For an action of a finite group $G$ on a curve $C/S$, is the induced action on $H^0(C,\omega_{C/X}^{\otimes m})$ "independent of the fiber"?

Question

Let $f : C\rightarrow S$ be a proper smooth morphism of relative dimension 1 over a connected scheme $S$.

Let $G$ be a finite group of order invertible on $S$ which acts faithfully and $\mathcal{O}_S$-linearly on $C/S$. Let $\omega_{C/S} = \Omega^1_{C/S}$ denote the dualizing sheaf of $C/S$, then the $G$-action on $C/S$ makes $\omega_{C/S}$ into an $(\mathcal{O}_C,G)$-module, or equivalently it makes $f_*\omega_{C/S}$ into a locally free $\mathcal{O}_S[G]$-module (is it projective in general?).

Proposition 3.2.5 of "Champs de Hurwitz" claims that:

"If $\mathcal{L}$ is an finite locally free $(\mathcal{O}_C,G)$-sheaf, then the Lefschetz trace $L_G(\mathcal{L}_s)$ is constant along geometric fibers ($s\in S$ a geom. point). In particular, the representation $H^0(C_s,(\omega_{C/S}^{\otimes m})_s)$ is independent of $s\in S$"

Here, if $k(s)$ is an algebraically closed field in which the order of $G$ is invertible, and $C_s/k(s)$ a smooth projective curve with an action by $G$, then for a locally free $(\mathcal{O}_{C_s},G)$-module $\mathcal{L}_s$, its "Lefschetz trace" is: $$L_G(\mathcal{L}_s) := \sum_{i=0}^1(-1)^i[H^i(C_s,\mathcal{L}_s)]$$ where the brackets "$[H^i(\cdots)]$" means the element of the representation ring (or Grothendieck group) $R_{k(s)}(G)$ corresponding to the representation of $G$ on $H^i(\cdots)$.

I'm trying to make sense of this proposition. The first question is - what does it mean to say that two representations over different fields are the same?

Ie, in general two geometric points $s_1,s_2\in S$ will have different "residue fields" - so the representations of $G$ on $H^i(C_{s_j},\mathcal{L}_{s_j})$ will be representations over the different residue fields $k(s_1),k(s_2)$.

From the perspective of character theory, we'd want to somehow identify the $n$th roots of unity in $k(s_1)$ and $k(s_2)$. If $H^0(S,\mathcal{O}_S)$ contains a primitive $n$th root of unity $\zeta_n$, then we can "identify" the images of $\zeta_n$ in $k(s_1),k(s_2)$, and since the character of any representation takes values in the additive monoid generated by $\zeta_n$, it would make sense to ask if $[H^i(C_{s_1},\mathcal{L}_{s_1})] = [H^i(C_{s_2},\mathcal{L}_{s_2})]$. However, I don't see why $\zeta_n$ need be a global section on $S$.

Perhaps there is a better way to think about this?

The second question is - Why is the proposition true? The authors do not give a proof or even a reference, but instead provide the cryptic comment: "...the following result (i.e., Prop 3.2.5) is an extension, to the equivariant case, of the classical result which establishes the constancy of the Euler characteristic $\chi(\mathcal{L}_s)$".

Another way to think about it, might be via the "cde" triangle in modular representation theory (c.f. Serre's book "Linear Representations of Finite Groups chapters 14-16, or this thesis). The idea here is to note that for any two points $x,y\in S$, by connectedness, one should be able to find a finite sequence: $$x = x_0 \leftarrow x_1\rightarrow x_2\leftarrow x_3\rightarrow \cdots \leftarrow x_{n-1} \rightarrow x_n = y$$ where the relation $a\rightarrow b$ means $a$ is a generization of $b$, or $b$ is a specialization of $a$. (One can definitely find such a sequence if $S$ is locally noetherian. Does anyone know if it exists for any connected scheme?)

Then, for every relation $a\rightarrow b$ in $S$, one should be able to transport a representation over the residue field $k(a)$ to a representation over $k(b)$ in the following manner: Firstly, we may assume that $a$ is "codimension 1" in $b$. Let $B$ be the closure of $\{b\}\subset S$, which is irreducible and contains $a$. Let $\widehat{O_{a,B}}$ be the complete local ring of $a$ inside $B$, then IF $\widehat{O_{a,B}}$ is a DVR, then $\text{Frac }O_{a,B}$ is the completion of $k(b)$, then by the "cde" triangle, we get maps $$P_{k(a)}(G)\rightarrow R_{k(b)}(G)\rightarrow R_{k(a)}(G)$$ where $P_{k(a)}(G)$ denotes the representation ring generated by projective $k(a)[G]$-modules. Here, the first map is "lift to $\widehat{O_{a,B}}$ and then localize", and the second map is "restrict to a $\widehat{O_{a,B}}$-lattice and then reduce modulo the maximal ideal. The fact that $n := |G|$ is invertible on $S$ implies that both maps are isomorphisms, which are inverse to each other on projective modules, and so we may view the representation $[H^i(C_a,\mathcal{L}_a)]$ as a representation over $k(b)$, and see if it agrees with the representation $[H^i(C_b,\mathcal{L}_b)]$. From this perspective, it seems that Proposition 3.2.5 becomes almost tautological, since both $[H^i(C_a,\mathcal{L}_a)]$ and $[H^i(C_b,\mathcal{L}_b)]$ come from restricting the representation on the free $\widehat{O_{a,B}}$-module $R^if_*\mathcal{L}_{\widehat{O_{a,B}}}$, and the method of transporting the representation from $a$ to $b$ is precisely via this module as an intermediary.

However, I'm still not sure what to do if $\widehat{O_{a,B}}$ is not a DVR. For example, what if $S$ is the affine plane, $b$ is the generic point of a singular planar curve on $S$, and $a$ is a singular point in $\overline{\{b\}}$?

Perhaps there is a better way of thinking about this?

Answer 1

Based on the comments, it is clear that this question is as much about modular representation theory (in the tame case) as about curves and pluricanonical sections.

Let $G$ be a finite group of order $\ell$. Let $\mathbb{Z}[1/\ell]\to R$ be a ring homomorphism. Let $M$ be an $R$-module. Let $G\to \text{Aut}_{R-\text{mod}}(M)$ be a group homomorphism. A pair $(e,f)$ of $G$-equivariant $R$-module homomorphisms $M\to M$ is a pair of $G$-typical idempotents if $e+f=\text{Id}_M$, $e\circ f = f\circ e=0$, and for every $R$-module $I$ (with trivial $G$-action), $$\text{Hom}_R(\text{Coker}(e),\text{Coker}(f)\otimes_R I)^G = \{0\}, $$ $$\text{Hom}_R(\text{Coker}(f),\text{Coker}(e)\otimes_R I)^G = \{0\}.$$ The basic example is the pair $(\epsilon_M,\phi_M)$, $$\epsilon_M(m):= \frac{1}{\ell}\sum_{g\in G}g\bullet m, \ \ \phi_M = \text{Id}_M-\epsilon_M.$$ For every $G$-equivariant $R$-module homomorphism $u:M\to M''$, $\epsilon_{M''}\circ u$ equals $u\circ \epsilon_M$. From this it quickly follows that the functor $M\mapsto \epsilon_M(M)$ is exact.

There is a set-valued covariant functor $S_M$ that associates to every commutative,unital $R$-algebra $A$ the set of pairs of $G$-typical idempotents of the $A$-module $M\otimes_R A$ with its induced $G$-action. When $M$ is a projective $R$-module of finite rank, this functor is representable by a locally closed subscheme of the affine space parameterizing pairs $(e,f)$ of elements of $\text{Hom}_R(M,M)$. The basic observation is that this $R$-scheme is étale.

To see this, let $I\subset R$ be a square-zero ideal. Denote $\overline{R}=R/I$. Denote $\overline{M}=M\otimes_R \overline{R}$. Let $(\overline{e},\overline{f})$ be a $G$-typical pair for the $\overline{R}$-module $\overline{M}$. Denote by $\overline{E}$, resp. $\overline{F}$, the image of $\overline{e}$, resp. the image of $\overline{f}$. These $\overline{R}$-submodules of $\overline{M}$ give a direct sum decomposition that is $G$-equivariant. Denote by $\pi:M\to \overline{M}$ the quotient homomorphism. Then $\pi^{-1}(\overline{E})$ fits into a short exact sequence, $$0 \to I\otimes_\overline{R} \overline{M} \to \pi^{-1}(\overline{E}) \to \overline{E} \to 0.$$ The quotient by the submodule $I\otimes_\overline{R} \overline{F}$ is a projective $R$-module $E$ with an induced $G$-action. Similarly define $F$. Because the $R$-modules are projective of finite rank, the quotient homomorpisms $$\text{Hom}_R(M,E) \to \text{Hom}_{\overline{R}}(\overline{M},\overline{E}), \ \text{Hom}_R(E,M)\to \text{Hom}_{\overline{R}}(\overline{E},\overline{M}),$$ are both surjective, and similarly for $F$. For an arbitrary lift of a $G$-equivariant $\overline{R}$-module homomorphism, after applying $\epsilon$, the lift is also $G$-equivariant. Thus, the $G$-equivariant direct sum decomposition, $$\overline{M}\twoheadrightarrow \overline{E}\hookrightarrow \overline{M}, \ \ \overline{M}\twoheadrightarrow \overline{F}\hookrightarrow \overline{M},$$ lifts to a $G$-equivariant direct sum decomposition of $M$, $$M\to E\oplus F.$$ Moreover, by hypothesis, $$\text{Hom}_\overline{R}(\overline{E},I\otimes_\overline{R}\overline{F})^G = \{0\}, $$ $$\text{Hom}_\overline{R}(\overline{F},I\otimes_\overline{R}\overline{E})^G = \{0\}. $$ Thus, this $G$-invariant direct sum decomposition is unique. Therefore the functor $S_M$ is étale.

The functor is also proper. Since the functor is representable by a finite type, locally closed subscheme of an affine space, it suffices to verify the valuative criterion of properness. Thus, assume that $R$ is a discrete valuation ring with maximal ideal $\mathfrak{m}$, residue field $k=R/\mathfrak{m}$ and fraction field $K$. For a direct sum decomposition $(E_K,F_K)$ of $M\otimes_R K$, define $E$ to be the kernel of the composition $$M \hookrightarrow M\otimes_R K \twoheadrightarrow (M\otimes_R K)/E_K.$$ In particular, $E$ is saturated. So $E$ is a free $R$-module and $F=M/E$ is also a free $R$-module. By construction, these are $G$-equivariant. By the same argument as above using $\epsilon$, every $G$-equivariant $k$-linear map from $E\otimes_R k$ to $F\otimes_R k$ or vice versa lifts to a $G$-equivariant $R$-module homomorphism from $E$ to $F$. This induces a $G$-equivariant $K$-linear map between $E_K$ and $F_K$. By hypothesis, every such map is the zero map. Thus, the same is true for $E\otimes_R k$ and $F\otimes_R k$. In particular, there is a unique $G$-equivariant, $k$-linear splitting of $M\otimes_R k \twoheadrightarrow F\otimes_R k$. This lifts to a $G$-equivariant $R$-linear splitting of $M\twoheadrightarrow F$. Thus, the generic splitting extends to a $G$-equivariant $R$-module splitting of $M$. So $S_M$ is proper over $R$. Therefore $S_M$ is finite over $R$.

Now apply all of this to the case that $R=\mathbb{Z}[1/\ell]$ and $M=R[G]$, the group algebra of $G$. This defines a finite, étale extension of $\mathbb{Z}[1/\ell]$, i.e., a product of finitely many irreducible domains each of which is a finite étale extension of $\mathbb{Z}[1/\ell]$. Define $K_G$ to be the Galois closure of the compositum of these finitely many finite extensions of $\mathbb{Q}$, and define $\mathfrak{o}_G$ to be the integral closure of $\mathbb{Z}[1/\ell]$ in $K_G$. (In fact, $\mathfrak{o}_G$ already equals the irreducible factor of maximal $R$-rank.) Then we can define the $G$-isotypic decomposition of $\mathfrak{o}_G[G]$, and this will be compatible with arbitrary base change.

Now all of the usual theory of irreducible representations, characters, orthogonality, etc., goes through for algebras over $\mathfrak{o}_G$. Now you can apply Lemma 4.2(3) from my article with Olsson (there are other references) to see that for every irreducible representation $V$ of $G$ over $\mathfrak{o}_G$, for the locally free sheaf $\omega_{C/S}^{\otimes n}\otimes_{\mathfrak{o}_G}V$ on the stack $[C/G]$ has locally free pushforward to the coarse moduli space $C//G$.