Loop spaces and descent

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 threefnew 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, hnews 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 new-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 $\mid Spec(\pi_0 E) \mid$ 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.