Let be a finite group, and let be a field of characteristic (posssibly ). If is a -vector space, recall that a representation of is a map .
Theorem (Maschke): Representation theory when does not divide (the nonmodular characteristic case) is good, i.e., the category is semisimple.
When , things become more interesting, but I won’t get into this. One important difference comes from the norm map.
Define a map by . Clearly this map kills anything of the form , and is invariant under the action of . It follows that this map factors as . This map is called the norm map.
Proposition: In nonmodular characteristic, the norm map is an equivalence; the inverse is induced by the map .
In chromatic homotopy theory, we have a whole collection of spectra which interpolate between characteristic zero and characteristic , namely the Morava -theories. One might ask how representation theory over a Morava -theory behaves. Here, we have the following preliminary result of Hovey-Sadofsky:
Theorem (Greenlees-Sadofsky): Suppose acts trivially on . Then the norm map is an equivalence.
A consequence of this theorem is the following.
Theorem (Hovey-Sadofsky): The -localization of the cofiber of the norm map on is trivial, i.e., is self-dual in the category of -local spectra.
We haven’t yet defined the norm map for group actions on spectra, but we will do so momentarily. Recently, Hopkins and Lurie have vastly generalized this result.
Theorem (Hopkins-Lurie): Let be a finite group acting on any -local spectrum . Then the norm map is an equivalence.
Woah! Their theorem is in fact much more general than this, but I don’t really know why they work so hard to prove it in generality. The goal of this post is to try to understand this theorem as a special case of Grothendieck-Serre duality.
The norm map in homotopy theory
We’d like to construct the mysterious norm map alluded to in the above two theorems. Let denote the cofiber of the canonical map . It turns out that if acts on , then . In particular, we have the following commuting diagram of cofiber sequences
and taking (strict) -fixed points gives
The map is the desired norm map.
Let’s say that we’re lazy (or ambitious?), and we want to construct the norm map without explicit reference to cofiber sequences, etc.
Perfect stacks are supposed to be generalizations of classifying stacks in nonmodular characteristic. In particular, they are stacks for which exactly the right equilibrium is reached when thinking about finiteness.
Definition: A derived stack with affine diagonal is said to be perfect if compact = perfect = dualizable in .
That’s a sensible definition, but it’s not how perfect stacks were originally defined. Let me state the definition that was originally provided by Ben-Zvi–Francis–Nadler in the form of theorem.
Theorem (Ben-Zvi–Francis–Nadler): A derived stack with an affine diagonal such that is generated under inductive limits (i.e., colimits) by the perfect complexes is a perfect stack.
Examples include the classifying stack over in nonmodular characteristic – this is not true in modular characteristic! (This is one of the reasons that the approach to representation theory via perfect stacks seems to be the ‘‘correct’’ one.) Another example is the moduli stack of elliptic curves away from characteristics and . This isn’t a very well-known result, sadly.
We will attempt to define a norm map that’ll be an equivalence for perfect stacks. It’s possible to do this in super-generality, using ‘‘Beck-Chevalley fibrations’’ (satisfying some technical conditions), but it’s not particularly enlightening. Most of our arguments will be categorical anyway, so if you know what Beck-Chevalley fibrations are, that’s fabulous; you’ll see how to generalize this straightaway.
We’ll need the following theorem of Ben-Zvi–Francis–Nadler.
Theorem: Let be a morphism of perfect stacks (this turns out automatically to be a ‘‘perfect morphism’’, and in fact the result stated here is true more generally for perfect morphisms). Then commutes with small colimits. Moreover, for any , we can construct the pullback (in derived stacks)
and the following canonical transformation is an equivalence
Constructing a norm for perfect stacks
Let denote the -category of perfect stacks and (perfect) morphisms. Any in gives rise to functors and such that is left adjoint to . This can be specified by a biCartesian (i.e., Cartesian and coCartesian) fibration , where is the -category with . Since each is perfect (it is a morphism between perfect stacks) the above theorem gives a right adjoint to . This is also classified by a biCartesian fibration whose Cartesian part agrees with the coCartesian part of . By the technical work of Barwick-Glasman-Nardin, this means that .
Lemma: Let be a perfect morphism. Then
Proof. To see this, suppose is dualizable with dual . Then we have a chain of equivalences
Thus, for dualizable , the projection formula is true. Since is perfect, every object of can be written as the colimit of dualizable objects. It follows that the projection formula holds true in general, so we’re done.
Here’s a cute result relating to this duality phenomenon.
Lemma: Let be a perfect morphism. If is a symmetric monoidal functor, then .
Proof. Suppose is dualizable; then:
The \dYoneda lemma finishes the proof. In general, we can write ; then
since is a left and right adjoint. QED.
We can now construct our norm map. Instead of constructing a map , we will construct a map . If all of these functors exist, and , this is clearly the same as a map . We’ll in fact prove something more – this map is an equivalence for perfect stacks.
Theorem: Let be a morphism of (nice enough) stacks. Then there is a norm map . Moreover, if is a morphism between perfect stacks, then the norm map where .
Proof. It’s hard to argue this directly, so we will show that the subcategory of for which is closed under small colimits and contains , which’ll complete the proof. We need to construct a map . We have a map , given by adjointing over the composite . (It’s not hard to see that it was this map that we proved was an equivalence.) This map gives . Adjointing over the inverse of this, and using the counit gives . This is our desired norm map.
To prove the equivalence for perfect stacks, we can reduce to the case that is affine. Fix . Let be the subcategory of spanned by those for which this map is an equivalence. If is the colimit of , then . This means that the -category is a stable -category closed under retracts. Since is clearly in (because is symmetric monoidal), we’re done.
This is the derived version of what’s known as Grothendieck-Serre duality. Note that if for a finite group , the norm map exactly gives us our classical norm map. (This is a horrible statement to make, because the right adjoint exists iff we are in nonmodular characteristic, which in turn happens iff we know that the norm map is already an equivalence. I’m saying it anyway just for comparison with what we already know.)
Question: What do we get for ? This is probably pretty easy, I haven’t thought about it yet.
A small result on Anderson duality
Using this theory, we can prove a generalization of a theorem of Heard-Stojanoska, stated in Theorem 3.10 of this paper. In order to recall their theorem, we need to define the Anderson dualizing spectrum.
You might know about the Brown-Comenetz dualizing spectrum , defined as follows. Define a functor on spectra by sending to . Since is an injective abelian group, this is representable, and the representing spectrum is denoted . The th homotopy group of this spectrum is the Pontryagin dual of the th homotopy group of the sphere. This spectrum is a standard source of counterexamples, but I’m not interested in that.
The map induces a map of spectra , the fiber of which is denoted . This is called the Anderson dualizing spectrum, and the Anderson dual of a spectrum is . Note that -locally, this is the same as , up to some shift.
Proposition: Let be the structure map of the derived lift of constructed in our first post. Then has a right adjoint , and
To explain our generalization, we need another definition.
Definition: Let be a derived stack. Say that a coherent sheaf on is a dualizing sheaf if the following conditions are satisfied.
- the sheaves vanish for
- the sheaf has finite injective dimension in
- the map is an equivalence
- define a functor on quasicoherent sheaves over by sending . then is an idempotent equivalence.
It turns out that the last condition is a consequence of the first three, but I don’t have a simple (or maybe even correct!) proof. Note that dualizing sheaves don’t have to exist in general.
Theorem: Let be a dualizing sheaf on . There’s an equivalence
Proof. Just work through the adjunctions:
And we’re done!
This recovers the Heard-Stojanoska result since when , the Anderson dualizing spectrum is a dualizing sheaf! The real hard work is computing .
I’ve got a lot more to say on Anderson duals in derived algebraic geometry, and Gross-Hopkins duality, but I’ll leave this for a future post.