The K-Theory of Endomorphisms

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

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

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

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 $\operatorname{Spec}(R)$, 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 $\operatorname{Spec}(R)$.

which corresponds to an element of $\operatorname{Ext}^1_R (M,M)$, 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)

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

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 $End(R)$

with exact rows, then

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

with the rows exact. There is a natural splitting

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

is an isomorphism

where

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

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

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

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 $n \mid N$, we have natural restriction maps between the fixed points

We then define

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 $p$-typical statement:

Theorem. (Hesselholt-Madsen) $\pi_0 (TR(R;p)) \simeq W_{(p)}(R),$

where the latter is a suitable version of $W(R;M)$ that takes a homotopy limit over only $U^{p^n}(R;M)^{C_{p^n}}$.

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

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.</span>

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.

Hypothetical abelian varieties from K-theory

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

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:

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

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

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

(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 $% $, the $p$-divisible group associated to $K(n)^* K(\mathbb{Q}/\mathbb{Z}, m)$ is (in a suitable sense) the $m$th 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 $% $, 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

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 twentieth 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.

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 on the MSRI website; 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

Some Thoughts on Descent and Descent Data

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!