Chromotopy

Watercooler research

March 13th, 2014

For the purposes of this post, I am going to assume that you are all familiar with the basics of Algebraic K-theory. If you’re not, just treat it as a black box, a gadget which takes in one of :

  • a (simplicial) ring (spectrum)
  • an augmented (simplicial) bimodule over a ring
  • an exact category
  • a Waldhausen category
  • a small, stable \infty-category

and spits out a spectrum K(\mathcal{C}), such that K(\mathcal{C}) does the job of  “storing all Euler characteristics” or “additive invariants”.

An important example is when \mathcal{C} is the category of finitely generated projective modules over a ring R, and then we call this K(R). It is a generally-accepted fact that the K-theory of rings is generically a very difficult and often useful thing to compute (knowledge of the K-groups of \mathbb{Z}, for example, would be quite valuable to many). In an ideal world, we would be able to understand the functor

A \mapsto \tilde{K}(A) = hofib(K(A) \rightarrow K(R))

on the category of algebras augmented over a given ring R, 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) R-bimodules

M \mapsto \tilde{K}(T_R (M)),

where T_R (M) is the tensor algebra on M. 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 R-algebras is given by

P_1 (id) (A) = A / I^2 ,

where I is the augmentation ideal of A. In the case where A = T_R (M) for some R-bimodule $M$, then the Goodwillie derivative P_1 (id)(T_R (M)) = R \oplus M, where R \oplus M is the the square-zero extension of R by M, the ring given by demanding that M^2 = 0. In studying the K-theory of square-zero extensions, then, we are also studying the K-theory of the linearization of the tensor algebra functor (note how many simplifications we have already done!).

A different strategy for dealing with this computational difficulty is to try and understand how the K-theory of a ring changes as we “perturb” the ring. To do this, we look at “parametrized K-theory” (or the “K-theory of parametrized endomorphisms”):

Definition. Let R be a ring, and let M be an R-bimodule. We define the parametrized K theory of R with coefficients in M, K(R;M), to be the K-theory of the exact category of pairs (P,f) where P is a finitely-generated projective R-module and f: P \rightarrow P \otimes_{R} M is a map of R-modules.

We think about K(R;M) as being the K-theory of endomorphisms with coefficients that are allowed to be in M. If our picture of a finitely-generated projective R-module P is as living as a summand of a rank n free R-module, then an element of the above exact category is an n \times n matrix with entries in M that commutes with the projection map R^n \rightarrow R^n defining P.

Why should we look towards endomorphisms as perturbations? Well, the picture is supposed to be the following: Let R be a ring and M an R-bimodule. This is the same data as a sheaf \mathcal{F}_M over Spec(R), and we would like to think of a deformation of this over, say, R[t]/t^2 as a flat sheaf \mathcal{F}_{M'} over R[t]/t^2 (a module) that restricts to \mathcal{F}_M over Spec(R).

Screen Shot 2014-03-13 at 2.21.02 PM

which corresponds to an element of Ext^1_R (M,M), or a derived endomorphism. The idea is that an extension of M corresponds to a deformation of M, which is a reasonable perspective.

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 K-theory, the K-theory of endomorphisms naturally comes up. In this paper they prove the following theorem:

Theorem. For R a ring and M a discrete R-bimodule)

\tilde{K}(R;B_\bullet M) \simeq \text{hofib} \left\{ K(R \oplus M) \rightarrow K(R) \right\} = \tilde{K}(R \oplus M),

where R \oplus M is again the the square-zero extension of R by M.
We think of R \oplus M as a perturbation of R by M, as the elements that we are adding on (those coming from the direct summand M) are “so small” that they multiply to zero. This result relates the “perturbation” approach to understanding K-theory to our earlier potential approach of understanding the free objects in the category of augmented R-algebras.

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.

K-Theory of Endomorphisms: The Classical Story

Definition. Let R be a ring and consider the category End(R), whose objects are pairs (P,f) with P a finitely-generated projective R-module and f:P \rightarrow P an endomorphism. The morphisms in this category are commutative diagrams of the appropriate type.

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:

Example. [The Characteristic Polynomial]
Let (P,f) \in End(R), then the characteristic polynomial of f is given by

\lambda_t (f) = \sum_{i \geq 0} Tr(\Lambda^i f)t^i.

This can also be obtained in the usual way as a determinant.

The important property of the characteristic polynomial is that if we have a commutative diagram in End(R)

Screen Shot 2014-03-13 at 2.20.50 PM

with exact rows, then

\lambda_t (f) = \lambda_t (f') \cdot \lambda_t (f''),

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).

We now know that a search for additive invariants is a desire to compute K-theory, and so we define:

Definition. K_0 (End(R)) is defined to be the free abelian group on isomorphism classes of objects in End(R) modulo the subgroup generated by the relations [(P,f)] = [(P',f')] + [(P'',f'')] if there is a commutative diagram

Screen Shot 2014-03-13 at 2.20.46 PM

with the rows exact. There is a natural splitting

K_0 (End(R)) \simeq K_0 (R) \times \tilde{K}_0 (End(R))

coming from thinking of the category of finitely generated projective R-modules as living in End(R) as the guys with 0 endomorphisms.

Of course, End(R) is an exact category, and we could define higher K-groups for this as well.

K_0 (End(R)) is the repository for additive invariants, in that it has the following universal property:

Proposition. Let F be a map from the commutative monoid of isomorphism classes of objects in End(R) to an abelian group A, such that F splits short exact sequences as above. Then F factors through K_0 (End(R)), or K_0(End(R)) is the initial abelian group for which short exact sequences of endomorphisms split.

What does this look like, though? Well, let’s go back to the characteristc polynomial:

Theorem. The map

c: [(P,f)] \mapsto ([P], \lambda_t (f))

is an isomorphism

K_0 (End(R)) \simeq K_0 (A) \times \tilde{W}(R),

where

\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 \}

is the multiplicative group of fractions with constant term 1.

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:

Proposition. The inclusion \tilde{W}(R) \rightarrow W(R) exhibits \tilde{K}_0 (R) as a dense \lambda-subring of W(R), where W(R) are the big Witt vectors of R, modelled as power series with constant term 1.

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.

It is also important to mention what some of the uses of this equivalence are:

  • 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;
  • Operations that exist on Witt vectors get turned into operations on K-theory, and vice-versa:
  • The n-th ghost map is given by
    gh_n ([P,f] = tr(f^n));
  • The Frobenius map is given by
    F_n ([P,f]) = [P, f^n];
  • The Verschiebung map is given by
    [V_n ([P,f]) = [P^{\oplus n}, v_n f],
    where v_n f is represented by a shift in the first n-1 factors and action by f in the last factor. This represents an n-th root of f, in that V_n^n looks like applying f to all of the blocks of P^{\oplus n}.

Lindenstrauss-McCarthy and Topological Witt Vectors

The first thing to do to start generalizing the previous construction is to allow for a “parametrization”, like we said before.

Definition. Let R be a ring and M an R-bimodule. End(R;M) is the category whose objects are pairs (P,f) with P a finitely generated projective R-module and f: P \rightarrow P \otimes_R M a map of R-modules. As in End(R), morphisms are commutative diagrams.

End(R;M) inherits an exact/Waldhausen structure from considering what’s happening in the “base”, and so it makes sense to define parametrized K-theory as

K(R;M) := K(End(R;M)).

There is a natural map from End(R;M) to the category of finitely generated projective R-modules that forgets the endomorphism, and so we can reduce and consider

\tilde{K}(R;M) := hofib(K(R;M) \rightarrow K(R)).

We can also consider $M$ to be simplicial by geometrically realizing, and this assembles to a functor

\tilde{K}(R;-): \left\{\text{simplicial } R\text{-bimodules} \right \}\; \rightarrow \text{Spectra},

which is a good setting to do Goodwillie calculus.

The remarkable result of Lindenstrauss-McCarthy is the following:

Theorem. The functors \tilde{K} (R; -), W(R;-) from simplicial R-bimodules to spectra have the same Taylor tower, where W(R;-) is a “topological Witt vectors” construction.

Now, this is only remarkable if I tell you what this W(R;-) 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.

Construction. Let R be a ring, M a bimodule. We define a bunch of spectra U^n (R;M) with C_n action by letting U^n (R;M) be the derived cyclic tensor product (over R) of n copies of M.

There is an evident C_n action on U^n (R;M) given by permuting the factors, and moreover when n | N, we have natural restriction maps between the fixed points

res : U^N (R;M)^{C_N} \rightarrow U^n (R;M)^{C_n}.

We then define

W(R;M) := holim_{res} U^n (R;M)^{C_n}.

Example. U^1 (R;M) \simeq THH(R;M) as normally defined, and if R = M then we have that W(R;M) \simeq TR(R;M).

This functor W(R;M) is meant to be an attempt to define TR in the absence of the cyclic symmetry that is present when M = R. The remarkable thing is that this actually works! The nomenclature is justified by the following:

Theorem. (Hesselholt-Madsen) \pi_0 (TR(R)) \simeq W(R).

The virtue of this setup is that we have explicit understanding of what the layers of the Taylor tower of W(R;-) look like, as well as various splitting theorems, e.g. we have the following “fundamental cofibration sequence”:

Theorem.  There is a homotopy fiber sequence

U^n (R; M)_{hC_n} \rightarrow W^{n}(R; M) \rightarrow W^{(n-1)}(R; M) ,

where W^{n}(R;M) are the functors obtained by truncating the homotopy limit defining W(R;M).

Moreover, we have that W^{n}(R;M) is the n-th polynomial approximation of W (R;M), so this is the fiber sequence that computes the layers of the Taylor tower.

Generalizations

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 R is a connective ring spectrum and M is a (simplicial) R-bimodule.

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 R, there is a id-Cartesian cube with initial vertex R, and such that all other vertices are canonically stably equivalent to the Eilenberg-MacLane spectrum of a simplicial ring.

The following fact lets us promote results about simplicial rings to those about connective ring spectra:

Theorem. If \chi is an (id)-Cartesian k-cube of connective ring spectra, then the cube K(\chi) is (k+1)-Cartesian.

Idea for proving generalization: Show that the functors W(-;-), \tilde{K}(-;-) have a similar property, so that the equivalence given by Lindenstrauss and McCarthy can be promoted to one for connective ring spectra.

This works just fine, and we arrive at:

Theorem. (P.) The Lindenstrauss-McCarthy result holds for connective ring spectra.

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 means. 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 K-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 K-theory, etc.).

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.

February 26th, 2014

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.

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. To correct for this, one takes the collection of all 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

D: \left\{\text{formal groups}\right\} \longrightarrow \left\{\begin{array}{c}\text{collections of}\;p\text{-typical curves} \\ + \\ \text{structure} \end{array}\right\}

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 perfect positive-characteristic field k. The three relevant pieces of structure are the actions of homotheties, of the Frobenius, and of the Verschiebung, which were all described way back in this post on Witt vectors. (In the notation of that post, we’re interested just in F_p and V_p.) Altogether, this gives a formula for the Cartier ring:

\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).

Say that a Dieudonné module is formal when it is: finite rank; free; reduced, meaning that it’s V-adically complete; and uniform, meaning that the natural map M / VM \to V^k M / V^{k+1} M 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 p-divisible group can be made sense of, and Dieudonné modules which are free of finite rank (but without reducedness or uniformity) are equivalent to p-divisible groups.

Dieudonné modules are thrilling because they are just modules, whereas p-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:

Theorem: Take \mathbb{G} to be a p-divisible group and M its Dieudonné module. There is a natural isomorphism of k-vector spaces T_0 \mathbb{G} \cong M / VM. In the case that \mathbb{G} is one-dimensional, then the rank of M as a \mathbb{W}(k)-module agrees with the height d of \mathbb{G}. Moreover, if \gamma is a coordinate on \mathbb{G} (considered as a curve), then \{\gamma, V\gamma, \ldots, V^{d-1} \gamma\} forms a basis for M.

There is also something like a classification of the simple Dieudonné modules over \bar{\mathbb{F}}_p (where, among other things, the étale component of a p-divisible group carries little data):

Theorem (Dieudonné): For m and n coprime and positive, set

G_{m,n} = \mathrm{Cart}(\bar{\mathbb{F}}_p) / (V^m = F^n).

Additionally, allow the pairs (m, n) = (0, 1) to get

G_{0,1} = D(\mathbb{G}_m[p^\infty])

and (m, n) = (1, 0) to get

G_{1,0} = D(\mathbb{Q} / \mathbb{Z}).

(All these modules G_{m,n} have formal dimension n and height (m+n).) For any simple Dieudonné module M, there is an isogeny M \to G_{m,n}, i.e., a map with finite kernel and cokernel. Moreover, up to isogeny every Dieudonné module is the direct sum of simple objects.

Every abelian variety A comes with a p-divisible group A[p^\infty] arising from its system of p-power order torsion points. This connection is remarkably strong; for instance, there is the following theorem:

Theorem (Serre–Tate): Over a p-adic base, the infinitesimal deformation theories of an abelian variety A and its p-divisible group A[p^\infty] agree naturally.[1]

For this reason and others, the p-divisible group of an abelian variety carries fairly strong content about the parent variety. On the other hand, it is not immediately clear which p-divisible groups arise in this way. Toward this end, there is a symmetry condition:

Lemma (“Riemann–Manin symmetry condition”): As every abelian variety is isogenous to its Poincaré dual and the corresponding (Cartier) duality on p-divisible groups sends (V to F and hence) G_{m,n} to G_{n,m}, these summands must appear in pairs in the isogeny type of the Dieudonné module of A[p^\infty].

As a simple example, this gives the usual categorization of elliptic curves: an elliptic curve is 1-dimensional, hence has p-divisible group of height 2. One possibility, called a supersingular curve, is for the Dieudonné module to be isogenous to G_{1,1}; this is a 1-dimensional formal group of height 2 and it satisfies the symmetry condition. The only other possibility, called an ordinary curve, is for the Dieudonné module to be isogenous to G_{1,0} \oplus G_{0,1}; this is the sum of a 1-dimensional formal group of height 1 with an étale component of height 1, and it too satisfies the symmetry condition.

A remarkable theorem is that the converse of the symmetry lemma holds as well:

Theorem (Serre, Oort; conjectured by Manin): If a Dieudonné module M satisfies the above symmetry condition, then there exists an abelian variety whose p-divisible group is isogenous to M.

Both proofs of this theorem are very constructive. Serre’s proof explicitly names abelian hypersurfaces whose p-divisible groups are of the form G_{m,n} \oplus G_{n,m}, for instance.

Now, finally, some input from algebraic topology. The Morava K-theories of Eilenberg–Mac Lane spaces give a collection of formal groups which can be interpreted in the following way:

Theorem (Ravenel–Wilson): For 1 < m ≤ n, the p-divisible group associated to K(n)^* K(\mathbb{Q}/\mathbb{Z}, m) is (in a suitable sense) the mth exterior power of the p-divisible group K(n)^* K(\mathbb{Q}/\mathbb{Z}, 1). It is smooth, has formal dimension \binom{n-1}{m-1}, and has height \binom{n}{m}. (Additionally, it is zero for m > n.)

Theorem (Buchstaber–Lazarev): For 1 < m ≤ n, the same p-divisible group K(n)^* K(\mathbb{Q}/\mathbb{Z}, m) has Dieudonné module isogenous to the product of \frac{1}{n_0} \cdot \binom{n}{m} copies of G_{n_0-m_0,m_0}, where m_0/n_0 is the reduced fraction of m/n.

The conclusion of Buchstaber and Lazarev is that this means that these p-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 n is even and m = n/2, 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:

Observation (I heard this from Ravenel, but surely Buchstaber and Lazarev knew of it): The sum of all of the Dieudonné modules

\bigoplus_{m=0}^{n\;\text{(or }\infty)} D(\mathrm{Spf}\;K(n)^* K(\mathbb{Q}/\mathbb{Z}, m))

satisfies the Riemann-Manin symmetry condition.

This is an interesting observation. In light of the comments at the end of the Buchstaber–Lazarev paper, one wonders: why privilege m = n/2? But, even more honestly, why privilege m = 1 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 p-divisible group realizes the large (2^{n-1}-dimensional!) formal group associated to the K-theoretic Hopf ring of Eilenberg–Mac Lane spaces?

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:

  • If such an abelian variety existed, there would be a strange filtration imposed on its p-divisible group arising from the degrees of the Eilenberg–Mac Lane spaces. The Riemann–Manin symmetry condition tells us that these G_{m,n} and G_{n,m} factors must come in pairs, but in almost all situations, one pair appears on one side of the middle-dimension n/2 and the other appears on the other side. What could this mean in terms of the hypothetical abelian variety?
  • Relatedly, this pairing arises from the “\circ-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?
  • There should be accessible examples of these hypothetical varieties at low heights. For instance, the one associated to height 1 via mod-p complex K-theory is (canonically, not merely isogenous to) a sum G_{1,0} \oplus G_{0,1} — i.e., it comes from an ordinary elliptic curve. Can we identify which elliptic curve — and in what sense we can even ask this question? Is there a naturally occurring map from the forms of \mathbb{G}_m to the ordinary locus on the (noncompactified?) moduli of elliptic curves? What about forms of \hat{\mathbb{G}}_m? What if we select a supersingular elliptic curve instead — is there an instructive assignment to abelian varieties whose Dieudonné module has isogeny type G_{1,0} \oplus G_{1,1} \oplus G_{0,1}? (On the face of it, this last bit doesn’t look so helpful, but maybe it is.)

[1] – Presumably this can be expressed by saying that the map from a moduli of abelian varieties to a moduli of p-divisible groups is formally étale, but no one seems to say this, so maybe I’m missing something.

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 20th century mathematics.

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 here: [link].

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 here: [link]; 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.

And, as an uninspired parting remark, we topologists do have access to the pro-system

\left( \cdots \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty\right)

November 27th, 2013

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…

Everyone’s pretty familiar with the (co)equalizer sheaf condition. That is, if we’ve got a map of rings R\to S and we want to know whether or not an S-module M has a (possibly unique) “representative” among R-modules, we need to make sure that when we tensor M up to being an S\otimes_R S-module along either the left unit \eta_L:S\cong S\otimes_R R\to S\otimes_R S or the right unit \eta_R:S\cong R\otimes_R S\to S\otimes_R S, we get the same S\otimes_R S module.

You might be more familiar with the opposite of this diagram that you get from thinking of such a module as a sheaf on Spec(S). 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 \infty).

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) \varphi:R\to S, the “descent problem” associated to this map is the question: Given an S-module M, when is it the case that there is an R-module M_0 such that M\cong M_0\otimes_R S? 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 Spec(S) and we want to “descend” it down to Spec(R). Note that one typically leaves unsaid exactly how S is an R-module, since it’s assumed to be clear that it’s along the map \varphi. However, sometimes I’ll use the notation M_0\otimes_\varphi S  to indicate the precise way in which S is an R module, and perhaps more importantly, exactly in what way we’re lifting up the module structure on M_0. Given the previous discussion, you can probably already see why this is going to be important to us.

The category of “descent data” for such a map \varphi should be, intuitively, things over S with some kind of information on how to produce a thing over R. There are many, many ways to formalize this. If you’re used to dealing with “covers” that look like \mathcal{U}=\{U_i\}_{i\in I}, 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.

But if our “things over S” 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 R\to S, and start with an S-module M. We can glue M along double intersections if, as we said above, the two possible ways of tensoring M up to S\otimes_R S-modules are isomorphic, and these isomorphisms satisfy a cocycle condition. Now from here, there are three ways to tensor M\otimes_S(S\otimes_R S) up to an S\otimes_R S\otimes_R S-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.

First we need the following:

Definition:  Let A^\bullet be a cosimplicial ring, and M^\bullet be a cosimplicial module over A. That is, A^\bullet is a functor A:\Delta\to CRng and M^\bullet is a functor M:\Delta\to CRng\times Mod whose value in CRng is A^\bullet. Then M^\bullet is said to be co-cartesian over A^\bullet if for every map \varphi:[n]\to[m] in \Delta, the map M(\phi):M^n\to M^m induces an isomorphism M^n\otimes_{A(\phi)} A^m\cong M^m

Let’s just unwind this definition for a second here. At level n\geq 0 in the cosimplicial module M, there are a whole bunch of maps M^n\to M^m for any other m\geq 0. What this condition is saying is the codomain of these maps is the same thing as just tensoring up to that cosimplicial level along the associated map in A^\bullet!

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 R\to S to be the cosimplicial modules over the cosimplicial ring S\to S\otimes_R S\to S\otimes_R S\otimes_R S\to\cdots which are cocartesian, where we replace the isomorphisms in our definition of cocartesian with homotopy equivalences.

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 S-module along a map \varphi:R\to S, and we do this by way of a Bousfield-Kan spectral sequence. Suppose we’re given an S-module M. Then what we’re interested now in computing are co-cartesian modules over the cosimplicial ring S\to S\otimes_R S\to\cdots whose bottom level is the module M. 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 M^n\simeq M\otimes_S (S\otimes_R\cdots \otimes_R S), where there are n copies of S on the right hand side.

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 R\to S, the category of descent data is equivalent to the category of S\otimes_R S comodules, where we regard S\otimes_R S as a coring, called the canonical descent coring. 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 S\otimes_R S in a homotopically coherent way. For instance, the first level of the cosimplicial module tells us that we have a map (well, two maps) M\to M\otimes_R S\cong M\otimes_S S\otimes_R S, which is a coaction of the canonical coring on M. The usefulness of the co-cartesian criterion is that we know that the two ways that M\otimes_S S\otimes_R S can be isomorphic to M\otimes_R S are equivalent. In other words, descent data with a fixed base S-module M are the same thing as S\otimes_R S-comodules structures on M. So how can we work out what possible such structures there are?

This is where the BKSS comes in. Notice that such a comodule structure on M is going to be a system of maps from the constant cosimplicial object on M, let’s denote it by \tilde{M}^\bullet, to the cosimplicial object which is just M tensored (over S!) with S\otimes_R S\to S\otimes_R S\otimes_R S\to\cdots. So, to compute the space of descent data on M, we need to attempt to compute the homotopy of the cosimplicial mapping space/spectrum/object of your favorite model category Hom(\tilde{M}^\bullet, M\otimes_S (S\otimes_R S)^\bullet).

I hope to write more soon about how this space we’re computing can be compared to the space of twisted forms for M along the map R\to S, 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.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!

 

November 16th, 2013

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 Complex Cobordism and Algebraic Topology. 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.

The story starts the usual way: there is a sense in which the homology groups MU_* X 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 \Lambda, thought of as change-of-coordinate maps. Equivalently, this can be packaged into saying that MU_* X forms a sheaf over the moduli stack of formal groups. After p-localization, the geometric points of this stack admit a nice description: the positive characteristic ones are indexed by “heights” 1 \le n \le \infty, 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 \Lambda as being “spent” on moving an arbitrary formal group law into one of these canonical forms. However, some of \Lambda will remain — the part which gives automorphisms of the Honda formal group law — and this part is referred to as the “stabilizer group” \mathbb{S}_n.

E-theory comes in by attempting to restrict the sheaf MU_* X 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 previous post) 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.

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 “p-series”, is given by [p]_n(x) = x^{p^n}. Any formal group law in positive characteristic carries a Frobenius endomorphism S: x \mapsto x^p, and so in the endomorphism ring of the Honda formal group law, there is a relation S^n = p. 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

\mathbf{o}_n = \mathbb{W}(k)\langle S \rangle / \left( \begin{array}{c} Sw = w^\varphi S \\ S^n = p \end{array} \right),

where the angle brackets denote a free noncommuting associative algebra. The stabilizer group \mathbb{S}_n can then be identified with the compositional units of the endomorphism ring: \mathbb{S}_n = \mathbf{o}_n^\times.

What about the point at infinite height? The notation is actually arranged so that you can sort of take limits in n: the p-series becomes [p]_\infty(x) = 0, and the endomorphism ring becomes

\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).

Much of the chromatic story is unable to be made explicit, but these two objects are quite familiar: the formal group law whose p-series vanishes is precisely the additive formal group law x +_\infty y = x + y. 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 previously done a questionable job in justifying a connection between this endomorphism algebra and the Steenrod algebra.

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 \mathbb{S}_n, and if we take these division algebras \mathbf{o}_n and quotient them by p, then we can pull back these E-theory representations to get a sequence of representations over the group

(\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.

Then, he says, these should “limit” in an appropriate sense to the representation that ordinary mod-p 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 K-theory of a finite spectrum from its ordinary mod-p homology will be affected by sparseness, and so for very large n, we wouldn’t expect the differential behavior of the E(n)-AHSS to differ from the E(n+1)-AHSS.

I would posit that something more might be happening.

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 p-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 x +_! y:

\log_!(x) = \int \left( \left.\frac{\partial(x +_! y)}{\partial y} \right|_{y=0} \right)^{-1} dx.

For finite height formal group laws, this integral equation has no solution — eventually you end up having to integrate a monomial like x^{p^n-1}. Indeed, the degree of the bottommost obstruction to this integral equation is in a degree of precisely that form, and that value n is referred to as the “height”.

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 x \mapsto x^{p^n} has the action of the zero map, which can be minimally justified by setting x^{p^n} itself to zero — i.e., restricting to the (p^n-1)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 space carries power operations, and in particular an action by \psi^p. Possibly some construction like E_n / \psi^p is relevant?) In any case, this is something I would like to know about.

This dishonest geometry also suggests another intriguing connection: there is a fast mnemonic for remembering the standard quotient algebras \mathcal{A}(n) 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, \mathcal{A}(n) corepresents precisely the power series which vanish above order 2^n, 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 x(n) with the property that H\mathbb{F}_2^* x(n) = \mathcal{A} /\!/ \mathcal{A}(n); such spectra would behave as though they were taking mod-2 cohomology, modified so that its coalgebra of cooperations was \mathcal{A}(n)_* (cf., the Adams spectral sequence method for computing the real K-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 n > 3. However, the spectra x(0), x(1), and x(2) all exist, and there is the following intriguing table:

\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}

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 x(n) side? Can partial theorems be proven about the nonexistent x(n) spectra, given these K(n)-local models? Can one see from the perspective of formal groups and some knowledge of the stabilizer groups this famous theorem that x(3) 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 x(3) that prevents it from existing and which they circumvent.

Hm!

——

This doesn’t really fit anywhere, but Morava has another article, titled Forms of K-theory, which has another really unique perspective on things. Published in 1989 and written in 1973, he manages to construct p-adic TMF 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.

In this same preparation spree, I also rediscovered a talk by Mike Hopkins, From Spectra to Stacks, where he produces stacks associated to any ring spectrum rather than just those which are complex oriented. He does this by passing them through MU in a certain sense, and what’s interesting is this remark of his that MU isn’t special in this regard; any spectrum satisfying a few nicety properties will do. Morava, in Forms of K-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 MU where the algebra of MU_* and MU_* MU 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 topology, however, is no less of a mystery.

Anyway, these are also worth reading.

October 20th, 2013

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.

Suppose f \colon X \to Y is a map of schemes, \mathcal{F} a quasicoherent sheaf on X. Descent in this setting takes the following form.

Question: When is \mathcal{F} isomorphic to f^*\mathcal{G} for some quasicoherent sheaf \mathcal{G} on Y?

Let’s first look for some necessary conditions. Suppose we have such a \mathcal{G}, in which case we say that \mathcal{F} satisfies descent. Write p_0, p_1 \colon X \times_Y X \to X for the projections onto the first and second factors, respectively. Then there necessarily exists an isomorphism \alpha \colon p_0^*\mathcal{F} \cong p_1^*\mathcal{F} by commutation of the relevant fiber square and naturality of the pullback functors. Going a step further and examining the threefold fiber product X \times_Y X \times_Y X, we see that \alpha 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 \mathcal{F} satisfies descent, it does so via descent data in the form of an \alpha satisfying the cocycle condition. The collection of such pairs (\mathcal{F}, \alpha) forms a category in the evident way, which we call the category of descent data for f. The question now becomes how close the category of descent data is to the category of quasicoherent sheaves on Y. A fairly general and important class of morphisms f are covered in the following theorem due to Grothendieck.

Theorem: (Grothendieck) If f \colon X \to Y is faithfully flat, i.e. flat and surjective, then the categories of descent data for f and quasicoherent sheaves on Y are equivalent through the above construction.

A nice, simple example of the theorem arises when X, Y are spectra of fields L, K, respectively, and f is induced by a Galois extension K \hookrightarrow L. Then Grothendieck’s theorem states that the category of vector spaces over K is equivalent to the category of vector spaces over L equipped with descent data \alpha as above. The usual statement of Galois descent, however, holds that this category is equivalent to the category of L-vector spaces with semilinear actions of the Galois group Gal(L/K). This arises from a different, equivalent formulation of descent data which we will now discuss.

The starting point of this alternative perspective is the observation that the fiber product X \times_Y X forms the morphisms of a groupoid scheme, with the objects given by X. That is, there are the two projection maps p_0, p_1 from before, which we will rename s, t, along with a unit morphism \eta \colon X \to X \times_Y X given by the diagonal, and an antipode \tau \colon X \times_Y X \to X \times_Y X given by interchanging the two factors. The subtlest bit of structure is in the composition (X \times_Y X) \times_X (X \times_Y X). This is given by first identifying this with the iterated fiber product X \times_Y X \times_Y X, applying f to the middle factor so as to map to X \times_Y Y \times_Y X, then identifying this with X \times_Y X. 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 \mathcal{F} is the same as an action of this so-called descent groupoid, 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 Spec(L) \times Gal(L/K), and the descent groupoid action translates to a semilinear Galois action.

Now let’s switch to some homotopy theory. Suppose the morphism we’re concerned with is the inclusion of a basepoint x \colon pt \to X into some space X, and we’re given a space Y over pt (that is, a space.)

Question: When is Y weakly equivalent to x^*Z for some space Z over X?

As stated, the answer to this question is “always,” since we may just take Z = Y \times X. A more interesting question is the following.

Question: Can we classify all Z over X with the property that Y \simeq x^* Z?

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 homotopical 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 homotopy fiber product pt \times_X^R pt with its associated projections p_0, p_1 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 \Omega_x X. Moreover, the appropriate notion of groupoid object here is a sort of \mathbb{E}_1-groupoid, and the discussion from above endows the pair (\Omega_x X, pt) with the structure of such a gadget. A groupoid with a single object is just a group, though, and correspondingly we obtain \Omega_x X as a grouplike \mathbb{E}_1-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 \Omega_x X. (Here we should really be saying “homotopical category” or “\infty-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.

Theorem: If X is connected, then the homotopy theories of spaces over X and \Omega_x X-spaces are equivalent.

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 \pi_0. The connectedness hypothesis ensures that x \colon pt \to X 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 X = BG for a discrete group G. Then the map x is homotopy equivalent to the universal cover of X, the based loop space is equivalent to G, and the descent statement is familiar from covering space theory.

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 \mathbb{E}_\infty-ring spectrum E. Then to first approximation, we should think of Spec(E) as being the ordinary scheme Spec(\pi_0 E) with a collection of quasicoherent sheaves Spec(\pi_n E) providing “higher nilpotents.” Alternatively, we have the topological space |Spec(\pi_0 E)| with a sheaf of \mathbb{E}_\infty-ring spectra on it given by localizing E appropriately over every open set. Quasicoherent sheaves on Spec(E) are just E-module spectra, and the terminal affine spectral scheme is given by Spec(S), the Zariski spectrum of the sphere spectrum. Consider the morphism Spec(H\mathbb{F}_p) \to Spec(S) given by the unit S \to H\mathbb{F}_p. Here the right-hand side is the Eilenberg-Mac Lane spectrum of the finite field \mathbb{F}_p as usual. The descent question here takes the following form.

Question: Given a H\mathbb{F}_p-module spectrum M, when is it weakly equivalent to H\mathbb{F}_p \wedge N for some spectrum N?

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 H\mathbb{F}_p \wedge H\mathbb{F}_p whose homotopy groups form the mod p dual Steenrod algebra. Thus the \infty-category of descent data is equivalent to the \infty-category of comodule spectra over the dual Steenrod algebra spectrum. Convergence of the Adams spectral sequence gives the following descent theorem.

Theorem: There is an equivalence of homotopy theories between that of comodule spectra over the dual Steenrod algebra spectrum that are compact as H\mathbb{F}_p-module spectra and of compact p-complete spectra.

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 (p) in Spec(S), which corresponds to p-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.