Generalizations of global Euler characteristic formula

Question

Let $ K $ be a number field, $ S $ a finite set of primes of $K $ including the archimedean primes and $ G_{K,S} $ be the Galois group of the maximal extension of $K$ unramified outside $ S $. Assume that $ M $ is a finite $ G_{K,S} $-module such that

1. $ S $ contains all the primes that divide the order of $ M $.

Then the global Euler characteristic formula says that $$ \chi(G_{K,S},M):=\dfrac{|H^{0}(G_{K,S},M)|\cdot |H^{2}(G_{K,S},M)|}{|H^{1}(G_{K,S},M)|}=\dfrac{1}{|M|^{d}}\prod_{v\in S_{\infty}}|H^{0}(G_{v},M)| $$ where the product is over all the archimedean primes, |-| denotes cardinality, $ d:=[K:\mathbb{Q}] $ and $ G_{v}:=\text{Gal}(\overline{K}_{v}/K_{v}) $ is the absolute Galois group of the completion $ K_{v} $ of $ K $ at $ v $.

My question is the following:

If we drop conditions 1 (i.e. $S$ does not contain all the primes that divide the order of $ M $) and assume that $ H^{i}(G_{K,S},M) $ is finite for $ i=0,1,2 $, then how to calculate $\chi(G_{K,S},M)$?

For example, we may ask:

• does the formula still hold?
• If not, is it always true that $$ \dfrac{|H^{0}(G_{K,S},M)|\cdot |H^{2}(G_{K,S},M)|}{|H^{1}(G_{K,S},M)|}\leq \dfrac{1}{|M|^{d}}\prod_{v\in S_{\infty}}|H^{0}(G_{v},M)|? $$
• Is it possible to generalize global Euler characteristic formula to this situation although it may not be in the above form? What results are known in its direction?

Any comments and reference would be highly appreciated.

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Update：

In the Proposition 9.4 of Presentations of Galois groups of maximal extensions with restricted ramification by Yuan Liu, if I understand correctly, it is proved that, for any finite $ \mathbb{Z}/\ell\mathbb{Z}[G_{K,S}] $-module $ M $ where $\ell$ a prime and $ S_{\ell}\cup S_{\infty}\nsubseteq S $, $$\log_{\ell}(\chi(G_{K,S},M))\leq \log_{\ell}(\chi(G_{K,T},M))+\dim_{\mathbb{F}_{\ell}}(M')^{G_{K,T}}+\epsilon(M)$$ where

• $ T=S\cup S_{\ell}\cup S_{\infty} $, here we denote by $ S_{\ell} $ the set of primes of $ K $ above $ \ell $ and $ S_{\infty} $ the set of primes at infinity.
• $ G_{K,T} $ be the Galois group of the maximal extension of $K$ unramified outside $ T $.
• $ M':=\text{Hom}(M,\overline{\mathbb{Q}}^{\times}) $.
• $ \epsilon(M):=-\sum_{v\in S_{\ell}\backslash S}\log_{\ell} \lVert |M|\rVert_v $, where $ \lVert x\rVert_v:=\ell^{-\text{ord}_{v}(x)} $ where $ \text{ord}_{v} $ is the additive valuation of $ K_{v} $.

For more details, please refer to the paper.

Answer 1

I'd say the right thing to do is to not look at Galois cohomology but étale cohomology instead, by geometrizing the situation. Let me explain. Let $U=\mathrm{Spec}(\mathcal{O}_{K,S})$ and denote $\eta=\mathrm{Spec}(K)$ its generic point. We have $\pi_1(U,\bar{\eta})=G_{K,S}$ and $M$ corresponds to a constructible (locally constant) sheaf on $U$. Using [1, Prop 2.9] we have $H^i(U,F)(l)=H^i(G_{K,S},M)(l)$ for all $l$ invertible on $U$.

The Galois cohomology groups are actually kind of bad to define an Euler characteristic as they are non-zero in arbitrary high degree and one has to artificially cut off. The problem is the same for étale cohomology of a constructible sheaf, but one can introduce a compactly supported cohomology (taking into account behavior at archimedean places) $H^\ast_c(U,F)$ for sheaves on $U$ that behaves better. The groups are the cohomology groups of the complex $R\Gamma_c(U,F)$ defined by the fiber sequence $$ R\Gamma_c(U,F) \to R\Gamma(U,F) \to \prod_{v\in S} R\Gamma(K_v,F_v) $$ where $K_v$ is the henselization at $v$ of $K$ if $v$ is non-archimedean, the completion if $v$ is archimedean, $R\Gamma(K_v,-)$ is Galois cohomology and $F_v$ is the $G_{K_v}$-module corresponding to the pullback of $F$ to $\mathrm{Spec}(K_v)$. This is a variant of the compactly supported cohomology of [1] that is actually bounded for $\mathbb{Z}$-constructible sheaves:

Proposition: If $F$ is a $\mathbb{Z}$-constructible sheaf on $U$, the groups $H^i_c(U,F)$ are $0$ for $i\neq 0,1,2,3$ and finite type for $i=0,1,2,3$. If $F$ is constructible then they are finite.

A good alternative to the global Euler characteristic of Tate is then $$ \chi_{c,U}(F)=\prod_i [H^i_c(U,F)]^{(-1)^i} $$ where $[-]$ denotes cardinality. It turns out that $\chi_{c,U}$ extends to an Euler characteristic $\chi_{W,U}$, the Weil-étale Euler characteristic, valued in $\mathbb{Q}_{>0}$ and defined for all $\mathbb{Z}$-constructible sheaves on $U$.

Let me invoke a powerful theorem of Swan:

Theorem [2, Cor. 1]: Let $G$ be a finite group. Then the kernel of $K_0(\mathbb{Z}[G]) \to K_0(\mathbb{Q}[G])$ is finite.

Now from a formal dévissage argument using explicit computations for sheaves supported on closed points and the theorem, it follows that $\chi_{W,U}(F)=\chi_{c,U}(F)=1$ for a constructible sheaf $F$. In my view, this is the right generalization. Note that if $F$ is constructible locally constant, corresponding to a $G_{K,S}$-module $M$ such that $l$ is invertible on $U$ for each $l$ dividing the order of $M$, then $H^i(U,F)=H^i(G_{K,S},M)$ and the statement $\chi_{c,U}(F)=1$ is equivalent, after unwinding the definitions, to the formula for the global Euler characteristic (thus by a dévissage one can also prove that $\chi_{c,U}(F)=1$ for $F$ constructible by reducing to the above case and the case of a sheaf supported on a closed point).

Since the groups $H^i_c(U,F)$ are finite, the groups $H^i(U,F)$ are also finite because Galois cohomology of $\mathbb{R}$ and of p-adic fields/henselian local fields of number fields with finite coefficients is finite. We can thus define for $F$ constructible: $$ \chi_U(F)=\prod_{i=0}^3 [H^i(U,F)]^{(-1)^i} $$ and the equality $\chi_c(U,F)=1$ becomes: $$ \chi_U(F)=\prod_{v\in S} \prod_{i=0}^3[H^i(K_v,F_v)]^{(-1)^i}=\prod_{v~\text{finite}} \chi_v(F_v) \prod_{v~\text{archimedean}} \frac{[H^0(K_v,F_v)]}{[\hat{H}^0(K_v,F_v)]} $$ where $\hat{H}^i(K_v,-)$ for $v$ archimedean denotes Tate cohomology and $\chi_v$ is the local Euler characteristic (I have used that the Herbrand quotient of a finite module is $1$). Using the computation of the local Euler characteristic [1, I.2.8] and the product formula, we find $$ \chi_U(F)= \prod_{v~\text{archimedean}} \frac{[H^0(K_v,F_v)]}{[\hat{H}^0(K_v,F_v)]|[M]|_v} $$ where $M$ is the $G_K$-module corresponding to the stalk $F_\eta$ at the generic point and $|-|_v$ is the normalized norm associated to the archimedean place $v$. In the case where $mF=0$ for $m$ invertible on $U$ this is [1, II.2.13].

When the invertibility condition for a locally constant constructible sheaf $F$ does not hold, the relation between $H^\ast(U,F)$ and $H^\ast(G_{K,S},M)$ is more complicated (and I don't even know if the latter are finite!): let $\bar{U}$ denote the normalization of $U$ in the fixed field of $G_{K,S}$. Then you have a spectral sequence $$ H^p(G_{K,S},H^q(\bar{U},F))\Rightarrow H^{p+q}(U,F) $$ One can say that $H^0(\bar{U},F)=M$, $H^1(\bar{U},F)=0$ and $H^i(\bar{U},F)(l)=0$ for $i\geq 0$ if $l$ is invertible on $U$ (see [1, the proof of I.2.9]); beyond that I don't know what can be said.

Side note: a lot of what I said is studied in my article Special values of L -functions on regular arithmetic schemes of dimension 1. I apologize for the self-plug but I feel like it was kind of inevitable

[1] Milne, J. S., Arithmetic duality theorems, Charleston, SC: BookSurge, LLC (ISBN 1-4196-4274-X/pbk). viii, 339 p. (2006). ZBL1127.14001.

[2]Swan, R. G., The Grothendieck ring of a finite group, Topology 2, 85-110 (1963). ZBL0119.02905.