# Norm maps

$\newcommand{\GL}{\mathrm{GL}}$ Let $G$ be a finite group, and let $k$ be a field of characteristic $p$ (posssibly $0$). If $V$ is a $k$-vector space, recall that a representation of $G$ is a map $G\to \GL(V)$.

Theorem (Maschke): Representation theory when $p$ does not divide $\# G$ (the nonmodular characteristic case) is good, i.e., the category $\mathrm{Rep}(G)$ is semisimple.

When $p|\# G$, things become more interesting, but I won’t get into this. One important difference comes from the norm map.

$\newcommand{\Nm}{\mathrm{Nm}}$ Define a map $V\to V$ by $v\mapsto \sum_{g\in G} gv$. Clearly this map kills anything of the form $gv-v$, and is invariant under the action of $G$. It follows that this map factors as $V\to V_G \xrightarrow{\Nm} V^G \to V$. This map $\Nm$ is called the norm map.

Proposition: In nonmodular characteristic, the norm map is an equivalence; the inverse is induced by the map $v\mapsto v/\# G$.

In chromatic homotopy theory, we have a whole collection of spectra which interpolate between characteristic zero and characteristic $p$, namely the Morava $K$-theories. One might ask how representation theory over a Morava $K$-theory behaves. Here, we have the following preliminary result of Hovey-Sadofsky:

Theorem (Greenlees-Sadofsky): Suppose $G$ acts trivially on $K(n)$. Then the norm map $K(n)_{hG} \to K(n)^{hG}$ is an equivalence.

A consequence of this theorem is the following.

Theorem (Hovey-Sadofsky): The $K(n)$-localization of the cofiber of the norm map on $L_{K(n)}S$ is trivial, i.e., $L_{K(n)}\Sigma^\infty_+ BG$ is self-dual in the category of $K(n)$-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 $G$ be a finite group acting on any $K(n)$-local spectrum $M$. Then the norm map $M_{hG} \to M^{hG}$ 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

$\newcommand{\dX}{\mathfrak{X}}$ We’d like to construct the mysterious norm map alluded to in the above two theorems. Let $\widetilde{EG}$ denote the cofiber of the canonical map $\Sigma^\infty_+ EG \to S$. It turns out that if $G$ acts on $X$, then $\Sigma^\infty_+ EG\wedge X \simeq \Sigma^\infty_+ EG\wedge F(\Sigma^\infty_+ EG, X)$. In particular, we have the following commuting diagram of cofiber sequences

and taking (strict) $G$-fixed points gives

The map $\Nm:X_{hG}\to X^{hG}$ is the desired norm map.

$\newcommand{\Sp}{\mathrm{Sp}}$ 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

$\newcommand{\QCoh}{\mathrm{QCoh}}$ 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 $\dX$ with affine diagonal is said to be perfect if compact = perfect = dualizable in $\QCoh(\dX)$.

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 $\dX$ with an affine diagonal such that $\QCoh(\dX)$ is generated under inductive limits (i.e., colimits) by the perfect complexes is a perfect stack.

Examples include the classifying stack $BG$ over $\text{Spec }k$ 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 $\mathcal{M}_\textbf{ell}[1/6]$ of elliptic curves away from characteristics $2$ and $3$. This isn’t a very well-known result, sadly.

$\newcommand{\dY}{\mathfrak{Y}}$ 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 $f:\dX\to \dY$ 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 $f_\ast$ commutes with small colimits. Moreover, for any $g:\dY^\prime\to \dY$, we can construct the pullback (in derived stacks)

and the following canonical transformation is an equivalence

# Constructing a norm for perfect stacks

$\newcommand{\cf}{\mathcal{F}}$ Let $S$ denote the $\infty$-category of perfect stacks and (perfect) morphisms. Any $f:\dX\to \dY$ in $S$ gives rise to functors $f^\ast:\QCoh(\dY)\to \QCoh(\dX)$ and $f_\ast:\QCoh(\dX)\to \QCoh(\dY)$ such that $f^\ast$ is left adjoint to $f_\ast$. This can be specified by a biCartesian (i.e., Cartesian and coCartesian) fibration $p:T\to S^{op}$, where $T$ is the $\infty$-category with $T\times_S\{\dX\}\simeq \QCoh(\dX)$. Since each $f$ is perfect (it is a morphism between perfect stacks) the above theorem gives a right adjoint $f^!:\QCoh(\dX)\to\QCoh(\dY)$ to $f_\ast$. This is also classified by a biCartesian fibration $q: P \to S$ whose Cartesian part agrees with the coCartesian part of $q:T\to S^{op}$. By the technical work of Barwick-Glasman-Nardin, this means that $q = p^\vee:T^\vee\to S$.

$\newcommand{\cG}{\mathcal{G}}$ Lemma: Let $f:\dX\to \dY$ be a perfect morphism. Then

Proof. To see this, suppose $\cG$ is dualizable with dual $\cG^\vee$. Then we have a chain of equivalences

Thus, for dualizable $\cG$, the projection formula is true. Since $\dY$ is perfect, every object of $\QCoh(\dY)$ 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.

$\newcommand{\Hom}{\mathrm{Hom}}$ Lemma: Let $f:\dX\to \dY$ be a perfect morphism. If $f_\ast$ is a symmetric monoidal functor, then $f^\ast = f^!$.

$\newcommand{\colim}{\text{colim}}$ Proof. Suppose $\cf$ is dualizable; then:

The \dYoneda lemma finishes the proof. In general, we can write $\cf = \colim_\alpha\cf_\alpha$; then

since $f_\ast$ is a left and right adjoint. QED.

$\newcommand{\co}{\mathcal{O}}$ We can now construct our norm map. Instead of constructing a map $f_! \to f_*$, we will construct a map $f^*\to f^!$. If all of these functors exist, and $f^\ast \dashv f_\ast \dashv f^! \dashv f_!$, this is clearly the same as a map $f_! \to f_*$. We’ll in fact prove something more – this map is an equivalence for perfect stacks.

Theorem: Let $f:\dX\to \dY$ be a morphism of (nice enough) stacks. Then there is a norm map $f^\ast \cf \otimes f^! \cG \to f^! (\cf\otimes \cG)$. Moreover, if $f:\dX\to \dY$ is a morphism between perfect stacks, then the norm map $f^!\cf\otimes f^\ast\cG\xrightarrow{\simeq} f^\ast(\cf\otimes \cG)$ where $\cf,\cG\in\QCoh(\dY)$.

Proof. It’s hard to argue this directly, so we will show that the subcategory of $\QCoh(\dY)$ for which $f^!(\cf\otimes\cG)\simeq f^!\cf\otimes f^\ast\cG$ is closed under small colimits and contains $\co_\dY$, which’ll complete the proof. We need to construct a map $f^!\cf\otimes f^\ast\cG\to f^!(\cf\otimes\cG)$. We have a map $f_\ast(\cf\otimes f^\ast\cG)\to f_\ast\cf\otimes \cG$, given by adjointing over the composite $f^\ast(f_\ast \cf\otimes \cG) \simeq f^\ast f_\ast \cf\otimes f^\ast \cG \to \cf\otimes f^\ast \cG$. (It’s not hard to see that it was this map that we proved was an equivalence.) This map gives $f_\ast f^!\cf\otimes \cG\to f_\ast(f^!\cf\otimes f^\ast\cG)$. Adjointing over the inverse of this, and using the counit $f_\ast f^!\to 1$ gives $\delta_{\cf,\cG}:f^!\cf\otimes f^\ast\cG\to f^!(f_\ast f^!\cf\otimes \cG)\to f^!(\cf\otimes\cG)$. This is our desired norm map.

$\newcommand{\cc}{\mathcal{C}}$ To prove the equivalence for perfect stacks, we can reduce to the case that $\dY$ is affine. Fix $\cf$. Let $\cc$ be the subcategory of $\QCoh(\dY)$ spanned by those $\cG$ for which this map is an equivalence. If $\cG$ is the colimit of $\{\cG_i\}$, then $\delta_{\cf,\cG} = \colim\delta_{\cf,\cG_i}$. This means that the $\infty$-category $\cc$ is a stable $\infty$-category closed under retracts. Since $\co_\dY$ is clearly in $\cc$ (because $f^\ast$ is symmetric monoidal), we’re done.

This is the derived version of what’s known as Grothendieck-Serre duality. Note that if $\dX = BG$ for a finite group $G$, the norm map exactly gives us our classical norm map. (This is a horrible statement to make, because the right adjoint $f^!$ 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 $\mathcal{M}_\textbf{ell}[1/6]$? This is probably pretty easy, I haven’t thought about it yet.

# A small result on Anderson duality

$\newcommand{\Z}{\mathbf{Z}}$ 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.

$\newcommand{\QQ}{\mathbf{Q}}$ You might know about the Brown-Comenetz dualizing spectrum $I_{\QQ/\Z}$, defined as follows. Define a functor on spectra by sending $M$ to $\Hom(\pi_{-\ast} M,\QQ/\Z)$. Since $\QQ/\Z$ is an injective abelian group, this is representable, and the representing spectrum is denoted $I_{\QQ/\Z}$. The $(-k)$th homotopy group of this spectrum is the Pontryagin dual of the $k$th homotopy group of the sphere. This spectrum is a standard source of counterexamples, but I’m not interested in that.

The map $\QQ\to \QQ/\Z$ induces a map of spectra $H\QQ\to I_{\QQ/\Z}$, the fiber of which is denoted $I_\Z$. This is called the Anderson dualizing spectrum, and the Anderson dual of a spectrum $M$ is $F(M,I_\Z)$. Note that $K(n)$-locally, this is the same as $I_{\QQ/\Z}$, up to some shift.

Proposition: Let $f:(BC_2,\co^\mathbf{top}) \to \text{Spec }S$ be the structure map of the derived lift of $BC_2$ constructed in our first post. Then $g_\ast$ has a right adjoint $g^!$, and

To explain our generalization, we need another definition.

$\newcommand{\DD}{\mathbf{D}}$ Definition: Let $\dX$ be a derived stack. Say that a coherent sheaf $I_\dX$ on $\dX$ is a dualizing sheaf if the following conditions are satisfied.

• the sheaves $\pi_k I_\dX$ vanish for $k\gg 0$
• the sheaf $I_\dX$ has finite injective dimension in $\QCoh(\dX)$
• the map $\co^\mathbf{top}\to \Hom(I_\dX,I_\dX)$ is an equivalence
• define a functor $\DD$ on quasicoherent sheaves over $\dX$ by sending $\cf\mapsto \Hom(\cf,I_\dX)$. then $\DD$ 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 $I_\dY$ be a dualizing sheaf on $\dY$. There’s an equivalence

Proof. Just work through the adjunctions:

And we’re done!

This recovers the Heard-Stojanoska result since when $\dX = \text{Spec }S$, the Anderson dualizing spectrum is a dualizing sheaf! The real hard work is computing $f^! I_\dY$.

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.