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?

algebraically closedfields, and every local noetherian ring $A$ admits a local injection into a discrete valuation ring (with the same fraction field, though bigger residue field by a finitely generated extension). So the formalism in Serre's book is sufficient. $\endgroup$