiber sequence

$$(L_{K(n)} \mathbb{S}^0 =) E_n^{h\mathbb{S}_n} \to E_n^{hS\mathbb{S}_n} \xrightarrow{\psi^\gamma - 1} E_n^{hS\mathbb{S}_n},$$

identifying $R_n$ as “half” of the K(n)-local sphere in this precise sense.

… And then you refuse to stop there. The space BU is constructed from the spectrum KU by the formula $BU = \Omega^\infty KU \langle 1 \rangle$, and so we too can build an analogue $W_n = \Omega^\infty R_n \langle 1 \rangle$. This turns out to have a natural map $J_n: W_n \to BGL_1\mathbb{S}^0$, 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 also yields a Thom spectrum $X_n = \mathrm{Thom}(J_n)$, in analogy to MU, and there is then a theory of orientations. The spectrum $R_n$ also 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.

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.

1. There is no mention of geometry in this discussion. In particular, there is no known analogue of the spaces $BU(m)$ for $m \neq 1, \infty$ — 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 $\Omega SU(m)$ — but doing so gives no indication of any underlying geometry.
2. There’s a second filtration that we may have accidentally confused with the skeletal filtration. Namely, there are equivalences $\mathbb{C}\mathrm{P}^\infty \simeq BU(1)$ and $\mathbb{R}\mathrm{P}^\infty \simeq BO(1)$, and the first filtration quotients of the bar construction for these two spaces are $S^2 \simeq \Sigma U(1)$ and $S^1 \simeq \Sigma O(1)$ respectively. So, we might further guess that $K(\mathbb{Z}, n+1)$ can be written as $BG(1)$ in the K(n)-local stable category in a nonstandard way — it may be possible to pick $G(1) = \Sigma^{-1} T_+ \Sigma^\infty_+ K(\mathbb{Z}, n+1)$ and produce an appropriate $A_\infty$ 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 $R_n$.

In any case, there seems to be an elephant out there, and we seem to have gotten a pretty good hnew of its tail. It will be really cool to see how this all sorts out in the distant future.

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