# Computing the cohomology of constructible étale sheaves on curves

Christophe Levrat

# Computing the cohomology of constructible étale sheaves on curves

Christophe Levrat

###### Abstract

We present an explicit expression of the cohomology complex of a constructible
sheaf of abelian groups on the small étale site of an irreducible curve over
an algebraically closed field, when the torsion of the sheaf is invertible in
the field. This expression only involves finite groups, and is functorial in
both the curve and the sheaf. In particular, we explain how to compute the
Galois action on this complex. We also present an algorithm which computes it
and study its complexity.

###### Contents

  1. 1 Introduction
  2. 2 The cover trivialising $\mathbb{Z}/n\mathbb{Z}$-torsors
  3. 3 Schemes of cohomological dimension 1
  4. 4 Explicit computation of $\operatorname{R\Gamma}$ of a (possibly singular) curve
  5. 5 Algorithmic aspects
  6. 6 First example: sheaves on subschemes of $\mathbb{P}^{1}$
  7. 7 Second example: sheaves on subschemes of an elliptic curve

##  1 Introduction

Let $X_{0}$ be an algebraic curve over a field $k$. Let $n$ be a positive
integer invertible in $k_{0}$, and $\mathcal{F}_{0}$ be a constructible sheaf
of $\mathbb{Z}/n\mathbb{Z}$-modules on the (small) étale site of $X_{0}$.
Denote by $k$ a separable closure of $X_{0}$, by $X$ the base change of
$X_{0}$ to $k$, and by $\mathcal{F}$ the restriction of $\mathcal{F}_{0}$ to
$X$. The étale cohomology complex $\operatorname{R\Gamma}(X,\mathcal{F})$ is
equipped with an action of $\operatorname{Gal}(k|k_{0})$. Given the curve $X$
and a suitable explicit description of the sheaf $\mathcal{F}$, we are
interested in computing a finite extension $k_{1}$ of $k_{0}$ and a complex of
$(\mathbb{Z}/n\mathbb{Z})[\operatorname{Gal}(k_{1}|k_{0})]$-modules which
represents $\operatorname{R\Gamma}(X,\mathcal{F})$.

The computability of étale cohomology groups of torsion sheaves on schemes of
finite type over algebraically closed fields was proved in 2014 by Poonen,
Testa and van Luijk [PTv15, Th. 7.9] in characteristic zero, and by Madore and
Orgogozo in arbitrary characteristic [MO15, Th. 0.1]. However, the algorithms
described in these articles are not efficient enough to be used in practice,
and the only known result about their complexity is that Madore and Orgogozo’s
algorithm is primitive recursive [Lev22, Prop. 4.1.9]. In the case of smooth
curves, some more efficient algorithms are known. When $X$ is a smooth
projective curve, $\operatorname{H}^{1}(X,\mu_{n})$ is canonically isomorphic
to the $n$-torsion of $\operatorname{Pic}(X)$; two algorithms, one developed
by Huang and Ierardi [HI98], the other by Couveignes [Cou09], compute this
group when the base field $k_{0}$ is finite. Jin’s algorithm [Jin20] computes
$\operatorname{H}^{1}(X,\mathcal{F})$ where $X$ is a smooth curve and
$\mathcal{F}$ is locally constant.

In this paper, we consider the case where $X_{0}$ is an integral curve over a
field, and $\mathcal{F}_{0}$ is a constructible sheaf (or even a complex of
such sheaves) on $X_{0}$. We present an explicit expression of the cohomology
complex $\operatorname{R\Gamma}(X,\mathcal{F})$ with the aforementioned Galois
action, as well as an algorithm which computes this complex (under some
classical computability assumptions on the field $k_{0}$). The latter makes
use of an existing algorithm computing $\operatorname{H}^{1}(X,\mu_{n})$, such
as those mentioned above. We also provide complexity bounds for this method.
In particular, in the case of locally constant sheaves on smooth projective
curves over finite fields, the complexity of computing
$\operatorname{H}^{1}(X,\mathcal{F})$ using this algorithm is lower than
Jin’s. In the case where the base field $k_{0}$ is finite, we present an idea
that should a allow to reduce the complexity of this algorithm, and explain
why this would be a crucial step towards developing a polynomial-time point
counting algorithm for surfaces.

In section 2, we investigate the properties of the minimal Galois cover of a
scheme trivialising the $\mathbb{Z}/n\mathbb{Z}$-torsors on this scheme, as
well as its construction in the case of curves. In section 3, we explain how
to compute the cohomology of a locally constant sheaf on a scheme of
cohomological dimension at most $1$. Section 4 contains the proof of the main
theorem: an explicit expression of $\operatorname{R\Gamma}(X,-)$ when $X$ is a
curve over a field of cohomological dimension at most $1$. We then present in
section 5 the algorithms used to compute the cohomology of a constructible
sheaf on such a curve, as well as their complexity. We also describe the
potential application of our algorithms to point counting on surfaces over
finite fields. In sections 6 and 7, we illustrate these algorithms in two
situations.

######  Terminology.

Unless explicitly stated otherwise, the word cover denotes a finite étale map.
A Galois cover is always supposed to be connected.

######  Acknowledgements.

The author would like to thank David Madore and Fabrice Orgogozo for their
continued help and support during his PhD thesis, whose main results led to
the ones presented in this article.

##  2 The cover trivialising $\mathbb{Z}/n\mathbb{Z}$-torsors

###  2.1 General construction and properties

Let $n$ be a positive integer. We will denote by $\Lambda$ the ring
$\mathbb{Z}/n\mathbb{Z}$. For a (discrete) $\Lambda$-module $M$, we will
denote by $M^{\vee}$ its $\Lambda$-dual.

######  Lemma 2.1.

Let $G$ be a locally compact topological group. Consider the abelian group
$\Lambda$, with the trivial action of $G$. Suppose
$\operatorname{H}^{1}(G,\Lambda)$ is finite. There is a unique closed normal
subgroup $S$ of $G$ such that $G/S$ is isomorphic to the $\Lambda$-dual
$\operatorname{H}^{1}(G,\Lambda)^{\vee}$ of the continuous cohomology group
$\operatorname{H}^{1}(G,\Lambda)$. Moreover, $S$ is a characteristic subgroup
of $G$.

###### Proof.

Define $S$ as the closure in $G$ of $G^{n}[G,G]$. This group $S$ is a
characteristic subgroup of $G$ because $G^{n}$ and $[G,G]$ are stable by
automorphisms. Since $\Lambda$ is an $n$-torsion abelian group, there is a
canonical isomorphism

$\operatorname{Hom}_{\mathrm{cont}}(G/S,\Lambda)\xrightarrow{\sim}\operatorname{Hom}_{\mathrm{cont}}(G,\Lambda)=\operatorname{H}^{1}(G,\Lambda).$

Pontryagin duality [HM20, Th. 7.63] applied to the locally compact abelian
group $G/S$ yields the following isomorphism:

$\operatorname{Hom}_{\mathrm{cont}}(G/S,\Lambda)^{\vee}\xrightarrow{\sim}G/S.$

Now let $H$ be a closed normal subgroup of $G$ such that $G/H$ is isomorphic
to $\operatorname{H}^{1}(X,\Lambda)^{\vee}$. It necessarily contains $S$ since
$G/H$ is a $\Lambda$-module. Since $G/S$ and $G/H$ have the same finite
cardinality, $H=S$. ∎

######  Corollary 2.2.

Let $X$ be an integral noetherian scheme such that
$\operatorname{H}^{1}(X,\Lambda)$ is finite. Up to isomorphism, there is a
unique étale Galois cover ${X^{\langle n\rangle}}$ of $X$ with automorphism
group isomorphic to $\operatorname{H}^{1}(X,\Lambda)^{\vee}$.

###### Proof.

This follows immediately from 2.1, the canonical isomorphism

$\operatorname{H}^{1}(\pi_{1}(X),\Lambda)\xrightarrow{\sim}\operatorname{H}^{1}(X,\Lambda)$

and Grothendieck-Galois theory. ∎

From now on, given such a scheme $X$, we will always denote by ${X^{\langle
n\rangle}}$ a Galois cover of $X$ as in the previous corollary.

######  Proposition 2.3.

Let $X$ be an integral noetherian scheme such that
$\operatorname{H}^{1}(X,\Lambda)$ is finite. For any finitely generated
$\Lambda$-module $M$, the morphism
$\operatorname{H}^{1}(X,M)\to\operatorname{H}^{1}({X^{\langle n\rangle}},M)$
is trivial.

###### Proof.

Recall that ${X^{\langle n\rangle}}$ corresponds to the open subgroup $S$ of
$\pi_{1}(X)$, which is the closure of the subgroup
$\pi_{1}(X)^{n}[\pi_{1}(X),\pi_{1}(X)]$. Since $M$ is $n$-torsion, any map
$\pi_{1}(X)\to M$ is trivial on $S$, hence
$\operatorname{Hom}(\pi_{1}(X),M)\to\operatorname{Hom}(\pi_{1}({X^{\langle
n\rangle}}),M)$ is trivial. ∎

###  2.2 Explicit construction in the case of curves

Let $U$ be a smooth integral curve over a field $k$. Denote by $K$ its
function field. Let $n$ be a positive integer invertible in $k$. Recall the
following description of $\operatorname{H}^{1}(U,\mu_{n})$ in terms of
divisors on $U$.

######  Lemma 2.4.

The group $\operatorname{H}^{1}(U,\mu_{n})$ is canonically isomorphic to the
quotient of

$\\{(D,f)\in\operatorname{Div}(U)\times K^{\times}\mid nD=\mathrm{div}(f)\\}$

by the subgroup of pairs $(D,f)$ with $f\in(K^{\times})^{n}$.

###### Proof.

This follows immediately from the corresponding description in terms of
invertible sheaves given in [Stacks, 040Q]. ∎

We will denote by $[D,f]$ the class of the pair $(D,f)$ in
$\operatorname{H}^{1}(U,\mu_{n})$. Now suppose that $k$ is separably closed.
Let $X$ be the smooth compactification of $U$. Denote by $g$ the genus of $X$.
Consider the closed complement $Z=\\{P_{1},\dots,P_{r}\\}$ of $U$ in $X$. When
$r=0$ the lemma above is the usual description of
$\operatorname{H}^{1}(X,\mu_{n})$ as the group of $n$-torsion points on the
Jacobian of $X$. When $r\geqslant 1$, the Gysin sequence

$0\to\operatorname{H}^{1}(X,\mu_{n})\to\operatorname{H}^{1}(U,\mu_{n})\to\operatorname{H}^{0}(Z,\Lambda)\to\Lambda\to
0$

shows that $\operatorname{H}^{1}(U,\mu_{n})$ is a free $\Lambda$-module of
rank $2g-1+r$. Consider a basis $([D_{i},g_{i}])_{1\leqslant i\leqslant s}$ of
$\operatorname{H}^{1}(U,\mu_{n})$. For every integer $j\in\\{1\dots s\\}$, we
denote by $V_{j}$ the normalisation of $U$ in
$K(\sqrt[n]{g_{1}},\dots,\sqrt[n]{g_{j}})$, by $\phi_{j}$ the induced cover
$U_{j}\to U$, and set $V=V_{s}$.

######  Proposition 2.5.

The cover $\phi\colon V\to U$ is isomorphic to ${U^{\langle n\rangle}}\to U$.

###### Proof.

First of all, each cover $V_{i}\to V_{i-1}$ is étale, because it is
constructed by taking an $n$-th root of a function whose valuation at each
point is a multiple of $n$. Let us check by induction on $j\in\\{1,\dots,r\\}$
that $V_{j}\to V_{j-1}$ is Galois with group $\mu_{n}(k)$. This is obviously
true for $j=1$. For any $i\in\\{1,\dots,j-1\\}$ the extension
$V_{i}=V_{i-1}(\sqrt[n]{g_{i}})$ of $V_{i-1}$ is Galois with group
$\mu_{n}(k)$ by induction hypothesis. The Hochschild-Serre spectral sequence
yields an exact sequence

$0\to\operatorname{H}^{1}(\mu_{n}(k),\Lambda)\to\operatorname{H}^{1}(V_{i-1},\Lambda)\to\operatorname{H}^{1}(V_{i},\Lambda).$

Therefore, the kernel of
$\phi_{i}^{\star}\colon\operatorname{H}^{1}(U,\Lambda)\to\operatorname{H}^{1}(V_{i},\Lambda)$
is the direct sum
$\Lambda[D_{1},g_{1}]\oplus\dots\oplus\Lambda[D_{i},g_{i}]\simeq\Lambda^{i}$,
and $[D_{j},g_{j}]$ is not in the kernel; the order of
$\phi_{i}^{\star}[D_{j},g_{j}]$ in $\operatorname{H}^{1}(V_{i},\mu_{n})$ is
still $n$. Thus, $V\to U$ is finite étale of order $n^{r}$. The field $k$
being separably closed, the extension $k(V)$ of $k(U)$ is the splitting field
of the polynomials $T^{n}-g_{1},\dots,T^{n}-g_{r}$, so it is Galois. The
morphism $V\to U$ is therefore an étale Galois covering. An element of the
group $\operatorname{Aut}(V|U)$ is an automorphism defined by
$(\sqrt[n]{g_{1}}\mapsto\zeta_{1}\sqrt[n]{g_{1}},\dots,\sqrt[n]{g_{r}}\mapsto\zeta_{r}\sqrt[n]{g_{r}})$,
where the $\zeta_{i}$ are $n$-th roots of unity in $k$; the group
$\operatorname{Aut}(V|U)$ is therefore canonically isomorphic to
$\operatorname{Hom}_{\Lambda}(\operatorname{H}^{1}(U,\mu_{n}),\mu_{n})=\operatorname{H}^{1}(U,\Lambda)^{\vee}$.
2.1 now ensures that $V$ is isomorphic to ${U^{\langle n\rangle}}$. ∎

######  Remark 2.6.

We will often consider the following situation. Let $V\to U$ be an étale
Galois cover of smooth integral curves over $k$. The cover ${V^{\langle
n\rangle}}\to U$ is still Galois because ${V^{\langle n\rangle}}\to V$ is
characteristic. Sometimes, we will need to compute a subcover $V^{\prime}\to
V$ of ${V^{\langle n\rangle}}$ by taking $n$-th roots of functions
$g_{1},\dots,g_{t}$ generating a submodule of
$\operatorname{H}^{1}(V,\mu_{n})$. In that case, $V^{\prime}\to U$ is still
Galois if and only if the submodule generated by $g_{1},\dots,g_{t}$ is stable
under the action of $\operatorname{Aut}(V|U)$. This is the case for instance
if we take the submodule of elements of $\operatorname{H}^{1}(V,\mu_{n})$
defined over some subfield of $k$ over which the elements of
$\operatorname{Aut}(V|U)$ are also defined.

###  2.3 Ramification at infinity

Let $U$ be an integral affine curve over an algebraically closed field $k$.
Denote by $X$ the smooth compactification of $U$. Let $n$ be an integer
invertible in $k$, and denote by $\Lambda$ the ring $\mathbb{Z}/n\mathbb{Z}$.
Let us study the ramification at infinity of ${U^{\langle n\rangle}}\to U$,
i.e. the ramification of the smooth compactification $X^{\prime}$ of
${U^{\langle n\rangle}}$ above the points $P_{0},\dots,P_{r}$ of $X-U$. The
map $\operatorname{H}^{2}_{Z}(X,\mu_{n})\to\operatorname{H}^{2}(X,\mu_{n})$
can be expressed, using the Gysin isomorphism
$\operatorname{H}^{0}(Z,\Lambda)\to\operatorname{H}^{2}_{Z}(X,\Lambda)$ and
the isomorphism $\operatorname{H}^{2}(X,\mu_{n})\to\Lambda$, as the sum map
$\operatorname{H}^{0}(Z,\Lambda)\to\Lambda$. A basis of its kernel is given by
$(P_{1}-P_{0},\dots,P_{r}-P_{0})$. Consider functions $g_{1},\dots,g_{r}\in
k(X)$ such that

$\mathrm{div}(g_{i})=nD_{i}+(P_{i}-P_{0})$

where $D_{i}\in\operatorname{Div}^{0}(X-\\{P_{0},\dots,P_{r}\\})$. The cover
$X^{\prime}\to{X^{\langle n\rangle}}$ corresponds to the function field
extension

$k({X^{\langle n\rangle}})(\sqrt[n]{g_{1}},\dots,\sqrt[n]{g_{r}})$

of $k({X^{\langle n\rangle}})$. Let $i\in\\{1\dots r\\}$. The extension
$k({X^{\langle n\rangle}})(\sqrt[n]{g_{j}},j\neq i)$ of $k({X^{\langle
n\rangle}})$ yields a cover $Y_{i}\to{X^{\langle n\rangle}}$ which is
unramified above $P_{i}$ because $v_{P_{i}}(g_{j})=0$. The cover
$X^{\prime}\to Y_{i}$, however, is ramified at $P_{i}$; let $Q_{i}$ be a
preimage of $P_{i}$ in $Y_{i}$. Since $v_{Q_{i}}(g_{i})=1$, the fibre
$X^{\prime}_{Q_{i}}$ is isomorphic to $k[x]/(x^{n})$, and the ramification
index of $X^{\prime}\to Y_{i}$ at any preimage of $Q_{i}$ is $n$. Above
$P_{i}$, there are exactly $\frac{1}{n}|\operatorname{H}^{1}(U,\Lambda)|$
points of $X^{\prime}$, each of with ramification index $n$. Let $R_{i}$ be a
preimage of $Q_{i}$ in $X^{\prime}$. The inertia subgroup
$I_{R_{i}|P_{i}}\subset\operatorname{Aut}({U^{\langle n\rangle}}|U)$ of
$R_{i}$ fits in the short exact sequence

$0\to I_{R_{i}|Q_{i}}\to I_{R_{i}|P_{i}}\to I_{Q_{i}|P_{i}}\to 0$

(see [Stacks, 0BU7]). As $I_{Q_{i}|P_{i}}=0$, there are isomorphisms

$I_{R_{i}|P_{i}}=I_{R_{i}|Q_{i}}=\operatorname{Aut}(X^{\prime}|Y_{i})\simeq\Lambda.$

${X^{\prime}}$${\bullet}$${\underset{R_{i}}{\bullet}}$${\bullet}$${\bullet}$${n^{2g+r-1}\text{
points}}$${Y_{i}}$${\bullet}$${\underset{Q_{i}}{\bullet}}$${\dots}$${\bullet}$${\bullet}$${n^{2g+r-1}\text{
points}}$${{X^{\langle
n\rangle}}}$${\bullet}$${\dots}$${\dots}$${\dots}$${\bullet}$${n^{2g}\text{
points}}$${X}$${\underset{P_{i}}{\bullet}}$$\scriptstyle{\Lambda}$$\scriptstyle{\Lambda^{r}}$$\scriptstyle{\operatorname{H}^{1}(X,\Lambda)}$
Figure 1: Ramification at infinity of the cover ${U^{\langle n\rangle}}\to U$

As a subgroup of $\operatorname{Aut}({U^{\langle n\rangle}}|U)$, the group
$I_{R_{i}|P_{i}}$ is generated by
$\sqrt[n]{g_{i}}\mapsto\zeta\sqrt[n]{g_{i}}$, where $\zeta$ is a primitive
$n$-th root of unity in $k$. The results regarding preimages and ramification
indices also apply to $P_{0}$, which had been chosen arbitrarily. With the
above notation, the subgroup $I_{R_{0}|P_{0}}$ is generated by
$(\sqrt[n]{g_{1}},\dots,\sqrt[n]{g_{r}})\mapsto(\zeta\sqrt[n]{g_{1}},\dots,\zeta\sqrt[n]{g_{r}})$.

###  2.4 Cohomology of the ramification groups

Let $n$ be a positive integer. Denote by $\Lambda$ the ring
$\mathbb{Z}/n\mathbb{Z}$. Let us consider an étale Galois cover $V\to U$ of
smooth integral curves over an algebraically closed field $k$ of
characteristic prime to $n$. Let $K$ be the function field of $U$. Denote by
$X$ (resp. $Y$) the smooth compactification of $U$ (resp. $V$). Set
$G=\operatorname{Aut}(V|U)$. Let $x$ be a closed point of $X-U$. Consider an
inertia group $I\subset\operatorname{Gal}(K^{\rm sep}|K)$ at $x$, and one of
its finite quotients $I_{y}\subset G$, which is the stabiliser in $G$ of a
closed point $y\in Y-V$ mapping to $x$. Denote by $P\triangleleft I$ and
$P_{y}\triangleleft I_{y}$ the wild inertia subgroups. There are canonical
isomorphisms [Stacks, 09EE]

$I/P\xrightarrow{\sim}\lim_{m\nmid p}\mu_{m}(k)\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
I_{y}/P_{y}\xrightarrow{\sim}\mu_{e}(k)$

where $e$ is the ramification index of $Y\to X$ at $y$. From now on, we assume
that $n$ divides $e$. Let $M$ be a $\Lambda[G]$-module.

######  Proposition 2.7.

In the situation described above, the canonical maps

$\tau_{\leqslant
1}\operatorname{R\Gamma}(I_{y}/P_{y},M^{P_{y}})\to\operatorname{R\Gamma}(I/P,M^{P})\to\operatorname{R\Gamma}(I,M)$

are quasi-isomorphisms.

###### Proof.

Let $\sigma$ denote a pro-generator of $I/P$, and $\sigma_{y}$ its image in
$I_{y}/P_{y}$. The actions of $\sigma$ and $\sigma_{y}$ on $M$ are equal. Set
$N_{y}=\sum_{i=1}^{e}\sigma_{y}^{i}$. The usual results on the cohomology of
(pro)-cyclic groups [Wei94, Calc. 6.2.1] [Fu15, Prop. 8.1.4] show that the map
on the left hand side is

${\operatorname{R\Gamma}(I/P,M^{P})}$ ${M^{P}}$${M^{P}}$${0}$${\tau_{\leqslant
1}\operatorname{R\Gamma}(I_{y}/P_{y},M^{P_{y}})}$
${M^{P}}$${\ker(N_{y})}$${0}$$\scriptstyle{\sigma-\operatorname{id}}$$\scriptstyle{\sigma_{y}-\operatorname{id}}$

Since $n$ divides $e$, the action of $N_{y}$ on $M$ is trivial and
$\ker(N_{y})=M$, which shows that the map above is a quasi-isomorphism. The
fact that the map on the right hand side is a quasi-isomorphism is shown in
[Fu15, Prop. 8.1.4] as well. ∎

The following lemma shows how to explicitly construct the inverse of the map
on the left hand side as a morphism of complexes; it will be used in Theorem
4.5. Recall that we denote by $\operatorname{Hom_{cr}}$ the groups of crossed
homomorphisms.

######  Lemma 2.8.

The canonical map
$\operatorname{Hom_{cr}}(I_{y}/P_{y},M^{P_{y}})\to\operatorname{Hom_{cr}}(I_{y},M)$
has a section.

###### Proof.

Let $u\colon I_{y}\to M$ be a crossed homomorphism. Consider the commutative
diagram

${I_{y}}$${M}$${M_{P_{y}}}$${I_{y}/P_{y}}$${M^{P_{y}}}$$\scriptstyle{u}$$\scriptstyle{q}$$\scriptstyle{\alpha}$

and set $f=q\circ u$. For all $x\in P_{y}$ and $g\in I_{y}$, the definition of
$M_{P_{y}}$ ensures that $f(xg)=f(x)+q(x\cdot u(g))=f(x)+f(g)$. Hence, for any
$x\in P_{y}$, $f(x^{|P_{y}|})=|P_{y}|f(x)$ is zero ; since multiplication by
$|P_{y}|$ is an automorphism of $M$, this means that $f(x)=0$. Therefore,
there is a quotient map $\bar{f}\colon I_{y}/P_{y}\to M_{P_{y}}$. Set
$\bar{u}=\alpha^{-1}\circ\bar{f}\colon I_{y}/P_{y}\to M^{P_{y}}$. The map
$u\mapsto\bar{u}$ is clearly linear. Moreover, $\bar{u}$ is still a crossed
homomorphism since
$\bar{u}(\bar{g}_{1}\bar{g}_{2})=\alpha^{-1}f(g_{1}g_{2})=\alpha^{-1}f(g_{1})+q(g_{1}\cdot\alpha^{-1}u(g_{2}))$.
The $I_{y}$-linearity of $\alpha^{-1}q$ concludes. ∎

###  2.5 A similar cover with Galois action

Let $k_{0}$ be a perfect field, and $k$ be an algebraic closure of $k_{0}$.
Let $n$ be an integer invertible in $k$. Denote by $\Lambda$ the ring
$\mathbb{Z}/n\mathbb{Z}$. Let $V_{0}$ be a geometrically integral smooth curve
over $k_{0}$. As usual, the base change $-\times_{k_{0}}k$ will be denoted by
removing the subscript 0. We wish to compute a (connected) characteristic
cover $V^{\prime}_{0}$ of $V_{0}$ such that the map
$\operatorname{H}^{1}(V,\Lambda)\to\operatorname{H}^{1}(V^{\prime},\Lambda)$
is trivial. In the case where $V$ is connected and the elements of
$\operatorname{H}^{1}(V,\mu_{n})$ are defined over $k_{0}$, the cover
${V^{\langle n\rangle}}$ constructed in the previous sections comes from
$k_{0}$, and we are done. However, this is not the case in general. The
construction below is a refinement of the simple idea of computing the Galois
orbit of ${V^{\langle n\rangle}}$.

#### Construction of $V^{\prime}_{0}$

The function field of $Y_{0}$ is of the form $k_{0}(x)[y]/(f)$, with $f\in
k(x)[y]$. Denote by $k_{1}$ the algebraic closure of $k_{0}$ in
$k_{0}(x)[y]/(f)$; since $k_{0}$ is perfect, $k_{1}$ is a separable extension.
Let $V_{1}$ be a connected component of $V$. Let
$([D_{1},g_{1}],\dots,[D_{r},g_{r}])$ be a basis of
$\operatorname{H}^{1}(V_{1},\mu_{n})$ as in 2.4. Denote by $k_{2}$ the minimal
extension of $k_{0}$ over which the $g_{i}$ are defined. Let $L$ be the Galois
closure of the extension of $k_{0}$ generated by $k_{1},k_{2}$ and
$\mu_{n}(k)$. Let $\alpha$ be a primitive element of the extension $L|k_{1}$,
and $m\in k_{1}[t]$ its minimal polynomial. For $i\in\\{1\dots r\\}$, write
$g_{i}=g_{i}^{\prime}(\alpha,x,y)$ with $g^{\prime}_{i}\in
k_{0}(x)[t,y]/(m(t),f(x,y))$. Let $V^{\prime}_{0}\to V_{0}$ be the
normalisation of $V_{0}$ in the function field
$k_{0}(\alpha,x,y)(\sqrt[n]{g^{\prime}_{1}},\dots,\sqrt[n]{g^{\prime}_{r}})$.
The curve $V^{\prime}$ is isomorphic to the
$\operatorname{Gal}(L|k_{0})$-orbit of $V_{1}^{\langle n\rangle}$; it has
$[L:k_{0}]$ connected components, and the map
$\operatorname{H}^{1}(V,\Lambda)\to\operatorname{H}^{1}(V^{\prime},\Lambda)$
is trivial.

Note that since the degree of $V^{\prime}_{0}\to(V_{0}\times_{k_{1}}L)$ is
$n^{r}$ and that of $V_{0}\times_{k_{1}}L\to V_{0}$ is $[L:k_{1}]$, the degree
of $V^{\prime}_{0}\to V_{0}$ is $n^{r}[L:k_{1}]$.

######  Proposition 2.9.

The cover $V^{\prime}_{0}\to V_{0}$ is characteristic, i.e. if $V_{0}$ is an
étale Galois cover of a curve $U_{0}$, then $V_{0}^{\prime}\to U_{0}$ is still
Galois.

###### Proof.

Consider the situtation where $V_{0}$ is an étale Galois cover of a smooth
integral $k_{0}$-curve $U_{0}$. Here is how to explicitly compute the
automorphism group of $V_{0}\to U_{0}$. The map $V^{\prime}_{0}\to V_{0}$ has
degree $n^{r}[L:k_{1}]$. Using the notations above, set
$z_{i}=\sqrt[n]{g^{\prime}_{i}}$. The elements of
$\operatorname{Aut}(k_{0}(V^{\prime}_{0})|k_{0}(V_{0}))$ are defined by
$t\mapsto\sigma(t),z_{i}\mapsto\zeta_{i}z_{i}$ where
$\sigma\in\operatorname{Gal}(L|k_{1})$ and $\zeta_{i}\in\mu_{n}(L)$. There are
$\deg(V^{\prime}_{0}\to V_{0})=n^{r}[L:k_{1}]$ such automorphisms since
$\mu_{n}(k)\subset L$, hence the cover $V^{\prime}_{0}\to V_{0}$ is Galois.
Let us now compute the automorphism group of $V^{\prime}_{0}\to U_{0}$. Its
elements are defined by

$(t,x,y,z_{1},\dots,z_{r})\mapsto(\sigma(t),x^{\prime},y^{\prime},z^{\prime}_{1},\dots,z^{\prime}_{r})$

where $\sigma\in\operatorname{Gal}(L|k_{0})$, the pair
$(x^{\prime},y^{\prime})$ is the image of $(x,y)$ under a $U_{0}$-automorphism
of $V_{0}$ whose image in $\operatorname{Gal}(k_{1}|k_{0})$ is the same as
that of $\sigma$, and the elements $z^{\prime}_{i}\in k_{0}(V^{\prime}_{0})$
satisfy ${z^{\prime}_{i}}^{n}=\phi(g^{\prime}_{i})$. As expected, there are
$\deg(V^{\prime}_{0}|U_{0})=n^{r}[L:k_{1}]\deg(V_{0}\to U_{0})$ elements in
$\operatorname{Aut}(V^{\prime}_{0}|U_{0})$, and $V^{\prime}_{0}\to U_{0}$ is
Galois. ∎

######  Remark 2.10.

If need be, the Galois extension $L$ of $k_{0}$ may be chosen to be a little
larger, for instance to make sure that the points at infinity of the curve
$V^{\prime}$ are defined over $L$. This does not affect any of the previous
proofs.

##  3 Schemes of cohomological dimension 1

Let $n$ be a positive integer, and $\Lambda=\mathbb{Z}/n\mathbb{Z}$. Let $X$
be an integral noetherian scheme, and ${\bar{\eta}}$ be a generic geometric
point of $X$.

######  Proposition 3.1.

Let $\mathscr{L}$ be a finite locally constant sheaf of
$\mathbb{Z}/n\mathbb{Z}$-modules on $X$. Let $Y\to X$ be an étale Galois cover
such that the morphism

$\operatorname{H}^{1}(X,\mathscr{L})\to\operatorname{H}^{1}(Y,\mathscr{L}|_{Y})$

is trivial. Then the morphism

$\tau_{\leqslant
1}\operatorname{R\Gamma}(\operatorname{Aut}(Y|X),\mathscr{L}_{{\bar{\eta}}})\to\tau_{\leqslant
1}\operatorname{R\Gamma}(X,\mathscr{L})$

is a quasi-isomorphism.

###### Proof.

Let $G$ be the automorphism group of $Y\to X$. The associated Hochschild-Serre
spectral sequence yields the following short exact sequence:

$0\to\operatorname{H}^{1}(G,\mathscr{L}_{{\bar{\eta}}})\to\operatorname{H}^{1}(X,\mathscr{L})\to\operatorname{H}^{0}(G,\operatorname{H}^{1}(Y,\mathscr{L}|_{Y}))$

Hence the map
$\operatorname{H}^{1}(G,\mathscr{L}_{{\bar{\eta}}})\to\operatorname{H}^{1}(X,\mathscr{L})$
is an isomorphism, and

$\operatorname{R\Gamma}(G,\mathscr{L}_{{\bar{\eta}}})\to\operatorname{R\Gamma}(X,\mathscr{L})$

yields isomorphisms on cohomology groups in degree 0 and 1. ∎

######  Remark 3.2.

If in addition $X$ has cohomological dimension 1 then

$\tau_{\leqslant
1}\operatorname{R\Gamma}(\operatorname{Aut}(Y|X),\mathscr{L}_{{\bar{\eta}}})\to\operatorname{R\Gamma}(X,\mathscr{L})$

is a quasi-isomorphism.

######  Remark 3.3.

Here is how to construct a cover $Y$ as in the proposition. Pick an étale
Galois cover $W\to X$ such that $\mathscr{L}|_{W}$ is a constant sheaf. If
$\operatorname{H}^{1}(W,\Lambda)$ is finite, set $Y=W^{\langle n\rangle}$:
since $Y\to W$ is characteristic, the cover $Y\to X$ is still Galois, and 2.3
ensures that
$\operatorname{H}^{1}(X,\mathscr{L})\to\operatorname{H}^{1}(Y,\mathscr{L}|_{Y})$
is trivial.

Given a profinite group $H$, we will denote by $P_{H}(\Lambda)$ the usual
projective resolution (sometimes called bar resolution) of the trivial
$\Lambda[[H]]$-module $\Lambda$.

######  Proposition 3.4.

Suppose $X$ is of cohomological dimension 1. Let
$\mathscr{L}=[\mathscr{L}^{0}\to\mathscr{L}^{1}\to\dots\to\mathscr{L}^{s}]$ be
a complex of finite locally constant sheaves of $\Lambda$-modules on $X$, and
$Y$ be an étale Galois cover of $X$ such that each map
$\operatorname{H}^{1}(X,\mathscr{L}^{i})\to\operatorname{H}^{1}(Y,\mathscr{L}^{i}|_{Y})$
is trivial. Write $G=\operatorname{Aut}(Y|X)$. Consider the double complex
$B^{\bullet,\bullet}$ defined by
$B^{i,j}=\operatorname{Hom}_{\Lambda[G]}(\tau_{\geqslant-1}P_{G}^{-j}(\Lambda),\mathscr{L}_{\bar{\eta}}^{i})$.
Then $\operatorname{R\Gamma}(X,\mathscr{L})$ is represented by the total
complex $\operatorname{Tot}B^{\bullet,\bullet}$.

###### Proof.

We wish to compute
$\operatorname{R\Gamma}(X,\mathscr{L})=\operatorname{R\Gamma}(\pi_{1}(X),\mathscr{L}_{\bar{\eta}})=\operatorname{RHom}_{\Lambda[[\pi_{1}(X)]]}(\Lambda,\mathscr{L}_{\bar{\eta}})$,
which is represented by the complex

$\operatorname{Hom}^{\bullet}_{\Lambda[[\pi_{1}(X)]]}(P_{\pi}(\Lambda),\mathscr{L}_{\bar{\eta}})=\operatorname{Tot}(A^{\bullet,\bullet})$

where
$A^{i,j}=\operatorname{Hom}_{\Lambda[[\pi_{1}(X)]]}(P_{\pi_{1}(X)}^{-j}(\Lambda),\mathscr{L}_{\bar{\eta}}^{i})$.
The map $B^{\bullet,\bullet}\to A^{\bullet,\bullet}$ induced by the quotient
map $\pi_{1}(X)\to G$ defines a morphism between the spectral sequence
associated to these quotients. Recall that for each $i\in\\{1\dots s\\}$, the
map $\tau_{\leqslant
1}\operatorname{Hom}_{\Lambda[G]}(P_{G}(\Lambda),\mathscr{L}^{i}_{\bar{\eta}})\to\operatorname{Hom}_{\Lambda[[\pi_{1}(X)]]}(P_{\pi_{1}(X)}(\Lambda),\mathscr{L}^{i}_{\bar{\eta}})$
is a quasi-isomorphism. Since the functor
$\operatorname{Hom}(-,\mathscr{L}^{i}_{\bar{\eta}})$ is left exact,
$\operatorname{Hom}_{\Lambda[G]}(\tau_{\geqslant-1}P_{G}(\Lambda),\mathscr{L}^{i}_{\bar{\eta}})=\tau_{\leqslant
1}\operatorname{Hom}_{\Lambda[H]}(P_{G}(\Lambda),\mathscr{L}^{i}_{\bar{\eta}})$.
Hence the map $B^{\bullet,\bullet}\to A^{\bullet,\bullet}$ defines, on each
column, a quasi-isomorphism of complexes. It therefore induces an isomorphism
between the first pages of the corresponding spectral sequences (for the
upward orientation) associated to $B^{\bullet,\bullet}$ and
$A^{\bullet,\bullet}$: in position $(i,j)$, it is the isomorphism
$\operatorname{H}^{j}(H,\mathscr{L}_{\bar{\eta}}^{i})\to\operatorname{H}^{j}(\pi_{1}(X),\mathscr{L}_{\bar{\eta}}^{i})$
for $j\leqslant 1$, and
$0\to\operatorname{H}^{j}(\pi_{1}(X),\mathscr{L}_{\bar{\eta}}^{i})=0$
otherwise. Therefore, the map between the abutments of these two spectral
sequences is an isomorphism, i.e. the map
$\operatorname{Tot}(B^{\bullet,\bullet})\to\operatorname{Tot}(A^{\bullet,\bullet})$
is a quasi-isomorphism. ∎

##  4 Explicit computation of $\operatorname{R\Gamma}$ of a (possibly
singular) curve

In this section, we are going to describe how to compute the cohomology of a
complex of constructible sheaves on a curve. Let $k_{0}$ be a field, and $k$
be a separable closure of $k_{0}$. Consider a geometrically irreducible curve
$X_{0}$ over $k_{0}$, and its base change $X$ over $k$. We are allowed to make
the following additional assumptions, which do not alter the computed
cohomology complex.

  * •

The field $k_{0}$ is perfect, hence $k$ is algebraically closed: the perfect
closure $k_{0}^{\rm pf}$ of $k_{0}$ being a purely inseparable extension, the
base change $-\times_{k_{0}}k_{0}^{\rm pf}$ induces an isomorphism on
cohomology.

  * •

The curve $X$ is reduced: being a universal homeomorphism, the map $X_{\rm
red}\to X$ induces an isomorphism on cohomology.

  * •

The curve $X$ has at worst ordinary (also called multicross, see [Stacks,
0C1P]) singularities: the seminormalisation map $X^{\rm sn}\to X$ being a
universal homeomorphism [Stacks, 0EUS], it induces an isomorphism on
cohomology, so we may assume $X$ is seminormal. Since a seminormal curve over
an algebraically closed field has at worst ordinary singularities [Dav78, §2,
Cor. 1], we may assume this is the case for $X$.

###  4.1 Cohomology with support in a 0-dimensional subscheme

Let $k$ be a algebraically closed field. Let $n$ be an integer invertible in
$k$, and $\Lambda=\mathbb{Z}/n\mathbb{Z}$. Let $X$ be an integral curve over
$k$. Consider a nonempty zero-dimensional subscheme $i\colon Z\to X$, and its
open complement $j\colon U\to X$. Let $\operatorname{R\Gamma}_{Z}(X,-)$ denote
cohomology with support in $Z$. Since
$\operatorname{R\Gamma}_{Z}(X,-)=\oplus_{z\in
Z}\operatorname{R\Gamma}_{z}(X,-)$, we consider a single point $z\in Z$ and
focus on computing $\operatorname{R\Gamma}_{z}(X,-)$.

######  Lemma 4.1.

Let $\mathscr{L}$ be a finite locally constant sheaf on $U$.

  * $\bullet$

$\operatorname{H}^{0}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{0}_{z}(X,j_{\star}\mathscr{L})=\operatorname{H}^{1}_{z}(X,j_{\star}\mathscr{L})=0$

  * $\bullet$

$\operatorname{H}^{1}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{0}(z,\mathscr{L})$

  * $\bullet$

$\operatorname{H}^{2}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{2}_{z}(X,j_{\star}\mathscr{L})$

  * $\bullet$

For all $i\geqslant 3$,
$\operatorname{H}^{i}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{i}_{z}(X,j_{\star}\mathscr{L})=0$.

###### Proof.

Since
$\operatorname{H}^{0}(X,j_{\star}\mathscr{L})\to\operatorname{H}^{0}(X-z,j_{\star}\mathscr{L})$
is an isomorphism and
$\operatorname{H}^{1}(X,j_{\star}\mathscr{L})\to\operatorname{H}^{1}(X-z,j_{\star}\mathscr{L})$
is always a monomorphism, the long exact sequence for cohomology with support
associated to $j_{\star}\mathscr{L}$ shows that
$\operatorname{H}^{0}_{z}(X,j_{\star}\mathscr{L})=\operatorname{H}^{1}_{z}(X,j_{\star}\mathscr{L})=0$.
Moreover, for any $j\geqslant 3$, the groups
$\operatorname{H}^{j-1}(X-z,j_{\star}\mathscr{L})$ and
$\operatorname{H}^{j}(X,\mathscr{L})$ are trivial, hence
$\operatorname{H}^{j}_{z}(X,j_{\star}\mathscr{L})=0$. Recall that
$\operatorname{H}^{i}_{z}(X,i_{\star}-)=\operatorname{H}^{i}(z,-)$. The long
exact sequence of $\operatorname{H}^{i}_{z}(X,-)$ associated to the short
exact sequence

$0\to j_{!}\mathscr{L}\to j_{\star}\mathscr{L}\to
i_{\star}i^{\star}j_{\star}\mathscr{L}\to 0$

shows that $\operatorname{H}^{0}_{z}(X,j_{!}\mathscr{L})=0$,
$\operatorname{H}^{1}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{0}(Z,\mathscr{L})$
and
$\operatorname{H}^{2}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{2}_{z}(X,j_{\star}\mathscr{L})$.
The groups $\operatorname{H}^{i}_{z}(X,j_{!}\mathscr{L})$ are also trivial as
soon as $i\geqslant 3$. ∎

Let us now compute the group
$\operatorname{H}^{2}_{z}(X,j_{!}\mathscr{L})=\operatorname{H}^{2}_{z}(X,j_{\star}\mathscr{L})$.
From now on, we assume $X$ to have only ordinary singularities. Let
$z_{1},\dots,z_{r}$ be the preimages of $z$ in the normalisation ${\tilde{X}}$
of $X$. Denote by $X_{\bar{z}}$ the strict henselisation of $X$ at $z$. It
contains one closed point $z^{\prime}$, as well as $r$ minimal primes
$z^{\prime}_{1},\dots,z^{\prime}_{r}$. Set $U_{\bar{z}}\coloneqq
U\times_{X}X_{\bar{z}}$. Consider the following cartesian diagram.

${U_{\bar{z}}}$${X_{\bar{z}}}$${z^{\prime}}$${U}$${X}$${z}$$\scriptstyle{g^{\prime}}$$\scriptstyle{j^{\prime}}$$\scriptstyle{g}$$\scriptstyle{i^{\prime}}$$\scriptstyle{j}$$\scriptstyle{i}$

The following proof is given in [Mil06, II, Prop. 1.1] in the special case of
smooth curves.

######  Lemma 4.2.

Let $\mathcal{F}$ be a sheaf on $U_{\bar{z}}$. For every positive integer $q$,
the group $\operatorname{H}^{q}(X_{\bar{z}},j^{\prime}_{!}\mathcal{F})$ is
trivial.

###### Proof.

The assertion holds for $q=0$ since
$\operatorname{H}^{0}(X_{\bar{z}},j^{\prime}_{!}\mathcal{F})$ is the kernel of
the map
$\operatorname{H}^{0}(X_{\bar{z}},j^{\prime}_{\star}\mathcal{F})\to\operatorname{H}^{0}(X_{\bar{z}},i^{\prime}_{\star}i^{\prime\star}j_{\star}\mathcal{F})$,
which is simply the identity map of
$\operatorname{H}^{0}(U_{\bar{z}},\mathcal{F})$. Let us first show that for
any injective sheaf $J$ on $U_{\bar{z}}$, the sheaf
$j^{\prime}_{!}\mathcal{F}$ on $X$ is acyclic. To this end, we are going to
prove that

$0\to j^{\prime}_{!}J\to j^{\prime}_{\star}J\to
i^{\prime}_{\star}i^{\prime\star}j^{\prime}_{\star}J\to 0$

is an injective resolution of $j^{\prime}_{!}J$ ; the long exact sequence
associated to this short exact sequence then shows that
$\operatorname{H}^{q}(X_{\bar{z}},j^{\prime}_{\star}J)=0$ for any $q\geqslant
1$. Fix separable closures of $k(z^{\prime}_{1}),\dots,k(z^{\prime}_{r})$, and
denote by $I_{1},\dots,I_{r}$ the associated Galois groups. The functor
$i^{\star}j_{\star}$ may be rewritten as follows:

$\begin{array}[]{rcl}\operatorname{Mod}_{I_{1}}\times\dots\times\operatorname{Mod}_{I_{r}}&\to&\operatorname{Ab}\\\
(M_{1},\dots,M_{r})&\mapsto&M_{1}^{I_{1}}\times\dots\times
M_{r}^{I_{r}}\end{array}$

This functor admits a left adjoint, which sends an abelian group $M$ to
$(M,\dots,M)$ with trivial $(I_{1},\dots,I_{r})$-action. Since this left
adjoint is exact, $i^{\prime\star}j^{\prime}_{\star}$ sends injectives to
injectives. The functors $i^{\prime}_{\star}$ and $j^{\prime}_{\star}$ also
send injectives to injectives, therefore the short exact sequence above is an
injective resolution of $j^{\prime}_{!}J$. We are now ready to prove the
result. Let $\mathcal{F}$ be a sheaf $\mathcal{F}$ on $U_{\bar{z}}$, and let
$J^{\bullet}$ be an injetive resolution of $\mathcal{F}$. Then
$j^{\prime}_{!}J^{\bullet}$ is an acyclic resolution of
$j^{\prime}_{!}\mathcal{F}$, and
$\operatorname{H}^{q}(X_{\bar{z}},j^{\prime}_{!}\mathcal{F})=\operatorname{H}^{q}(\Gamma(X_{\bar{z}},j^{\prime}_{!}J^{\bullet}))$.
The latter group is the image of $\mathcal{F}$ under the $q$-th right derived
functor of $\Gamma(X_{\bar{z}},j^{\prime}_{!}-)$, which is zero as proven
above. ∎

Let $\nu\colon{\tilde{X}}\to X$ be the normalisation map. Set
$\tilde{z}\coloneqq z\times_{X}{\tilde{X}}$. Denote by
$\eta_{1},\dots,\eta_{r}$ the generic points of the strict henselisations of
${\tilde{X}}$ at the preimages $z_{1},\dots,z_{r}$ of $z$ in ${\tilde{X}}$.
Denote by $\widetilde{X_{\bar{z}}}$ the normalisation of the strict
henselisation of $X$ at $z$. The normalisation of $U_{\bar{z}}$ is

$U_{\bar{z}}\times_{X_{\bar{z}}}\widetilde{X_{\bar{z}}}=U\times_{X}({\tilde{X}}_{\bar{z}_{1}}\sqcup\dots\sqcup{\tilde{X}}_{\bar{z}_{r}})=\eta_{1}\sqcup\dots\sqcup\eta_{r}$

and since normalisation is birational, the map

$\eta_{1}\sqcup\dots\sqcup\eta_{r}\to U_{\bar{z}}$

is an isomorphism.

######  Proposition 4.3.

Let $\mathscr{L}$ be a finite locally constant sheaf on $U$. The map

$\operatorname{R\Gamma}_{z}(X,j_{\star}\mathscr{L})\to\operatorname{R\Gamma}_{\tilde{z}}({\tilde{X}},\nu^{\star}j_{\star}\mathscr{L})$

is a quasi-isomorphism.

###### Proof.

4.1 shows that it suffices to prove that

$\operatorname{H}^{2}_{z}(X,j_{\star}\mathscr{L})\to\operatorname{H}^{2}_{\tilde{z}}({\tilde{X}},\nu^{\star}j_{\star}\mathscr{L})=\operatorname{H}^{2}_{z_{1}}({\tilde{X}},\nu^{\star}j_{\star}\mathscr{L})\times\dots\times\operatorname{H}^{2}_{z_{r}}({\tilde{X}},\nu^{\star}j_{\star}\mathscr{L})$

is an isomorphism. Let us compute
$\operatorname{H}^{2}_{z}(X,j_{\star}\mathscr{L})$. By excision, there is a
canonical isomorphism

$\operatorname{H}^{2}_{z}(X,j_{\star}\mathscr{L})\xrightarrow{\sim}\operatorname{H}^{2}_{z^{\prime}}(X_{\bar{z}},g^{\star}j_{!}\mathscr{L})=\operatorname{H}^{2}_{z^{\prime}}(X_{\bar{z}},j^{\prime}_{!}g^{\prime\star}\mathscr{L}).$

The long exact sequence in cohomology with support on $z^{\prime}$ for the
sheaf $j^{\prime}_{!}g^{\prime\star}\mathscr{L}$ reads:

$\operatorname{H}^{1}(X_{\bar{z}},j^{\prime}_{!}g^{\prime\star}\mathscr{L})\to\operatorname{H}^{1}(U_{\bar{z}},g^{\prime\star}\mathscr{L})\to\operatorname{H}^{2}_{z^{\prime}}(X_{\bar{z}},j^{\prime}_{!}g^{\prime\star}\mathscr{L})\to\operatorname{H}^{2}(X_{\bar{z}},j^{\prime}_{!}g^{\prime\star}\mathscr{L}).$

4.2 ensures that the map

$\operatorname{H}^{1}(U_{\bar{z}},g^{\prime\star}\mathscr{L})\to\operatorname{H}^{2}_{z^{\prime}}(X_{\bar{z}},j^{\prime}_{!}g^{\prime\star}\mathscr{L})$

is an isomorphism. Since the scheme $U_{\bar{z}}$ is the coproduct of the
points $z^{\prime}_{1},\dots,z^{\prime}_{r}$, the group
$\operatorname{H}^{1}(U_{\bar{z}},g^{\prime\star}\mathscr{L})$ is simply
$\operatorname{H}^{1}(z^{\prime}_{1},\mathscr{L}_{z^{\prime}_{1}})\times\dots\times\operatorname{H}^{1}(z^{\prime}_{r},\mathscr{L}_{z^{\prime}_{r}})$.
On the other hand, for $t\in\\{1\dots r\\}$,
$\operatorname{H}^{2}_{z_{t}}({\tilde{X}},\nu^{\star}j_{\star}\mathscr{L})$ is
isomorphic to $\operatorname{H}^{1}(\eta_{t},\mathscr{L}_{\eta_{t}})$, and the
map

$\operatorname{H}^{1}(z^{\prime}_{t},\mathscr{L}_{z^{\prime}_{t}})\to\operatorname{H}^{1}(\eta_{t},\mathscr{L}_{\eta_{t}})$

is simply the isomorphism induced by
$\eta_{t}\xrightarrow{\sim}z^{\prime}_{t}$. ∎

######  Corollary 4.4.

For each point $z\in Z\times_{X}{\tilde{X}}$, choose a place of $K^{\rm sep}$
above $z$ and denote by $I_{z}$ the corresponding inertia group. The one-term
complex

$0\to 0\to\bigoplus_{z\in
Z\times_{X}{\tilde{X}}}\operatorname{H}^{1}(I_{z},M)\to 0\to\cdots$

represents $\operatorname{R\Gamma}_{Z}(X,j_{\star}\mathscr{L})$.

###### Proof.

This is merely rephrasing 4.3 using the fact that only
$\operatorname{H}^{2}_{z}$ is nonzero, which was proven in 4.1. ∎

###  4.2 Cohomology of constructible sheaves

Let $k$ be an algebraically closed field. Let $n$ be an integer invertible in
$k$. Denote by $\Lambda$ the ring $\mathbb{Z}/n\mathbb{Z}$. Let $X$ be an
integral curve with ordinary singularities over $k$. Denote by
$\nu\colon\tilde{X}\to X$ its normalisation. Let
$\mathcal{F}^{\bullet}=[\mathcal{F}^{0}\to\cdots\to\mathcal{F}^{t}]$ be a
complex of constructible sheaves of $\Lambda$-modules on $X$. This section
aims to give an explicit description of the cohomology complex
$\operatorname{R\Gamma}(X,\mathcal{F})$.

Let $j\colon U\to X$ be the inclusion of a regular open affine subscheme of
$X$ on which every sheaf $\mathcal{F}^{s}$ is locally constant. Let $i\colon
Z\to X$ be the inclusion of its closed complement with the reduced subscheme
structure. Set $\mathscr{L}^{\bullet}\coloneqq
j^{\star}\mathcal{F}^{\bullet}$, and
$M^{\bullet}\coloneqq\mathscr{L}^{\bullet}_{\bar{\eta}}$. Let $V\to U$ be an
étale Galois cover such that:

  * •

each sheaf $\mathscr{L}^{s}|_{V}$, $s\in\\{1\dots t\\}$ is constant;

  * •

each map
$\operatorname{H}^{1}(U,\mathscr{L}^{s})\to\operatorname{H}^{1}(V,\mathscr{L}^{s}|_{V})$,
$s\in\\{1\dots t\\}$ is trivial;

  * •

the ramification index of $V\to U$ above every point of ${\tilde{X}}-U$ is
divisible by $n$.

Denote by $G$ the group $\operatorname{Aut}(V|U)$. Let $\tilde{Z}$ be the
preimage of $Z$ in ${\tilde{X}}$. Given $\tilde{z}\in\tilde{Z}$, denote by
$I_{\tilde{z}}\subset G$ its inertia group, and by $P_{\tilde{z}}\triangleleft
I_{\tilde{z}}$ its wild inertia subgroup. For each integer
$s\in\\{1,\dots,t\\}$, let $\phi^{s}\colon\mathcal{F}^{s}\to
j_{\star}\mathscr{L}^{s}$ be the adjunction unit. The remainder of this
section will be dedicated to the proof of the following result.

######  Theorem 4.5.

In this situation,
$\operatorname{R\Gamma}(X,\mathcal{F}^{\bullet})[1]\in\operatorname{D}^{b}_{c}(\Lambda)$
is represented by the cone of the following morphism of complexes, whose terms
are indexed by $s\geqslant 0$:

${\cdots}$${M^{s}\oplus\operatorname{Hom_{cr}}(G,M^{s-1})\oplus\bigoplus_{z\in
Z}\mathcal{F}_{z}^{s}\oplus\bigoplus_{\tilde{z}\in{\tilde{Z}}}\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M_{P_{\tilde{z}}}^{s-2})}$${\cdots}$${\cdots}$${\bigoplus_{\tilde{z}\in{\tilde{Z}}}\left(M^{s}\oplus\operatorname{Hom_{cr}}(I_{\tilde{z}}/P_{\tilde{z}},M^{s-1}_{P_{\tilde{z}}})\oplus\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M^{s-2}_{P_{\tilde{z}}})\right)}$${\cdots}$

Here, $(a,b,c_{z},d_{\tilde{z}})$ is sent to
$(a-\phi_{z}(c),\operatorname{res}_{G}^{I_{z}}(b),d)$ where $\nu(\tilde{z})=z$
and $\operatorname{res}_{G}^{I_{z}}$ denotes the composite map
$\operatorname{Hom_{cr}}(G,M^{s})\to\operatorname{Hom_{cr}}(I_{z},M)^{s}\to\operatorname{Hom_{cr}}(I_{z}/P_{z},M^{s}_{P_{z}})$
defined in 2.8. The transition map of the top complex is
$(a,b,c_{z},d_{\tilde{z}})\mapsto(\partial_{M}(a),\partial_{M}(b)-\partial_{G}(a),\partial_{\mathcal{F}}(c_{z}),\partial_{M}(d_{\tilde{z}})-\operatorname{res}_{G}^{I_{z}}(b))$.
The transition map of the bottom complex is
$(a,b,c)\mapsto(\partial_{M}(a),\partial_{M}(b)-\partial_{G}(a),\partial_{M}(b)-\partial_{G}(a),\partial_{M}(c)-\partial_{G}(b))$.

######  Corollary 4.6.

When computing the cohomology of a single constructible sheaf $\mathcal{F}$ on
$X$ with generic geometric fibre $M$, the complex
$\operatorname{R\Gamma}(X,\mathcal{F})[1]$ is the cone of the following
morphism of complexes:

${(\oplus_{z\in Z}\mathcal{F}_{z})\oplus
M}$${\operatorname{Hom_{cr}}(G,M)}$${\oplus_{\tilde{z}\in\tilde{Z}}\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M_{P_{\tilde{z}}})}$${\oplus_{\tilde{z}\in\tilde{Z}}M}$${\oplus_{\tilde{z}\in\tilde{Z}}\operatorname{Hom_{cr}}(I_{\tilde{z}}/P_{\tilde{z}},M_{P_{\tilde{z}}})}$${\oplus_{\tilde{z}\in\tilde{Z}}\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M_{P_{\tilde{z}}})}$$\scriptstyle{(0,\partial_{G})}$$\scriptstyle{((\phi_{z}-\operatorname{id})_{\tilde{z}\to
z})_{z}}$$\scriptstyle{(\operatorname{res}_{G}^{I_{\tilde{z}}})_{\tilde{z}}}$$\scriptstyle{(\operatorname{res}_{G}^{I_{\tilde{z}}})_{\tilde{z}}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\partial_{I_{z}}}$

###### Proof of the theorem.

The functors $j_{\star},j^{\star},i_{\star},i^{\star}$ can be extended to the
(non-derived) categories of complexes of constructible sheaves on $X$. These
functors will allow us to compute $\operatorname{R\Gamma}(X,\mathcal{F})$ in
the following way: first consider a short exact sequence involving
$\mathcal{F}$ in the (non-derived) category of complexes of constructible
sheaves on $X$, then compute the associated distinguished triangle in
$\operatorname{D}^{b}_{c}(\Lambda)$.

Consider the following short exact sequence of complexes of constructible
sheaves on $X$:

$0\to\mathcal{F}\to j_{\star}\mathscr{L}\oplus
i_{\star}i^{\star}\mathcal{F}\to i_{\star}\mathcal{Q}\to 0$

where $\mathcal{Q}\simeq i^{\star}j_{\star}\mathscr{L}$. The object
$\operatorname{R\Gamma}(X,\mathcal{F})[1]$ of
$\operatorname{D}^{b}_{c}(\Lambda)$ is the cone of the morphism

$\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})\oplus\operatorname{R\Gamma}(X,i_{\star}i^{\star}\mathcal{F})\to\operatorname{R\Gamma}(X,i_{\star}\mathcal{Q}).$

#### Computing $\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})$

The following distinguished triangle in $\operatorname{D}^{b}_{c}(X,\Lambda)$
[Stacks, 09XP]:

$\operatorname{R\Gamma}_{Z}(X,j_{\star}\mathscr{L})\to\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})\to\operatorname{R\Gamma}(U,\mathscr{L})\xrightarrow{+1}$

shows that $\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})[1]$ is the cone of
$\operatorname{R\Gamma}(U,\mathscr{L})\to\operatorname{R\Gamma}_{Z}(X,j_{\star}\mathscr{L})[1]$.
Let us now turn to the computation of $\operatorname{R\Gamma}(U,\mathscr{L})$
and $\operatorname{R\Gamma}_{Z}(X,j_{\star}\mathscr{L})$. According to 3.4,
$\operatorname{R\Gamma}(U,\mathscr{L})$ is represented by the total complex
associated to the double complex

${\operatorname{Hom_{cr}}(G,M^{0})}$${\operatorname{Hom_{cr}}(G,M^{1})}$${\operatorname{Hom_{cr}}(G,M^{2})}$${\cdots}$${M^{0}}$${M^{1}}$${M^{2}}$${\cdots}$

which is

$M^{0}\to M^{1}\oplus\operatorname{Hom_{cr}}(G,M^{0})\to
M^{2}\oplus\operatorname{Hom_{cr}}(G,M^{1})\to\dots$

For each integer $s\in\\{1\dots t\\}$,
$\operatorname{R\Gamma}_{Z}(X,j_{\star}\mathscr{L}^{s})$ is represented by the
one-term complex

$\bigoplus_{\tilde{z}\in\tilde{Z}}\operatorname{H}^{2}_{\tilde{z}}(X,\nu^{\star}j_{\star}\mathscr{L}^{s})[-2]$

according to 4.1. By 4.4 and 2.7,
$\operatorname{H}^{2}_{\tilde{z}}(X,\nu^{\star}j_{\star}\mathscr{L}^{s})=\operatorname{H}^{1}(I_{\tilde{z}},M^{s})=\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M^{s}_{P_{\tilde{z}}})$.
It follows that $\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})$ is
represented by the complex

$M^{0}\xrightarrow{(\partial_{M}^{0},\partial_{G}^{0})}M^{1}\oplus\operatorname{Hom_{cr}}(G,M^{0})\xrightarrow{(\partial_{M}^{1},\partial_{G}^{0}-\partial_{M}^{0},\operatorname{res}_{G}^{I_{\tilde{z}}})}M^{2}\oplus\operatorname{Hom_{cr}}(G,M^{1})\oplus\bigoplus_{\tilde{z}\in\tilde{Z}}\operatorname{H}^{1}(I_{\tilde{z}},M^{0})\to\dots$

#### Representing $\operatorname{R\Gamma}(X,i_{\star}i^{\star}\mathcal{F})$

For each integer $s\in\\{1\dots t\\}$,
$\operatorname{R\Gamma}(X,i_{\star}i^{\star}\mathcal{F}^{s})$ is simply
represented by the one-term complex
$\operatorname{H}^{0}(Z,\mathcal{F}^{s})[0]$. Therefore
$\operatorname{R\Gamma}(X,i_{\star}i^{\star}\mathcal{F})$ is represented by
the complex

$\bigoplus_{z\in Z}\mathcal{F}^{0}_{z}\to\bigoplus_{z\in
Z}\mathcal{F}^{1}_{z}\to\bigoplus_{z\in Z}\mathcal{F}^{2}_{z}\to\cdots$

whose differentials are those of $\mathcal{F}^{\bullet}$.

#### Representing $\operatorname{R\Gamma}(X,i_{\star}\mathcal{Q})$

We know that
$\operatorname{R\Gamma}(X,i_{\star}\mathcal{Q}^{s})=\operatorname{R\Gamma}(X,i_{\star}i^{\star}j_{\star}\mathscr{L}^{s})$
is represented by the one-term complex
$\bigoplus_{z\in{\tilde{Z}}}\operatorname{H}^{0}(I_{z},M^{s})$ concentrated in
degree zero. However, in order to be able to express the map
$\operatorname{R\Gamma}_{Z}(X,j_{\star}\mathscr{L})\to\operatorname{R\Gamma}(X,i_{\star}\mathcal{Q}^{s})$
as a morphism of complexes, we will write
$\operatorname{R\Gamma}(X,i_{\star}\mathcal{Q}^{s})$ as the following three-
term complex:

$\bigoplus_{{\tilde{z}}\in{\tilde{Z}}}M^{s}\to\bigoplus_{{\tilde{z}}\in{\tilde{Z}}}\operatorname{Hom_{cr}}(I_{\tilde{z}}/P_{\tilde{z}},M^{s}_{P_{\tilde{z}}})\to\bigoplus_{{\tilde{z}}\in{\tilde{Z}}}\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M^{s}_{P_{\tilde{z}}})$

whose first differential is the one sending $(m^{s}_{\tilde{z}})_{\tilde{z}}$
to $(g_{\tilde{z}}\mapsto(g_{\tilde{z}}\cdot
m_{\tilde{z}}^{s}-m_{\tilde{z}}^{s}))_{\tilde{z}}$, and the second one is the
usual quotient map. By the same arguments as previously,
$\operatorname{R\Gamma}(X,i_{\star}\mathcal{Q})$ is thus represented by the
complex

$\bigoplus_{{\tilde{z}}\in{\tilde{Z}}}M^{0}\to\bigoplus_{{\tilde{z}}\in{\tilde{Z}}}(M^{1}\oplus\operatorname{Hom_{cr}}(I_{\tilde{z}}/P_{\tilde{z}},M^{0}_{P_{\tilde{z}}}))\to\bigoplus_{{\tilde{z}}\in{\tilde{Z}}}(M^{2}\oplus\operatorname{Hom_{cr}}(I_{\tilde{z}}/P_{\tilde{z}},M^{1}_{P_{\tilde{z}}})\oplus\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M^{0}_{P_{\tilde{z}}}))\to\cdots$

#### Putting everything together

The complex $\operatorname{R\Gamma}(X,\mathcal{F})[1]$ is the cone of the
following morphism of complexes of $\Lambda$-modules:

${\cdots}$${M^{s}\oplus\operatorname{Hom_{cr}}(G,M^{s-1})\oplus\bigoplus_{z\in
Z}\mathcal{F}_{z}^{s}\oplus\bigoplus_{\tilde{z}\in{\tilde{Z}}}\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M_{P_{\tilde{z}}}^{s-2})}$${\cdots}$${\cdots}$${\bigoplus_{\tilde{z}\in{\tilde{Z}}}\left(M^{s}\oplus\operatorname{Hom_{cr}}(I_{\tilde{z}}/P_{\tilde{z}},M^{s-1}_{P_{\tilde{z}}})\oplus\operatorname{H}^{1}(I_{\tilde{z}}/P_{\tilde{z}},M^{s-2}_{P_{\tilde{z}}})\right)}$${\cdots}$

sending $(a,b,c_{z},d_{\tilde{z}})$ to
$(a-\phi_{z}(c),\operatorname{res}_{G}^{I_{z}}(b),d)$ where $\nu(\tilde{z})=z$
and $\operatorname{res}_{G}^{I_{z}}$ denotes the composite map
$\operatorname{Hom_{cr}}(G,M)\to\operatorname{Hom_{cr}}(I_{z},M)\to\operatorname{Hom_{cr}}(I_{z}/P_{z},M^{s}_{P_{z}})$
defined in 2.8. ∎

######  Remark 4.7.

A cover $V\to U$ as in the theorem may be computed in the following way:
consider an étale Galois cover $W\to U$ such that $\mathscr{L}|_{W}$ is
constant, and set $V=W^{\langle n\rangle}$. While $W^{\langle n\rangle}$ is
the canonical choice, we may for complexity reasons choose any of its
subcovers that still satisfy the three required properties listed above. Here
is one example of such a subcover. The action of $\operatorname{Gal}(k|k_{0})$
on $\operatorname{H}^{1}(U,\mathscr{L})$ factors through a finite quotient
$\operatorname{Gal}(k_{1}|k_{0})$. The image of
$\operatorname{H}^{1}(U,\mathscr{L})$ in $\operatorname{H}^{1}(W,\Lambda)$ is
still defined over $k_{1}$, and we may construct a Galois subcover of
$W^{\langle n\rangle}\to U$ by taking $n$-th roots only of functions that are
defined over $k_{1}$. This will be used in the case of finite fields in
section 5.6.

###  4.3 Computing the Galois action

Let $k_{0}$ be a perfect field, let $k$ be an algebraic closure of $k_{0}$ and
$\mathfrak{G}_{0}=\operatorname{Gal}(k|k_{0})$. Consider a curve $X_{0}$ over
$k_{0}$. For the sake of simplicity, we consider a single sheaf
$\mathcal{F}_{0}$ on $X_{0}$. The general case of complexes of sheaves is
handled in the same way. Let $X=X_{0}\times_{k_{0}}k$ and
$\mathcal{F}=\mathcal{F}_{0}|_{X}$. This section is dedicated to the
description of a complex of $\Lambda[\mathfrak{G}_{0}]$-modules representing
$\operatorname{R\Gamma}(X,\mathcal{F})$. Let $U_{0}$ be an affine open subset
of $X_{0}$ on which $\mathcal{F}$ is locally constant, and $W_{0}\to U_{0}$ be
a Galois cover such that $\mathcal{F}_{0}|_{W_{0}}$ is constant. Let $Z_{0}$
be the reduced closed complement of $U_{0}$ in $X_{0}$. Denote by $U,W,Z$
their base changes to $k$. Consider the cover $V=W^{\langle n\rangle}$, and
the complex representing $\operatorname{R\Gamma}(X,\mathcal{F})$ computed
above using $V$.

#### The easy case: when $W_{0}$ has a rational point $w_{0}$

In that case, $W$ is connected. The terms of
$\operatorname{R\Gamma}(X,\mathcal{F})$ consist of cohomology groups of
$G=\operatorname{Aut}(V|U)$ or $I_{z}/P_{z}$, where $z$ is a closed point at
infinity of $U$, with values in (a suitable quotient of) $M$. Understanding
the action of $\mathfrak{G}_{0}$ on $G$ is straightforward: let $\bar{w}$ be a
geometric point of $W$ whose image in $W$ is $w_{0}$, let $\bar{u}$ be its
image in $U$ and $\bar{v}$ a preimage of $\bar{w}$ in $V=W^{\langle
n\rangle}$. The group $\mathfrak{G}_{0}$ acts on
$\pi_{1}(W,\bar{w})\subset\pi_{1}(U,\bar{u})$ by functoriality, and since
$W^{\langle n\rangle}\to W$ is characteristic, this action restricts to
$\pi_{1}(V,\bar{v})$. Moreover, for $z\in X(k_{0})$, the group
$\mathfrak{G}_{0}$ acts naturally on $I_{z}/P_{z}$, which is canonically
isomorphic to $\mu_{e}(k)$, where $e$ is the ramification index of $V^{\langle
n\rangle}\to U$ above $z$. Finally, the group $M^{P_{z}}$ does not depend on
the choice of preimage of $Z$ in the compactification of $V^{\langle
n\rangle}$. For a point $z$ defined over an extension $k_{1}$ of $k_{0}$, we
need to consider its Galois orbit $T$: then $\mathfrak{G}_{0}$ acts naturally
on $\oplus_{t\in T}\operatorname{R\Gamma}(I_{t}/P_{t},M^{P_{t}})$, where the
preimages of $t$ whose inertia groups we compute have been chosen in the same
$\mathfrak{G}_{0}$-orbit. These considerations allow us to compute the action
of $\mathfrak{G}_{0}$ on each of the terms of the complex representing
$\operatorname{R\Gamma}(X,\mathcal{F})$.

#### The general case

In general, $W$ need not be connected, and the function field $k_{0}(W_{0})$
may contain a finite nontrivial Galois extension of $k_{0}$. Section 2.5 shows
how to construct a Galois covering $V_{0}\to U_{0}$ whose function field
contains a sufficiently large Galois extension $L$ of $k_{0}$ over which the
elements of $\operatorname{H}^{1}(W,\mu_{n})$ as well as the points at
infinity of $V\coloneqq V_{0}\times_{k_{0}}k$ are defined. This ensures that
any closed point of the smooth compactification of $V_{0}$ above a point of
$Z_{0}$ is exactly $L$. Consider a connected component $V_{c}$ of $V$. The
group
$G_{c}\coloneqq\operatorname{Aut}(V_{c}|U)=\ker(\operatorname{Aut}(V_{0}|U_{0})\to\operatorname{Gal}(L|k_{0}))$
is the stabiliser of $V_{c}$ in $G\coloneqq\operatorname{Aut}(V_{0}|U_{0})$.
Set $M_{0}=\operatorname{H}^{0}(V_{0},\mathcal{F}_{0})$ and
$M=\operatorname{H}^{0}(V,\mathcal{F})$; then $M$ is the induced
representation $\operatorname{ind}_{G_{c}}^{G}(M_{0})$, and Shapiro’s lemma
[Neu13, Th. 4.19] shows that the map

$\tau_{\leqslant 1}\operatorname{R\Gamma}(G_{c},M_{0})\to\tau_{\leqslant
1}\operatorname{R\Gamma}(G,M)$

in $\operatorname{D}^{b}_{c}(\Lambda)$ is a quasi-isomorphism. As an abelian
group, $M=M_{0}^{d}$ where $d$ is the number of connected components of $V$.
The group $\mathfrak{G}_{0}$ acts naturally on $M$ as it does on set the
connected components of $V$, in a way that is compatible with its action on
$G$. This allows to compute the action of $\mathfrak{G}_{0}$ on
$\tau_{\leqslant 1}\operatorname{R\Gamma}(G,M)$. Let $z\in Z_{0}$ be a closed
point with residue field $k_{1}$. Let $v$ be a preimage of $z$ in $V_{0}$.
Denote by $T\coloneqq\\{\tau(z),\tau\in\operatorname{Gal}(k_{1}|k_{0})\\}$ the
$\mathfrak{G}_{0}$-orbit of $z$. For each
$\tau\in\operatorname{Gal}(k_{1}|k_{0})$, consider a preimage $v_{\tau}$ of
$\tau(z)$ in $V_{0}$; we choose the $v_{\tau}$ to be in the same
$\mathfrak{G}_{0}$-orbit. Let $D_{\tau}\triangleleft G$ be the decomposition
group of $v_{\tau}$. The map $D_{\tau}\to\operatorname{Gal}(L^{\prime}|k_{0})$
is surjective, and its kernel is the inertia group $I_{\tau}$ of $v_{\tau}$,
so that $M=\operatorname{ind}_{I_{\tau}}^{D_{\tau}}M_{0}$. Shapiro’s lemma now
ensures that

$\operatorname{R\Gamma}(D_{\tau},M)=\operatorname{R\Gamma}(I_{\tau},M_{0})=\operatorname{R\Gamma}(I_{\tau}/P_{\tau},M_{0}^{P_{\tau}})$

in $\operatorname{D}^{b}_{c}(\Lambda)$. The group $\mathfrak{G}_{0}$ acts
naturally on the complex $\bigoplus_{\tau}\operatorname{R\Gamma}(D_{\tau},M)$,
whose cohomology groups are
$\operatorname{H}^{0}(T,j_{\star}\mathcal{F}|_{U})$ and
$\operatorname{H}^{2}_{T}(X,j_{\star}\mathcal{F}|_{U})$. Using these
notations, $\operatorname{R\Gamma}(X,\mathcal{F})[1]$ is isomorphic in
$\operatorname{D}^{b}_{c}(\Lambda[\mathfrak{G}_{0}])$ to the cone of the
following morphism of complexes:

${0}$${M\oplus\bigoplus_{T}\operatorname{H}^{0}(T,\mathcal{F})}$${\operatorname{Hom_{cr}}(G,M)}$${\bigoplus_{T}\bigoplus_{\tau}\operatorname{H}^{1}(I_{\tau}/P_{\tau},M_{P_{\tau}})}$${0}$${0}$${\bigoplus_{T}\bigoplus_{\tau}M_{P_{\tau}}}$${\bigoplus_{T}\bigoplus_{\tau}\operatorname{Hom_{cr}}(I_{\tau}/P_{\tau},M_{P_{\tau}})}$${\bigoplus_{T}\bigoplus_{\tau}\operatorname{H}^{1}(I_{\tau}/P_{\tau},M_{P_{\tau}})}$${0}$$\scriptstyle{\bigoplus_{T,\tau}(\operatorname{res}_{I_{\tau}}^{G}-\phi_{z})}$$\scriptstyle{(\partial_{G},0)}$$\scriptstyle{\bigoplus_{T,\tau}\operatorname{res}_{I_{\tau}}^{G}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\bigoplus_{z}\partial_{I_{\tau}}}$

where $T$ runs through the $\mathfrak{G}_{0}$-orbits in $Z$, and $\tau$ runs
through the $k_{0}$-automorphisms of the residue field of the closed points of
$T$.

###  4.4 Functoriality over $\operatorname{Spec}k$

Consider a morphism $\phi\colon X^{\prime}\to X$ of integral curves over an
algebraically closed field $k$. As usual, let $n$ be an integer invertible in
$k$, and denote by $\Lambda$ the ring $\mathbb{Z}/n\mathbb{Z}$. Let
$\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X$. Here is
how to compute a morphism of complexes of $\Lambda$-modules representing
$\operatorname{R\Gamma}(X,\mathcal{F})\to\operatorname{R\Gamma}(X^{\prime},f^{\star}\mathcal{F})$.
Let $U$ be an affine open subset of $X$ on which $\mathcal{F}$ is locally
constant. Let $W\to U$ be an étale Galois cover such that $\mathcal{F}|_{W}$
is constant, and $W^{\prime}\to U^{\prime}$ be the Galois closure of a
connected component of $W\times_{X}X^{\prime}$. Consider the Galois covers
$V=W^{\langle n\rangle}\to W$ and $V^{\prime}=(W^{\prime})^{\langle
n\rangle}\to W^{\prime}$. Given the construction of $V$ and $V^{\prime}$, the
map
$\operatorname{H}^{1}(W,\mu_{n})\to\operatorname{H}^{1}(W^{\prime},\mu_{n})$
defines a map $V^{\prime}\to V$.

By elementary Galois theory, there is a map
$\operatorname{Aut}(V^{\prime}|U^{\prime})\subseteq\operatorname{Aut}(V^{\prime}|U)\to\operatorname{Aut}(V|U)$.
For each point $z\in X-U$, choose a preimage $z_{V}$ of $z$ in the smooth
compactification of $V$. For each preimage $z^{\prime}$ of $z$ in
$X^{\prime}$, consider a preimage $z_{V}^{\prime}$ of $z^{\prime}$ in the
smooth compactification of $V^{\prime}$ whose image in $V$ is $z_{V}$.
Consider the inertia groups $P_{V}\triangleleft
I_{V}\subseteq\operatorname{Aut}(V|U)$ of $z_{V}$ and
$P_{V^{\prime}}\triangleleft
I_{V^{\prime}}\subseteq\operatorname{Aut}(V^{\prime}|U^{\prime})$ of
$z_{V}^{\prime}$. The map
$\operatorname{Aut}(V^{\prime}|U^{\prime})\to\operatorname{Aut}(V|U)$ induces
for each choice of $z,z^{\prime},z_{V},z_{V}^{\prime}$ a map
$I_{V^{\prime}}/P_{V^{\prime}}\to I_{V}/P_{V}$. The functoriality of the bar
resolution thus allows to compute the maps
$\operatorname{R\Gamma}(\operatorname{Aut}(V^{\prime}|U^{\prime}),M)\to\operatorname{R\Gamma}(\operatorname{Aut}(V|U),M)$
and
$\operatorname{R\Gamma}(I_{V}/P_{V},M^{P_{V}})\to\operatorname{R\Gamma}(I_{V^{\prime}}/P_{V^{\prime}},M^{P_{V^{\prime}}})$
that are needed to compute the morphism of complexes representing
$\operatorname{R\Gamma}(X,\mathcal{F})\to\operatorname{R\Gamma}(X^{\prime},\phi^{\star}\mathcal{F})$.

###  4.5 Cohomology of curves over a field of cohomological dimension 1

Consider a perfect field $k_{0}$ of cohomological dimension 1 such that
$\operatorname{H}^{1}(k_{0},\mathbb{Z}/n\mathbb{Z})$ is finite, e.g. a finite
field or the fraction field of a strictly henselian local DVR. Let $n$ be an
integer invertible in $k_{0}$. Denote by $\Lambda$ the ring
$\mathbb{Z}/n\mathbb{Z}$. Let $X_{0}$ be an integral curve over $k_{0}$, and
$\mathcal{F}_{0}^{\bullet}$ be a complex of constructible sheaves of
$\Lambda$-modules on $X_{0}$. Denote by $k$ the algebraic closure of $k_{0}$,
by $\mathfrak{G}_{0}$ the Galois group $\operatorname{Gal}(k|k_{0})$ and by
$X,\mathcal{F}$ the base change of $X_{0},\mathcal{F}_{0}$ to $k$. Theorem 4.5
shows how to compute a complex $M^{\bullet}$ of $\mathfrak{G}_{0}$-modules
representing $\operatorname{R\Gamma}(X,\mathcal{F})$. Recall that

$\operatorname{R\Gamma}(X_{0},\mathcal{F}_{0})=\operatorname{R\Gamma}(\mathfrak{G}_{0},\operatorname{R\Gamma}(X,\mathcal{F})).$

We described in section 3 how to determine a complex representing this object.
Let $k_{1}$ be a Galois extension of $k$ such that the action of
$\mathfrak{G}_{0}$ on $\operatorname{R\Gamma}(X,\mathcal{F})$ factors through
$\operatorname{Gal}(k_{1}|k_{0})$. Consider the extension $k_{1}^{\langle
n\rangle}$ of $k_{0}$ with Galois group
$\operatorname{H}^{1}(k_{1},\Lambda)^{\vee}$, and the Galois group
$G=\operatorname{Gal}(k_{1}^{\langle n\rangle}|k_{0})$. Then
$\operatorname{R\Gamma}(X_{0},\mathcal{F}_{0})$ is represented by the total
complex associated to the double complex
$B^{i,j}=\operatorname{Hom}_{\Lambda[G]}(\tau_{\geqslant-1}P_{G}^{-j}(\Lambda),M^{i})$,
where $P_{G}$ is the usual projective resolution of $\Lambda$ as a
$\Lambda[G]$-module.

######  Remark 4.8.

The same method also applies in theory to the general case where
$\operatorname{H}^{1}(k_{0},\Lambda)$ is infinite, using continuous group
cohomology and $\Lambda[[G]]$-modules; one particularly interesting case to
consider would be when $k_{0}$ is the function field of a curve over an
algebraically closed field. However, to go any further in practical
computations, one is quickly confronted with the issue of computing a
generating set of $\operatorname{H}^{1}(k_{0},\mathbb{Z}/n\mathbb{Z})\simeq
k_{0}^{\times}/(k_{0}^{\times})^{n}$.

##  5 Algorithmic aspects

###  5.1 Representing curves and sheaves

#### Representing curves

A smooth projective curve over a field $k_{0}$ is defined by a (possibly
singular) plane model given by a polynomial in two variables. A closed
subscheme of such a curve is defined by equations; an open subscheme is
defined by its closed complement. A morphism of smooth curves is given by a
morphism of plane models, i.e. by two polynomials in two variables. A curve
$X$ with ordinary singularities is defined by its normalisation ${\tilde{X}}$,
as well as the subsets of points of ${\tilde{X}}$ that have the same singular
image in $X$.

#### Representing sheaves

Let $X$ be a smooth curve over an algebraically closed fied $k$. A
constructible sheaf of abelian groups on the étale site of $X$ will be
described by the following gluing data:

  * $\bullet$

A closed 0-dimensional subscheme $Z$ of $X$, defined by an equation;

  * $\bullet$

A finite locally constant sheaf $\mathscr{L}$ on the open complement $U$ of
$Z$, defined by a Galois cover $V\to U$ and the action of the group
$G=\operatorname{Aut}(V|U)$ on the $\Lambda$-module
$M=\operatorname{H}^{0}(V,\mathscr{L})$;

  * $\bullet$

For each point $z\in Z$, the $\Lambda$-module $\mathcal{F}_{z}$ defined by
generators and relations;

  * $\bullet$

For each point $z\in Z$, the gluing morphism
$\phi_{z}\colon\mathcal{F}_{z}\to(j_{\star}\mathscr{L})_{z}=\operatorname{H}^{0}(I_{z},M)$,
where $I_{z}\subset G$ is the stabiliser of a preimage of $z$ in $V$.

This data also allows to represent morphisms and direct sums of sheaves in a
very straightforward manner, as well as to compute tensors products and
$\operatorname{Hom}$-sheaves; see [Lev22, §III.4] for more details. While this
representation of constructible sheaves might not be the first that comes to
mind, it is well suited to our computation of cohomology groups. The usual
ways of defining constructible sheaves (as cokernel of $f_{!}\Lambda\to
g_{!}\Lambda$ with $f,g$ étale or kernel of $f_{\star}\Lambda\to
g_{\star}\Lambda$ with $f,g$ finite) also admit an algorithmic representation,
which can be converted into this one (see [Lev22, §III.3]).

###  5.2 Computing the cohomology of $\mu_{n}$: existing algorithms

Our methods rely on existing algorithms which, given a smooth integral curve
$X$ over an algebraically closed field $k$, compute
$\operatorname{H}^{1}(X,\mu_{n})$. Recall that we need to be able to compute
it even for affine curves, which can prove to be a bit trickier than in the
projective case.

The most efficient algorithm computing $\operatorname{H}^{1}(X,\mu_{n})$,
developed by Couveignes, only applies to projective curves over finite fields,
and actually requires prior knowledge of the characteristic polynomial of the
Frobenius endomorphism of $X$; since it makes use of some properties of the
Frobenius and the group structure of $\operatorname{Pic}^{0}(X)[n]$, adapting
it to the cohomology of affine curves, or of curves over other types of fields
does not seem easy. Given a curve of genus $g$ over $\mathbb{F}_{q}$,
described by an ordinary plane model of degree $d$, it computes
$\operatorname{Pic}^{0}(X)(\mathbb{F}_{q})[n]$ in time polynomial in $d,g,\log
q,n$ [Cou09, Th. 1].

While it was also first described only for projective curves over finite
fields, Huang and Ierardi’s method [HI98] applies to more general settings.
Their algorithm constructs an affine scheme whose points correspond divisors
whose $n$-th power is the divisor of a rational function, and then finds a
point in each irreducible component of this scheme, which is enough to find a
representative of every $n$-torsion class in $\operatorname{Pic}^{0}(X)$. This
strategy readily adapts to the computation of division by $n$ in
$\operatorname{Pic}^{0}(X)$, thus allowing to compute the cohomology of
$\mu_{n}$ on an open subset of $X$. It is also independent of the chosen base
field. The complexity of their algorithm, computed when the base field is
$\mathbb{F}_{q}$, is polynomial in $n^{g},n^{d},\log q$ [Lev22, Prop. 4.3.3].

In the remainder of this article, we will denote by
$\mathsf{H1Const}(k_{0},n,d,g,r)$ or simply $\mathsf{H1Const}(X,n)$ the
complexity of the computation of $\operatorname{H}^{1}(X,\mu_{n})$, where $X$
is a smooth integral curve of genus $g$ over $k$, given by a degree $d$
polynomial in $k_{0}[x,y]$, with $r$ points at infinity. We will also denote
by $\mathsf{Root(k_{0},n,d)}$ the complexity of computing an $n$-th root of an
element in a degree $d$ extension of $k_{0}$.

###  5.3 Computing in $\operatorname{H}^{1}(X,\mu_{n})$

Let $U_{0}$ be a smooth integral geometrically irreducible curve over a
perfect field $k_{0}$, and $U$ its base change to an algebraic closure $k$ of
$k_{0}$. Let $X$ be the smooth compactification of $U$. Let $n$ be an integer
invertible in $k$, and $\Lambda$ be the ring $\mathbb{Z}/n\mathbb{Z}$. Denote
by $P_{0},\dots,P_{r}$ the points of $X-U$. Recall that the elements of
$\operatorname{H}^{1}(U,\mu_{n})$ are (equivalence classes of) pairs $(D,f)$
where $f$ is a rational function on $U$ such that $\mathrm{div}_{U}(f)=nD$.
The pair $(D,f)$ is trivial in $\operatorname{H}^{1}(U,\mu_{n})$ precisely
when $f$ is an $n$-th power. The sum $(D,f)+(D^{\prime},f^{\prime})$ is
defined by $(D+D^{\prime},ff^{\prime})$.

Here is how to compute the coordinates of an element of
$\operatorname{H}^{1}(U,\mu_{n})$ in a given basis using the Weil pairing, as
is done in the projective case in [Cou09, §8]. The Weil pairing

$e_{n}\colon\operatorname{H}^{1}(U,\mu_{n})\times\operatorname{H}^{1}_{c}(U,\mu_{n})\to\mu_{n}$

is nondegenerate. In this context,

$\operatorname{H}^{1}_{c}(U,\mu_{n})=\frac{\\{(D,f)\in\operatorname{Div}(U)\times
k(X)^{\times}\mid
nD=\mathrm{div}(f),f(P_{0})=...=f(P_{r})=1\\}}{\\{(D,f^{n})\text{ where
}f(P_{0})=...=f(P_{r})=1\\}}$

sits in the short exact sequence

$0\to\frac{\mu_{n}(k)^{r}}{\mu_{n}(k)}\to\operatorname{H}^{1}_{c}(U,\mu_{n})\to\operatorname{H}^{1}(X,\mu_{n})\to
0$

and may be computed in the following way. Choose a primitive $n$-th root of
unity $\zeta\in k$. For each $i\in\\{1\dots r\\}$, consider a function
$g_{i}\in k(X)$ such that $g_{i}(P_{0})=\zeta^{-1}$, $g_{i}(P_{i})=\zeta$, and
$g_{i}(P_{j})=1$ for all $j\neq 0,i$. Then
$((\mathrm{div}(g_{1}),g_{1}^{n}),\dots,(\mathrm{div}(g_{r},g_{r}^{n}))$ is a
basis of the image of $\mu_{n}(k)^{r}/\mu_{n}(k)$ in
$\operatorname{H}^{1}_{c}(U,\mu_{n})$. The Weil pairing is computed as usual:
given $v=(D,f)\in\operatorname{H}^{1}(U,\mu_{n})$ and
$w=(E,g)\in\operatorname{H}^{1}_{c}(U,\mu_{n})$ where $f$ and $g$ are suitably
normalised,

$e_{n}(v,w)=\frac{f(E)}{g(D)}.$

Data: Smooth integral curve $U$ over alg. closed field $k$, smooth
compactification $X$ of $U$

The points $P_{0},\dots,P_{r}$ of $X-U$

Positive integer $n$ invertible in $k$

Basis $B=(v_{1},\dots,v_{2g+r})$ of $\operatorname{H}^{1}(U,\mu_{n})$, where
$v_{i}=(D_{i},f_{i})\in\operatorname{Div}(U)\times K^{\times}$ and
$(v_{1},\dots,v_{2g})$ is a basis of $\operatorname{H}^{1}(X,\mu_{n})$

Element $v_{0}\in\operatorname{H}^{1}(U,\mu_{n})$ represented by
$(D_{0},f_{0})\in\operatorname{Div}(U)\times K^{\times}$

Primitive $n$-th root of unity $\zeta\in k$

Result: Coordinates $(\alpha_{1},\dots,\alpha_{2g+r})\in\Lambda^{2g+r}$ of $v$
w.r.t. $B$

Function $h\in k(X)^{\times}$ such that $f=h^{n}f_{1}^{\alpha_{1}}\dots
f_{2g+r}^{\alpha_{2g+r}}$

for  _$i\in\\{1\dots r\\}$_ do

Compute function $f_{i}$ such that $f_{i}(P_{0})=\zeta^{-1}$,
$f_{i}(P_{i})=\zeta$, and $f_{i}(P_{j})=1$ for $j\neq 0,i$

Set $v_{1-i}:=(\mathrm{div}(g_{i}),g_{i}^{n})$

end for

Compute matrix $M=e_{n}(v_{i},v_{j})_{0\leqslant i\leqslant 2g+r,1-r\leqslant
i\leqslant 2g}\in\operatorname{Mat}_{(2g+r+1)\times(2g+r)}(\mu_{n}(k))$

Compute an element $(1,-\alpha_{1},\dots,-\alpha_{2g})\in\ker(M)$: then
$v_{0}=\sum_{i}{\alpha_{i}}v_{i}$ in $\operatorname{H}^{1}(U,\mu_{n})$

Compute Riemann-Roch space $L$ of $D_{0}-\sum_{i}\alpha_{i}D_{i}$

Pick $h\in L$: then $D_{0}-\sum_{i}\alpha_{i}D_{i}=\mathrm{div}(h^{-1})$

Compute $n$-th root $c\in k$ of $f_{0}h^{n}\prod_{i}f_{i}^{-\alpha_{i}}\in k$

return $\alpha_{1},\dots,\alpha_{2g+r}\in\Lambda$ and function $ch\in
k(X)^{\times}$

Algorithm 1 CoordinatesInBasis

######  Lemma 5.1.

Suppose all of the divisors $D_{0},\dots,D_{2g+r}$ are given as difference of
two effective divisors of degree $\leqslant m$, and the curve $X$ is given by
a plane model of degree $d$. This algorithm returns the coordinates of $v$ in
time
$\operatorname{Poly}(d,m,g,r)+\mathsf{Root}\left(k_{0},n,n^{(2g+r)^{2}}\right)$.

###### Proof.

Computing the functions $f_{i}$ using Lagrange interpolation, as well as the
matrix $M$ using the definition above, is straightforward. The kernel of $M$
is computed using standard linear algebra techniques over $\Lambda$ (using the
isomorphism $\mu_{n}(k)\to\Lambda$ given by $\zeta\mapsto 1$), which run in
polynomial time in the size of $M$. ∎

###  5.4 Construction of ${V^{\langle n\rangle}}$

Let $k_{0}$ be a perfect field, and $k$ be its algebraic closure. Let $X_{0}$
be an integral smooth projective curve over $k_{0}$, and $U$ an affine open
subscheme of $X$. Set $U\coloneqq U_{0}\times_{k_{0}}k$. Let $V$ be an étale
Galois cover of $U$. The following algorithm computes the cover ${V^{\langle
n\rangle}}\to V$ defined in section 2.1, as well as the group
$\operatorname{Aut}({V^{\langle n\rangle}}|U)$.

Data: Galois cover $V\to U$ of smooth integral curves over alg. closed field
$k$

Generating set $S$ of $\operatorname{Aut}(V|U)$

Integer $n$ invertible in $k$

Result: Generating set of $\operatorname{Aut}({V^{\langle n\rangle}}|U)$

Compute basis $(D_{i},f_{i})_{1\leqslant i\leqslant s}$ of
$\operatorname{H}^{1}(V,\mu_{n})$ (see section 5.2)

for  _$\sigma\in S$_ do

for  _$i\in\\{1\dots s\\}$_ do

Compute $\sigma^{\star}(D_{i},f_{i})$

Compute $h,\alpha_{1},\dots,\alpha_{s}$ such that
$\sigma^{\star}f_{i}=h_{i}^{n}f_{1}^{\alpha_{1}}\dots f_{s}^{\alpha_{s}}$
using Algorithm 1

end for

Define $\rho_{\sigma}\colon(x,y)\mapsto\sigma(x,y),z_{j}\mapsto
h_{j}z_{1}^{\alpha_{1}}\dots z_{s}^{\alpha_{s}}$

end for

for  _$i\in\\{1\dots s\\}$_ do

Define $\phi_{i}\colon(x,y)\mapsto(x,y),z_{i}\mapsto\zeta z_{i},(z_{j})_{j\neq
i}\mapsto(z_{j})_{j\neq i}$

end for

return $\\{\rho_{\sigma}\\}_{\sigma\in S}\cup\\{\phi_{i}\\}_{1\leqslant
i\leqslant s}$

Algorithm 2 nTorsCover

######  Proposition 5.2.

If $U$ has $r$ points at infinity and $V$ is given by an ordinary plane model
of degree $d$, Algorithm 2 computes a generating set of
$\operatorname{Aut}({V^{\langle n\rangle}}|U)$ in

$\mathsf{H1Const}(k_{0},n,d,(2g+r)[V:U],r[V:U])$

elementary operations.

###### Proof.

The genus of $V$ is bounded by $(2g+r)[V:U]$. The complexity of computing the
coordinates of the pullback of the divisors is polynomial-time in
$n,[V:U](2g+r),d$, hence dominated by that of computing a basis of
$\operatorname{H}^{1}(V,\mu_{n})$. ∎

Here is how, once $G^{\langle n\rangle}=\operatorname{Aut}(V^{\langle
n\rangle}\to U)$ has been computed, to compute the preimages of points of $X$
in the smooth compactification of ${V^{\langle n\rangle}}$, as well as their
inertia group. This is done by considering a suitable explicit model of
${V^{\langle n\rangle}}$. Recall that the function field of ${V^{\langle
n\rangle}}$ is obtained from that of $V$ by adjoining $n$-th roots of
functions $f_{1},\dots,f_{t}\in k(V)$. Write $f_{i}=\frac{g_{i}}{h_{i}}$,
where $g_{i},h_{i}\in k[x,y]$. Denote by $Z$ (resp. $W$, resp. $W^{\langle
n\rangle}$) the sets of points at infinity of $U$ (resp. $V$, resp.
$V^{\langle n\rangle}$). Consider a point $z\in Z$, and a preimage $w$ of $z$
in $W$. Replacing $f_{i}$ with $h_{i}^{n}f_{i}$ if necessary, we may suppose
$h_{i}(w)\neq 0$. Then the affine curve given by the equation of $V$ and
$h_{i}z_{i}^{n}-g_{i}$ contains $n^{t-1}$ preimages of $w$, which are
nonsingular. Given one of these preimages $w$, the inertia subgroup $I_{w}$
can be computed simply by evaluating the elements of $G^{\langle n\rangle}$ at
$w$.

Data: Galois cover $V\to U$ of smooth integral curves over alg. closed field
$k$, with smooth compactification $Y\to X$ Integer $n$ invertible in $k$

Group $\operatorname{Aut}({V^{\langle n\rangle}}|U)$ and basis
$(D_{i},f_{i}=\frac{g_{i}}{h_{i}})_{1\leqslant i\leqslant s}$ of
$\operatorname{H}^{1}(V,\mu_{n})$

Point $z$ in compactification of $U$, preimage $w$ of $z$ in compactification
of $V$

Result: Preimage $w^{\langle n\rangle}$ of $w$ in compactification of
${V^{\langle n\rangle}}$

Generating set of inertia group $I^{\langle
n\rangle}\subset\operatorname{Aut}({V^{\langle n\rangle}}|U)$ of $w^{\langle
n\rangle}$

for  _$i\in\\{1\dots s\\}$_ do

if  _$h_{i}(w)=0$_ then

$g_{i}\leftarrow h_{i}^{n-1}g_{i},h_{i}\leftarrow 1$

end if

Compute root $t_{i}$ of $h_{i}(z)T^{n}-g_{i}(z)\in k[T]$

end for

Set $w^{\langle
n\rangle}=(w,t_{1},\dots,t_{s})\in\operatorname{Spec}Y[z_{1},\dots,z_{n}]/(h_{i}z_{i}^{n}-g_{i})$

$I^{\langle n\rangle}\coloneqq\\{\sigma\in G^{\langle
n\rangle}\mid\sigma(w^{\langle n\rangle})=w^{\langle n\rangle}\\}$

return $w^{\langle n\rangle},I^{\langle n\rangle}$

Algorithm 3 InertiaGroup

######  Lemma 5.3.

Algorithm 3 returns a preimage $w^{\langle n\rangle}$ of $w$ in the smooth
compactification of ${V^{\langle n\rangle}}$ and its stabiliser in
$[V:U]\left((n^{2g+r}+(2g+r)\mathsf{Root}\left(k_{0},n,n^{([V:U](2g+r))^{2}}\right)\right)$
elementary operations.

###### Proof.

Computing $w^{\langle n\rangle}$ requires $[V:U](2g+r)$ computations of $n$-th
roots in $k$. Computing $I^{\langle n\rangle}$ requires $[V:U]n^{2g+r}$
function evaluations. ∎

###  5.5 Computation of $\operatorname{R\Gamma}$

Let $k_{0}$ be a perfect field, and $k$ be its algebraic closure. Let $n$ be
an integer invertible in $k$, and denote by $\Lambda$ the ring
$\mathbb{Z}/n\mathbb{Z}$. Let $X_{0}$ be an integral curve over $k_{0}$ with
ordinary singularities, and $X=X_{0}\times_{k_{0}}k$. Consider a complex
$\mathcal{F}_{0}=[\mathcal{F}_{0}^{0}\to\dots\to\mathcal{F}_{0}^{t}]$ of
constructible sheaves of $\mathbb{Z}/n\mathbb{Z}$-modules on $X_{0}$, and set
$\mathcal{F}\coloneqq(\mathcal{F}_{0})|_{X}$. Let $U$ be a smooth open affine
subscheme of $X$ such that $\mathcal{F}|_{U}$ is a complex of locally constant
sheaves. Let $r$ be the number of points of $X-U$. Let $V\to U$ be an étale
Galois cover such that, for each integer $i\in\\{0\dots t\\}$, the sheaf
$\mathcal{F}^{i}|_{U}$ is constant. Denote by $G$ the automorphism group of
$V\to X$. The following algorithm computes the cohomology complex
$\operatorname{R\Gamma}(X,\mathcal{F})$ described in Theorem 4.5.

Data: Integral curve $X$ over alg. closed field $k$

Integer $n$ invertible in $k$

Constructible sheaf $\mathcal{F}$ on $X$ as described in section 5.1:

affine open $U\subset X$ where $\mathcal{F}$ is locally constant,

Galois cover $V\to U$ which trivialises
$\mathscr{L}\coloneqq\mathcal{F}|_{U}$,

Galois group $\operatorname{Aut}(V|U)$ and inertia subgroups $I_{z}\subset G$
for $z\in X-U$,

Generic fibre $M$ of $\mathcal{F}$ with action of $\operatorname{Aut}(V|U)$,

fibres $\mathcal{F}_{z}$ for $z\in X-U$,

adjunction units $\phi_{z}\colon\mathcal{F}_{z}\to M^{I_{z}}$.

Result: Complex of $\Lambda$-modules representing
$\operatorname{R\Gamma}(X,\mathcal{F})$

Compute $\operatorname{Aut}({V^{\langle n\rangle}}|U)$ using Algorithm 2

Compute inertia subroups $I_{z}^{\prime}\subset\operatorname{Aut}({V^{\langle
n\rangle}}|U)$ using Algorithm 3

Using linear algebra, compute
$\operatorname{Hom_{cr}}(\operatorname{Aut}({V^{\langle n\rangle}}|U),M)$ and
$\operatorname{Hom_{cr}}(I_{z}^{\prime},M)$ for $z\in X-U$

Compute the morphism
$\Psi\colon\operatorname{R\Gamma}(X,i_{\star}i^{\star}\mathcal{F})\oplus\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})\to\operatorname{R\Gamma}(X,i_{\star}i^{\star}j_{\star}\mathscr{L})$
of Theorem 4.5

return $\operatorname{Cone}(\Psi)[-1]$

Algorithm 4 RGamma

######  Theorem 5.4.

Let $m$ be an integer such that $M$ and the fibres $\mathcal{F}_{z}^{i}$,
$z\in Z$, $i\in\\{0\dots t\\}$ are given by at most $m$ generators. Denote by
$d$ the degree of an ordinary plane model of $V$. Algorithm 4 computes a
complex of $\Lambda$-modules representing
$\operatorname{R\Gamma}(X,\mathcal{F})$ in

$\mathsf{H1const}(k_{0},n,d,[V:U](2g+r))+\operatorname{Poly}\left((n^{[V:U](2g+r))^{2}},m,t\right)+[V:U](2g+r)\mathsf{Root}\left(k_{0},n,n^{([V:U](2g+r))^{2}}\right)$

elementary operations. When $k_{0}=\mathbb{F}_{q}$, this number is bounded by

$\operatorname{Poly}\left(n^{([V:U](2g+r))^{2}},n^{d},m,\log q,t\right).$

###### Proof.

This is just putting together the complexities of the previous algorithms,
taking into account that the computation of modules of crossed homomorphisms
is done using linear algebra over $\Lambda$. In order to bound the number in
the case of a finite field, we use the complexity of Huang and Ierardi’s
algorithm to compute $\operatorname{H}^{0}(V,\mu_{n})$. ∎

######  Remark 5.5.

The only existing algorithm computing $\operatorname{H}^{1}(X,\mathcal{F})$
when $\mathcal{F}$ is locally constant is Jin’s algorithm; its complexity is
exponential in $|M|^{\log|M|}$ [Jin20, Th. 1.2], to which the complexity of
our algorithm compares favourably.

###  5.6 Improving complexity when $k_{0}$ is finite

Here we consider the case where $k_{0}=\mathbb{F}_{q}$ is a finite field,
$X_{0}$ is a smooth curve over $k_{0}$, and $\mathcal{F}_{0}$ is a
constructible sheaf of $\mathbb{F}_{\ell}$-vector spaces on $X_{0}$, where
$\ell$ is a prime not dividing $q$. Let $U_{0}$ be an open subset of $X_{0}$
on which $\mathcal{F}_{0}$ is locally constant. Denote by $V_{0}\to U_{0}$ an
étale Galois cover such that $\mathcal{F}_{0}|_{U_{0}}$ is constant with fibre
$M$. Write $m=\dim_{\mathbb{F}_{\ell}}(M)$. Denote by
$k=\overline{\mathbb{F}_{q}}$ an algebraic closure of $k_{0}$, and by $X,U,V$
the base changes of $X_{0},U_{0},V_{0}$ to $k$. Denote by $g$ the genus of $X$
and by $r$ the number of points $X-U$.

######  Lemma 5.6.

Set $D=|\operatorname{GL}_{m(2g+r)}(\mathbb{F}_{\ell})|$ and $Q=q^{D}$.
Consider a basis $(D_{1},f_{1}),\dots,(D_{s},f_{s})$ of the
$\mathbb{F}_{\ell}$-vector space
$\operatorname{H}^{1}(V,\mu_{\ell})(\mathbb{F}_{Q})$, that is, the subspace of
elements of $\operatorname{H}^{1}(V,\mu_{\ell})$ invariant under the action of
$\operatorname{Gal}(k|\mathbb{F}_{Q})$. Denote by $V_{D}^{\langle\ell\rangle}$
the étale Galois covering of $V$ defined by the function field extension
$k(V)(\sqrt[\ell]{f_{1}},\dots,\sqrt[\ell]{f_{r}})$. The map
$\operatorname{H}^{1}(U,\mathcal{F}|_{U})\to\operatorname{H}^{1}(V_{D}^{\langle\ell\rangle},\mathcal{F}|_{V_{D}^{\langle\ell\rangle}})$
is trivial.

###### Proof.

We know that the action of $\operatorname{Gal}(k|k_{0})$ factors through a
finite quotient $\operatorname{Gal}(k_{1}|k_{0})$, where $[k_{1}:k_{0}]$
divides
$\operatorname{Aut}_{\mathbb{F}_{\ell}}(\operatorname{H}^{1}(U,\mathcal{F}))$.
Recall the prime-to-$p$ fundamental group of $U$ is generated by at most
$2g+r$ elements. Now $\operatorname{H}^{1}(U,\mathcal{F})$ is a quotient of
$\operatorname{Hom_{cr}}(\pi_{1}(U)^{(p^{\prime})},M)$; its dimension as an
$\mathbb{F}_{\ell}$-vector space is bounded above by $m(2g+r)$, where $g$ is
the genus of $U$. Therefore,
$\operatorname{Aut}_{\mathbb{F}_{\ell}}(\operatorname{H}^{1}(U,\mathcal{F}))$
injects into $\operatorname{GL}_{m(2g+r)}(\mathbb{F}_{\ell})$. Note that since
$\ell$ divides $Q-1$, the field $\mathbb{F}_{Q}$ contains a primitive
$\ell^{\rm th}$ root of unity $\zeta$, and the isomorphism
$\operatorname{H}^{1}(V,\mathbb{F}_{\ell})\to\operatorname{H}^{1}(V,\mu_{\ell})$
defined by $1\mapsto\zeta$ is
$\operatorname{Gal}(k|\mathbb{F}_{Q})$-equivariant. Since the
$U$-automorphisms of $V$ are defined over $\mathbb{F}_{q}$, the set
$\operatorname{H}^{1}(V,\mu_{\ell})(\mathbb{F}_{Q})$ is stable under the
action of $\operatorname{Aut}(V|U)$, and $V^{\prime}\to U$ is still Galois.
The construction of $V_{D}^{\langle\ell\rangle}$ ensures that
$\operatorname{H}^{1}(V,\mathbb{F}_{\ell})(\mathbb{F}_{Q})\to\operatorname{H}^{1}(V_{D}^{\langle\ell\rangle},\mathbb{F}_{\ell})$
is trivial. The $m$ copies of $\mathcal{F}|_{V}\simeq\mathbb{F}_{\ell}^{m}$
being stable under the action of $\operatorname{Gal}(k|k_{0})$, the map

$\operatorname{H}^{1}(U,\mathcal{F})\to\operatorname{H}^{1}(V_{D}^{\langle\ell\rangle},\mathcal{F})\xrightarrow{\sim}\operatorname{H}^{1}(V_{D}^{\langle\ell\rangle},\mathbb{F}_{\ell})^{m}$

factors through
$\operatorname{H}^{1}(V,\mathbb{F}_{\ell})(\mathbb{F}_{Q})^{m}$, and is also
trivial. ∎

Hence, we may use $V_{D}^{\langle\ell\rangle}$ instead of
$V^{\langle\ell\rangle}$ in Algorithm 4. Note that
$D\leqslant\ell^{(m(2g+r))^{2}}$. For the moment, we do not know how to
compute $\operatorname{H}^{1}(V,\mathbb{F}_{\ell})(\mathbb{F}_{Q})$ when $V$
is affine. In the projective case, Couveignes’ algorithm does so in time
polynomial in $\ell$, the genus of $Y$ and $\log(Q)$, assuming the
characteristic polynomial of the $Q$-Frobenius on $V$ is known.

###  5.7 Potential application: point counting on surfaces

Let $X_{0}$ be a smooth projective surface over a finite field
$k_{0}=\mathbb{F}_{q}$. Denote by $k$ its algebraic closure, and set
$X=X_{0}\times_{k_{0}}k$. Consider the problem of computing $|X(k_{0})|$. The
usual $\ell$-adic approach, as in Schoof’s algorithm, is to compute this
number modulo $\ell$ for enough primes $\ell$ up to $O(\log q)$.

The Lefschetz theorem reduces this question to computing the trace of the
Frobenius on $\operatorname{H}^{i}(X,\mathbb{F}_{\ell})$. The classical way of
computing these groups, as in [Mil80, §V.3], is by using a Lefschetz pencil,
which yields a fibration $\pi\colon\tilde{X}\to\mathbb{P}^{1}$, where
$\tilde{X}$ is a blowup of $X$ at a finite number of points. Edixhoven
conjectured in [EC11, Epilogue] that this strategy might allow to compute
$|X(k_{0})|$ in time polynomial in $\log(q)$. Here is where we stand on this
conjecture. The sheaf $\mathcal{F}\coloneqq{\rm
R}^{1}\pi_{\star}\mathbb{F}_{\ell}$ is a constructible sheaf on the projective
line. It is locally constant on the open subset $U$ of $\mathbb{P}^{1}$ over
which the fibres of $\pi$ are smooth curves. For $z\in\mathbb{P}^{1}-U$, we
know how to compute $\operatorname{R\Gamma}(X_{z},\mathbb{F}_{\ell})$.
Moreover, given an explicit description of $\mathcal{F}$, we know how to
compute $\operatorname{R\Gamma}(\mathbb{P}^{1},\mathcal{F})$. Denote by
$X_{\bar{\eta}}$ the generic fibre of $\pi$, and by $g_{\bar{\eta}}$ its
genus. A trivialising cover $V\to U$ of $\mathcal{F}|_{U}$ is given by the
normalisation of $U$ in an extension of $K$ of $k(t)$ over which
$\operatorname{Pic}(X_{{\bar{\eta}}})[\ell]$ is defined. The following data
helps us estimate the complexity we need for the different steps of the
algorithms:

  * •

the degree $[K:k(t)]$ is smaller than $\ell^{4g_{\bar{\eta}}^{2}}$;

  * •

the number $r=|\mathbb{P}^{1}-U|$ only depends on $\pi$ and not on $\ell$;

  * •

the genus $g_{V}$ of $V$ is bounded by $r\ell^{4g_{\bar{\eta}}^{2}}$.

Computing the whole cover $V^{\langle\ell\rangle}$ would be too costly.
However, as suggested in section 5.6, it is sufficient to compute compute a
subcover $V_{D}^{\langle\ell\rangle}$ of $V^{\langle\ell\rangle}$ using only
elements of $\operatorname{H}^{1}(V,\mu_{\ell})$ defined over a degree
$D=O(\ell^{4g_{\bar{\eta}}^{2}r^{2}})$ extension $\mathbb{F}_{Q}$ of
$\mathbb{F}_{q}$. Hence, if we could compute:

  * •

$\operatorname{H}^{1}(X_{\bar{\eta}},\mu_{\ell})$ in time polynomial in $\ell$
and $\log(q)$,

  * •

$\operatorname{H}^{1}(V,\mu_{\ell})(\mathbb{F}_{Q})$ in time polynomial in
$\ell,\log Q$ and the genus of $V$,

then we should be able to compute the $\operatorname{H}^{i}(X,\mathcal{F})$
with their Frobenius action in time $\operatorname{Poly}(\ell,\log q)$. Mascot
recently described an algorithm to deal with the first item of the list
[Mas23, Alg. 2.2] using $p$-adic approximation; however, parts of his method
are not yet rigorous [Mas23, Rk. 4.3].

For the moment, this is nothing more than wishful thinking: all existing
algorithms to compute $\operatorname{H}^{1}(V,\mu_{\ell})$, even for
projective curves, have complexity exponential either in $\log(q)$ or in the
genus of $V$. However, there is some hope. Harvey’s algorithm [Har14, Th. 1],
which computes the zeta function of hyperelliptic curves, reaches an average
polynomial-time complexity. Combined with Couveignes’ algorithm, this allows
for an average polynomial-time complexity for the computation of
$\operatorname{H}^{1}(V,\mu_{\ell})(\mathbb{F}_{Q})$ in the case where $V$ is
a (projective) hyperelliptic curve.

##  6 First example: sheaves on subschemes of $\mathbb{P}^{1}$

###  6.1 The covering

Take $n=2$. Let $k_{0}$ be a field of odd characteristic. Consider the degree
2 (ramified) Galois covering

$\bar{f}\colon\begin{array}[t]{rcl}\mathbb{P}^{1}&\longrightarrow&\mathbb{P}^{1}\\\
y&\longmapsto&y^{2}\end{array}$

whose automorphism group is generated by $\tau\colon y\mapsto-y$. Set
$U=\mathbb{P}^{1}-\\{0,1,\infty\\}$ and $V=f^{-1}U=\mathbb{P}^{1}-\\{0,\pm
1,\infty\\}$, and consider the étale covering $f\colon V\to U$ induced by
$\bar{f}$.

#### Computation of ${V^{\langle n\rangle}}$

The group $\operatorname{H}^{1}(V,\mu_{2})\simeq\Lambda^{3}$ is generated by
the divisor-function pairs $(0-\infty,x),(1-\infty,x-1),(-1-\infty,x+1)$. The
covering ${V^{\langle n\rangle}}\to V$ with group
$\operatorname{H}^{1}(V,\Lambda)^{\vee}$ corresponds to the field extension
$k(\sqrt{x},\sqrt{x-1},\sqrt{x+1})/k(x)$. The corresponding cover of
$\bar{V}=\mathbb{P}^{1}$ is the map

$\begin{array}[]{ccc}\operatorname{Proj}k[y,z,t,h]/(z^{2}-(y^{2}-h^{2}),t^{2}-(y^{2}+h^{2}))&\longrightarrow&\operatorname{Proj}k[y,h]\\\
(y:z:t:h)&\longmapsto&(y^{2}:h^{2})\end{array}$

which is ramified above $0,\pm 1,\infty$.

#### Computation of $\operatorname{Aut}({V^{\langle n\rangle}}|U)$

The automorphism group $G\coloneqq\operatorname{Aut}({V^{\langle
n\rangle}}|U)$ has order 16; in order to compute all of its elements, it
suffices to compute a preimage of $\tau$ in $\operatorname{Aut}({V^{\langle
n\rangle}}|U)$. Such a preimage is given by
$\gamma\colon(y:z:t:h)\mapsto(\sqrt{-1}y:\sqrt{-1}t:\sqrt{-1}z:h)$. Let
$\sigma_{1}\colon y\mapsto-y$, $\sigma_{2}\colon z\mapsto-z$,
$\sigma_{3}\colon t\mapsto-t$ bethe obvious generators of
$\operatorname{Aut}({V^{\langle n\rangle}}|V)\triangleleft G$. Then
$\gamma_{2}=\sigma_{3}\gamma$ and $\gamma_{3}=\sigma_{3}\gamma$, which implies
that $\langle\sigma_{2},\sigma_{3}\rangle$ is normal in $G$. It is easy to
check that the composite map

$\langle\gamma\rangle\to G/\langle\sigma_{2},\sigma_{3}\rangle$

is an isomorphism; therefore,

$G=\langle\sigma_{2},\sigma_{3}\rangle\rtimes\langle\gamma\rangle.$

###  6.2 Ramification

#### Ramification

Let $Z,W,W^{\prime}$ denote the sets of points at infinity of $U,V,{V^{\langle
n\rangle}}$ respectively. The following table gives an overview of the
situation.

Points in $Z$ | $0$ | $1$ | $\infty$
---|---|---|---
| Preimages in $W$
---
Ramification
| $0$
---
index $2$
| $-1$
---
index $1$
| $1$
---
index $1$
| $\infty$
---
index $2$
| Preimages in $W^{\prime}$
---
Ramification
| 4 points
---
index 4
| 4 points
---
index 2
| 4 points
---
index 2
| 4 points
---
index 4
| A preimage in $W^{\prime}$
---
Its inertia group
| $P_{0}=(0,\sqrt{-1},1)$
---
$\langle\gamma\sigma_{2}\rangle\simeq\mu_{4}(k)$
| $P_{-1}=(\sqrt{-1},\sqrt{-2},0)$
---
$\langle\sigma_{3}\rangle\simeq\mu_{2}(k)$
| $P_{1}=(1,0,\sqrt{2})$
---
$\langle\sigma_{2}\rangle\simeq\mu_{2}(k)$
| $P_{\infty}=(1:1:1:0)$
---
$\langle\gamma\rangle\simeq\mu_{4}(k)$

The canonical isomorphism $I_{P_{0}}\to\mu_{4}(k)$ can be described explicitly
as follows. The function $y$ is a uniformiser of ${V^{\langle n\rangle}}$ at
$P_{0}=(0,\sqrt{-1},1)$. The orbit of $y$ under the action of
$I_{P_{0}}=\langle\gamma\sigma_{2}\rangle$ is $\\{\pm y,\pm\sqrt{-1}y\\}$.
Hence the set $\\{\frac{\sigma(y)}{y}(P_{0})\mid\sigma\in I_{P_{0}}\\}$ is
exactly $\mu_{4}(k)$, and the isomorphism $I_{P_{0}}$ to $\mu_{4}(k)$ sends an
element $\sigma\in I_{P_{0}}$ to $\frac{\sigma(y)}{y}(P_{0})$.

The generator $\sqrt{-1}$ of $\mu_{4}(k)$ exchanges the two copies of
$\Lambda$ in $M=\Lambda^{2}$. The $\Lambda$-module of crossed homomorphisms
$\mu_{4}(k)\to M$ is isomorphic to $\Lambda^{2}$, and $\tau_{\leqslant
1}\operatorname{R\Gamma}(I_{P_{0}},M)$ is represented by the following
complex.

$\begin{array}[]{rcl}\Lambda^{2}&\to&\Lambda^{2}\\\
(a,b)&\mapsto&\left[\sqrt{-1}\mapsto(a+b,a+b)\right]\end{array}$

The group $\mathcal{F}_{0}=\operatorname{H}^{0}(I_{P_{0}},M)$ is generated by
$(1,1)$, and
$\operatorname{H}^{2}_{0}(X,j_{\star}\mathcal{F})=\operatorname{H}^{1}(I_{P_{0}},M)$
is generated by the class of $(0,1)$. The complex $\tau_{\leqslant
1}\operatorname{R\Gamma}(I_{P_{\infty}},M)$ can be computed similarly. The
group $I_{P_{1}}$ is canonically isomorphic to $\mu_{2}(k)$, and acts
trivially on $M$. Therefore, $\tau_{\leqslant
1}\operatorname{R\Gamma}(I_{P_{1}},M)$ is represented by the following
complex.

$\begin{array}[]{rcl}\Lambda^{2}&\to&\Lambda^{2}\\\
(a,b)&\mapsto&0\end{array}$

###  6.3 Cohomology of a locally constant sheaf

The sheaf $\mathcal{F}\coloneqq f_{\star}\Lambda$ is locally constant on $U$,
trivialised by the covering $f\colon V\to U$ since
$f^{\star}f_{\star}\Lambda\simeq\Lambda^{2}$. It corresponds to the
$\operatorname{Aut}(V|U)$-module $\Lambda^{2}$, where the non-trivial element
of $\operatorname{Aut}(V|U)$ exchanges the two copies of $\Lambda$. Since $f$
is finite, ${\rm R}f_{\star}\Lambda=(f_{\star}\Lambda)[0]$ and there is a
canonical isomorphism

$\operatorname{R\Gamma}(U,f_{\star}\Lambda)=\operatorname{R\Gamma}(V,\Lambda).$

We therefore expect to find

$\operatorname{H}^{1}(U,\mathcal{F})=\operatorname{H}^{1}(V,\Lambda)\simeq\Lambda^{3}.$

#### Computing $\operatorname{R\Gamma}(U,\mathcal{F})$

We know that $\operatorname{R\Gamma}(U,\mathcal{F})$ is represented by the
following two-term complex:

$\Lambda^{2}\to\operatorname{Hom_{cr}}(G,\Lambda^{2}).$

A crossed homomorphism $f\colon G\to\Lambda^{2}$ is determined by the images
of $\sigma_{1},\sigma_{2},\sigma_{3},\gamma$. Using the relations
$\gamma\sigma_{1}=\sigma_{1}\gamma,\gamma\sigma_{2}=\sigma_{3}\gamma$ and
$\gamma^{2}=\sigma_{1}\sigma_{2}\sigma_{3}$, we see that such a map is
uniquely determined by a tuple $(a,a_{1},a_{2},a_{3})\in\Lambda^{4}$; the
corresponding map $f$ is defined by $f(\sigma_{1})=(a_{1},a_{1})$,
$f(\sigma_{2})=(a_{2},a_{3})$, $f(\sigma_{3})=(a_{3},a_{2})$ and
$f(\gamma)=(a,a+a_{1}+a_{2}+a_{3})$. The principal crossed homomorphisms
correspond to $(0,0,0,0)$ and $(1,0,0,0)$. Hence the complex above is
isomorphic to

$\begin{array}[]{rcl}\Lambda^{2}&\longrightarrow&\Lambda^{4}\\\
(a,b)&\longmapsto&(a+b,0,0,0)\end{array}$

and its cohomology groups are $\Lambda$ et $\Lambda^{3}$, as expected.

#### Computing the Galois action

The action of $\mathfrak{G}_{0}=\operatorname{Gal}(k|k_{0})$ on
$\operatorname{Aut}(Y|X)$ clearly factors through the quotient
$\operatorname{Gal}(k_{0}(\sqrt{-1})|k_{0})$. The latter group is generated by
$\phi\colon\sqrt{-1}\mapsto-\sqrt{-1}$. The automorphism $\phi$ acts trivially
on $\sigma_{1},\sigma_{2},\sigma_{3}$, and
$\phi\cdot\gamma=\sigma_{1}\sigma_{2}\sigma_{3}\gamma\colon(y,z,t)\mapsto(-\sqrt{-1}y,-\sqrt{-1}z,-\sqrt{-1}t)$.
The action of $\phi$ on $\Lambda^{2}$ is trivial, and its action on
$\operatorname{Hom_{cr}}(G,\Lambda^{2})\simeq\Lambda^{4}$ is $(\phi\cdot
f)(x)=\phi f(\phi^{-1}x)=f(\phi^{-1}x)$. Explicitly, since
$\phi\cdot\gamma=\sigma_{1}\sigma_{2}\sigma_{3}\gamma$, this action is given
by

$\phi\cdot(a,a_{1},a_{2},a_{3})=(a+a_{1}+a_{2}+a_{3},a_{1},a_{2},a_{3}).$

###  6.4 Cohomology of a constructible sheaf

We still consider a field $k_{0}$ of odd characteristic, where $-1$ is not a
square. Define $k_{1}=k_{0}(\sqrt{-1})$. Let us now consider the ramified
cover of projective curves $\bar{f}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$
defined by $f$, and the sheaf $\mathcal{F}\coloneqq\bar{f}_{\star}\Lambda$ on
$\mathbb{P}^{1}$. Since $\bar{f}$ is unramified outside $0,\infty$, the sheaf
$\mathcal{F}$ is locally constant on the open subset
$\mathbb{G}_{m}=\mathbb{P}^{1}-\\{0,\infty\\}$, and
$\mathscr{L}\coloneqq\mathcal{F}|_{\mathbb{G}_{m}}$ is trivialised by
$\bar{f}|_{\mathbb{G}_{m}}\colon\mathbb{G}_{m}\to\mathbb{G}_{m},y\mapsto
y^{2}$. The covering $\mathbb{G}_{m}^{\langle n\rangle}$ is also
$\mathbb{G}_{m}\to\mathbb{G}_{m},z\mapsto z^{2}$, and the composite

$\mathbb{G}_{m}^{\langle
n\rangle}\to\mathbb{G}_{m}\xrightarrow{f}\mathbb{G}_{m}$

is given by $z\mapsto z^{4}$. Its automorphism group is $G=\langle\gamma\colon
z\mapsto\sqrt{-1}z\rangle\simeq\mathbb{Z}/4\mathbb{Z}$, and the inertia
subgroups at $0$ and $\infty$ are both equal to $G$. Denote by $j$ the
inclusion $\mathbb{G}_{m}\to\mathbb{P}^{1}$. We have
$(j_{\star}\mathscr{L})_{0}=\Lambda$ and
$(j_{\star}\mathscr{L})_{\infty}=\Lambda$. The adjunction units
$\mathcal{F}_{0}\to(j_{\star}\mathscr{L})_{0}$ and
$\mathcal{F}_{\infty}\to(j_{\star}\mathscr{L})_{\infty}$ are the identity maps
$\Lambda\to\Lambda$.

#### Computing $\operatorname{R\Gamma}(\mathbb{G}_{m},\mathscr{L})$

The crossed homomorphisms $G\to M=\mathscr{L}_{{\bar{\eta}}}\simeq\Lambda^{2}$
are uniquely determined by the image of $(a,b)\in\Lambda^{2}$ under $\gamma$.
The usual cochain complex representing
$\operatorname{R\Gamma}(G,M)=\operatorname{R\Gamma}(\mathbb{G}_{m},\mathscr{L})$
is the following.

$\begin{array}[]{rcl}\Lambda^{2}&\longrightarrow&\operatorname{Hom_{cr}}(G,\Lambda^{2})\\\
(a,b)&\longmapsto&\left[\gamma\mapsto(a+b,a+b)\right]\end{array}$

Therefore $\operatorname{H}^{1}(G,M)$ is isomorphic to $\Lambda$, and the
kernel of the map $\Lambda^{2}\to\operatorname{H}^{1}(G,M)$ sending a crossed
homomorphism to its cohomology class is $\langle(1,1)\rangle$ ; this map can
be rewritten as

$\begin{array}[]{rcl}\Lambda^{2}&\longrightarrow&\Lambda\\\
(a,b)&\longmapsto&a+b.\end{array}$

#### Computing $\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})$

The element
$\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})[1]\in\operatorname{D}^{b}_{c}(X,\Lambda)$
is the cone of

$\tau_{\leqslant
1}\operatorname{R\Gamma}(G,M)\to\operatorname{H}^{1}(I_{0},M)[-1]\oplus\operatorname{H}^{1}(I_{\infty},M)[-1]=\Lambda^{2}[-1].$

Therefore, $\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})$ is represented by
the complex

$\Lambda^{2}\longrightarrow\Lambda^{2}\longrightarrow\Lambda^{2}$

where both morphisms are given by $(a,b)\mapsto(a+b,a+b)$.

#### Computing $\operatorname{R\Gamma}(X,\mathcal{F})$

Let us now turn to the computation of the map

$\operatorname{R\Gamma}(X,j_{\star}\mathscr{L})\oplus\operatorname{R\Gamma}(Z,i^{\star}\mathcal{F})\to\operatorname{R\Gamma}(Z,i^{\star}j_{\star}\mathscr{L}).$

On the onde hand,
$\operatorname{R\Gamma}(Z,i^{\star}\mathcal{F})=\operatorname{H}^{0}(Z,i^{\star}\mathcal{F})[0]=\mathcal{F}_{0}[0]\oplus\mathcal{F}_{\infty}[0]$.
On the other hand, $\operatorname{R\Gamma}(Z,i^{\star}j_{\star}\mathscr{L})$
is represented by the complex

${\Lambda^{2}\oplus\Lambda^{2}}$${\Lambda^{2}\oplus\Lambda^{2}}$${\Lambda^{2}}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$

where the arrows are given by
$\alpha^{\prime}\colon(a,b,c,d)\mapsto(a+b,a+b,c+d,c+d)$ and
$\beta^{\prime}\colon(a,b,c,d)\mapsto(a+b,c+d)$. Hence the map we were looking
for is

${\Lambda^{4}}$${\Lambda^{2}}$${\Lambda^{2}}$${\Lambda^{4}}$${\Lambda^{4}}$${\Lambda^{2}}$$\scriptstyle{\alpha}$$\scriptstyle{u}$$\scriptstyle{\beta}$$\scriptstyle{v}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$

where, writing the upper left-hand side term as
$\mathcal{F}_{0}\oplus\mathcal{F}_{\infty}\oplus M$, the arrows are given by

  * $\bullet$

$u\colon(a,b,c,d)\mapsto(a+c,a+d,b+c,b+d)$

  * $\bullet$

$\alpha\colon(a,b,c,d)\mapsto(c+d,c+d)$

  * $\bullet$

$\beta\colon(a,b)\mapsto(a+b,a+b)$

  * $\bullet$

$v\colon(a,b)\mapsto(a,b,a,b)$

  * $\bullet$

$\alpha^{\prime}\colon(a,b,c,d)\mapsto(a+b,a+b,c+d,c+d)$

  * $\bullet$

$\beta^{\prime}\colon(a,b,c,d)\mapsto(a+b,c+d)$.

Computing the cone of this morphism and shifting by 1 yields the following
complex, which represents $\operatorname{R\Gamma}(X,\mathcal{F})$.

${\Lambda^{4}}$${\Lambda^{6}}$${\Lambda^{6}}$${\Lambda^{2}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{2}}$

  * $\bullet$

$\partial_{0}\colon(a,b,c,d)\mapsto(c+d,c+d,a+c,b+c,a+d,b+d)$

  * $\bullet$

$\partial_{1}\colon(a,b,c,d,e,f)\mapsto(a+b,a+b,a+c+d,b+c+d,a+e+f,b+e+f)$

  * $\bullet$

$\partial_{2}\colon(a,b,c,d,e,f)\mapsto(a+c+d,b+e+f)$.

The cohomology groups of this complex are
$\operatorname{H}^{0}=\langle(1,1,1,1)\rangle$, $\operatorname{H}^{1}=0$ et
$\operatorname{H}^{2}=\langle\overline{(1,0,1,0,0,0)}\rangle\simeq\Lambda$.
This result was to be expected: we have computed the cohomology of
$(\mathbb{P}^{1}\xrightarrow{x\mapsto x^{2}}\mathbb{P}^{1})_{\star}\Lambda$,
which is the cohomology of the constant sheaf $\Lambda$ on $\mathbb{P}^{1}$.

#### Computing the Galois action

The action of $\operatorname{Gal}(k|k_{0})$ on
$\operatorname{R\Gamma}(X,\mathcal{F})$ factors through the quotient
$\operatorname{Gal}(k_{0}(\sqrt{-1})|k_{0})$ of $\operatorname{Gal}(k|k_{0})$.
Denote by $\sigma\colon\sqrt{-1}\mapsto-\sqrt{-1}$ the nontrivial element of
$\operatorname{Gal}(k_{0}(\sqrt{-1})|k_{0})$. The group groupe
$\operatorname{Gal}(k|k_{0})$ acts on $G$ by $\sigma\cdot\gamma=\gamma^{3}$,
and trivially on $M$. Its action on $\tau_{\leqslant
1}\operatorname{R\Gamma}(G,M)=\Lambda^{2}\to\Lambda^{2}$ is trivial on the
first term, and $(a,b)\mapsto(b,a)$ on the second. In particular, it acts
trivially on $\operatorname{H}^{1}(G,M)=\Lambda$. The action of $\sigma$ on
$\mathcal{F}_{0},\mathcal{F}_{\infty}$ is also trivial. hence the action of
$\operatorname{Gal}(k|k_{0})$ on the complex

$\Lambda^{4}\to\Lambda^{6}\to\Lambda^{6}\to\Lambda^{2}$

representing $\operatorname{R\Gamma}(X,\mathcal{F})$ is trivial on the first
and last terms,

$\sigma\cdot(a,b,c,d,e,f)=(b,a,c,d,e,f)$

on the second term, and

$\sigma\cdot(a,b,c,d,e,f)=(a,b,d,c,f,e)$

on the third term. In particular, $\operatorname{Gal}(k|k_{0})$ acts trivially
on $\operatorname{H}^{0}(X,\Lambda)$ and $\operatorname{H}^{2}(X,\Lambda)$, as
expected.

##  7 Second example: sheaves on subschemes of an elliptic curve

The examples in this section illustrate a non-trivial case in which we can
easily compute the cohomology of sheaves using only functions that are already
available in current computer algebra systems: when the locally constant part
of the sheaf is trivialised by a subscheme of a hyperelliptic curve.

###  7.1 The covering

Consider the finite field $k_{0}=\mathbb{F}_{11}$, and the integer $n=2$
invertible in $k_{0}$. Consider the field extension
$\mathbb{F}_{121}=\mathbb{F}_{11}(a)$, where $a$ generates the cyclic group
$\mathbb{F}_{121}^{\times}$ and $a^{2}+7a+2=0$. Denote by $\bar{E}$ the
elliptic curve over $k=\overline{\mathbb{F}_{11}}$ defined by the affine
Weierstrass equation $y^{2}=(x-1)(x-2)(x-3)$. Let $\bar{C}$ be the genus 2
curve over $k$ given by the affine equation
$y^{2}=(x^{2}-1)(x^{2}-2)(x^{2}-3)$. The curve $\bar{C}$ has two points at
infinity $\infty_{+},\infty_{-}$ which do not lie on the affine open defined
by this equation. Consider the degree 2 covering
$\bar{f}\colon\bar{C}\to\bar{E}$ given by $(x,y)\mapsto(x^{2},y)$. It is
ramified at the affine points $P=(0,4)$ and $Q=(0,7)$ of $\bar{C}$, whose
images in $E$ are respectively $(0,4)$ and $(0,7)$. Denote by
$C=\bar{C}-\\{P,Q\\}$ and $E=\bar{E}-\\{\bar{f}(P),\bar{f}(Q)\\}$ the affine
curves obtained from $\bar{C}$ and $\bar{E}$ by removing the ramification
locus. Denote by $f\colon C\to E$ the étale Galois covering induce by
$\bar{f}$. Both curves $C$ and $E$ are obtained by base change from curves
$C_{0}$ and $E_{0}$ defined over $k_{0}=\mathbb{F}_{11}$.

Computing the Galois group of $C^{\langle n\rangle}\to E$ will allow us to
determine the cohomology of any locally constant sheaf on $E$ trivialised by
$f$. First, we need to compute $\operatorname{H}^{1}(C,\mu_{2})$. This is
where we need to cheat a little since no algorithm performing this computation
for a general curve has been implemented yet; fortunately, $\bar{C}$ being a
genus two curve, we have other means of finding a generating set for this
group.

#### Computing $\operatorname{H}^{1}(C,\mu_{2})$

Denote by

$P_{1}^{\pm}=(\pm 1,0),\qquad P_{2}^{\pm}=(\pm a^{6},0),\qquad
P_{3}^{\pm}=(\pm 5,0)$

the points of $C$ with $y$-coordinate 0. A basis of
$\operatorname{Jac}(\bar{C})[2]$ is given by the classes of the divisors

$D_{1}\coloneqq P_{1}^{+}-P_{1}^{-},\qquad D_{2}\coloneqq
P_{2}^{+}-P_{2}^{-},\qquad D_{3}\coloneqq P_{2}^{+}-P_{3}^{+},\qquad
D_{4}\coloneqq P_{1}^{+}-P_{3}^{-}.$

The rational functions

$f_{1}\coloneqq\frac{x-1}{x+1},\qquad
f_{2}\coloneqq\frac{x-a^{6}}{x+a^{6}},\qquad
f_{3}\coloneqq\frac{x-a^{6}}{x-5},\qquad f_{4}\coloneqq\frac{x-1}{x+5}$

all satisfy $2D_{i}=\mathrm{div}(f_{i})$. Denote by $D_{5}$ the divisor $P-Q$,
which is linearly equivalent to two times the divisor

$\bar{D}_{5}\coloneqq(a^{41},a^{29})+(-a^{41},a^{29})-(\infty_{+}+\infty_{-}).$

We found this divisor $\bar{D}_{5}$ using a brute-force search on
$\operatorname{Jac}(\bar{C})(\mathbb{F}_{121})$; for hyperelliptic curves such
as $\bar{C}$, there should be a more elegant way of doing this using division
polynomials (see e.g. [Eid23, Theorem C]). The divisor of the rational
function

$f_{5}=\frac{y+a^{8}x^{2}+7}{x}$

is $2\bar{D}_{5}-D_{5}$. Therefore, an $\mathbb{F}_{2}$-basis of
$\operatorname{H}^{1}(C,\mu_{2})$ is given by

$(D_{1},f_{1}),\dots,(D_{4},f_{4}),(\bar{D}_{5},f_{5}).$

#### Computing $\operatorname{Aut}(C^{\langle n\rangle}|E)$

The covering $C^{\langle n\rangle}\to C$ with group
$\operatorname{H}^{1}(C,\Lambda)^{\vee}$ is defined by its function field
$k(C)(z_{1},\dots,z_{5})$ where $z_{i}^{2}=f_{i}$. Recall that we never need
to compute a smooth model of $C^{\langle n\rangle}$. For reasons explained in
section 7.2, we choose to replace $f_{5}$ with $x^{2}f_{5}$, which still
yields the same function field. The group $G=\operatorname{Aut}(C^{\langle
n\rangle}|E)$ has order 64 ; it contains the normal subgroup
$H=\operatorname{Aut}(C^{\langle
n\rangle}|C)\simeq(\mathbb{Z}/2\mathbb{Z})^{5}$ generated by the elements
$\gamma_{i}\colon z_{i}\mapsto-z_{i}$. Let us now determine a preimage in $G$
of the generator $\sigma\colon(x,y)\mapsto(-x,y)$ of
$\operatorname{Aut}(C|E)$. First, we compute the divisors
$\sigma^{\star}D_{i}$.

$\begin{array}[]{rllll}\sigma^{\star}D_{1}&=&-D_{1}&&\\\
\sigma^{\star}D_{2}&=&-D_{2}&&\\\
\sigma^{\star}D_{3}&=&D_{1}+D_{3}+\mathrm{div}(h_{3})&\text{where}&h_{3}=\frac{y}{x^{3}+a^{58}x^{2}+a^{2}x+a^{54}}\\\
\sigma^{\star}D_{4}&=&D_{2}+D_{4}+\mathrm{div}(h_{4})&\text{where}&h_{4}=\frac{y}{x^{3}+a^{80}x^{2}+a^{103}x+a^{114}}\\\
\sigma^{\star}D_{5}&=&D_{5}\end{array}$

Note that $\sigma^{\star}f_{3}=h_{3}^{2}f_{1}f_{3}$,
$\sigma^{\star}f_{4}=h_{4}^{2}f_{2}f_{4}$ and $\sigma^{\star}f_{5}=-f_{5}$.
Since $a^{30}$ is a square root of $-1$ in $\mathbb{F}_{121}$, the
automorphism $\delta\in G$ given by

$x\mapsto-x,\quad y\mapsto y,\quad z_{1}\mapsto\frac{1}{z_{1}},\quad
z_{2}\mapsto\frac{1}{z_{2}},\quad z_{3}\mapsto h_{3}(x,y)z_{1}z_{3},\quad
z_{4}\mapsto h_{4}(x,y)z_{2}z_{4},\quad z_{5}\mapsto a^{30}z_{5}$

is a preimage of $\sigma$. In particular, since
$h_{3}(x,y)h_{3}(-x,y)=h_{4}(x,y)h_{4}(-x,y)=-1$, the automorphism
$\delta^{2}$ is given by

$x\mapsto x,\quad y\mapsto y,\quad z_{1}\mapsto z_{1},\quad z_{2}\mapsto
z_{2},\quad z_{3}\mapsto-z_{3},\quad z_{4}\mapsto-z_{4},\quad z_{5}\mapsto-
z_{5}.$

Hence $\delta^{2}=\gamma_{3}\gamma_{4}\gamma_{5}$ and the order of $\delta$ as
an element of $G$ is 4. Furthermore,
$\delta\gamma_{1}=\gamma_{1}\gamma_{3}\delta$ et
$\delta\gamma_{2}=\gamma_{2}\gamma_{4}\delta$. The elements
$\gamma_{3},\gamma_{4},\gamma_{5}$ generate the center of $G$. Its commutator
subgroup is
$\langle\gamma_{3}=[\gamma_{1},\delta],\gamma_{4}=[\gamma_{2},\delta]\rangle$.
Finally, let us compute the action of
$\operatorname{Gal}(\mathbb{F}_{121}|\mathbb{F}_{11})=\langle\phi\colon
a\mapsto 2a^{-1}\rangle$ on $G$, which can be done very easily since the
elements of $G$ are defined by their action on coordinates of points.
Predictably, $\phi$ acts trivially on the $\gamma_{i}$; it also sends $\delta$
to $\gamma_{5}\delta$.

###  7.2 Ramification

Here is how to compute the preimages of the points $P$ and $Q$ in $C^{\langle
n\rangle}$. We have computed its function field as $k(C^{\langle
n\rangle})=k(C)(z_{1}^{2}-f_{1},\dots,z_{5}^{2}-f_{5})$, and would now like to
compute an actual affine model of $C^{\langle n\rangle}$. For each
$i\in\\{1\dots 5\\}$, write $f_{i}=\frac{g_{i}}{h_{i}}$. For $P$ and $Q$ to
have easily computable preimages, we can replace $f_{i}$ with $h_{i}^{2}f_{i}$
when $h_{i}(P)=0$ or $h_{i}(Q)=0$. This is only the case for $f_{5}$, which
now reads $x(y+a^{8}x^{2}+7)$. The following affine curve is birational to
$C^{\langle n\rangle}$:

$\begin{array}[]{rl}\operatorname{Spec}k[x,y,z_{1},\dots,z_{5}]/&(y^{2}-(x^{2}-1)(x^{2}-2)(x^{2}-3),\\\
&(x+1)z_{1}^{2}-(x-1),\\\ &(x+a^{6})z_{2}^{2}-(x-a^{6}),\\\
&(x-5)z_{3}^{2}-(x-a^{6}),\\\ &(x+5)z_{4}^{2}-(x-1),\\\
&z_{5}^{2}-x(y+a^{8}x^{2}+7)).\end{array}$

The $\frac{1}{2}|\operatorname{H}^{1}(C,\mu_{2})|=16$ preimages of $P=(0,4)$
in $C^{\langle n\rangle}$ are the points $(0,4,\pm 1,\pm a^{30},\pm 3a^{3},\pm
3a^{30},0)$. The preimages of $Q=(0,7)$ are the points $(0,7,\pm 1,\pm
a^{30},\pm 3a^{3},\pm 3a^{30},0)$. Choose two preimages $P_{C^{\langle
n\rangle}}=(0,4,1,a^{30},3a^{3},3a^{30},0)$ and $Q_{C^{\langle
n\rangle}}=(0,7,1,a^{30},3a^{3},3a^{30},0)$ of $P$ and $Q$ in this affine
curve birational to $C^{\langle n\rangle}$, and denote by
$P_{E}=(0,4),Q_{E}=(0,7)$ their respective images in $E$. The inertia group
$I_{P_{C^{\langle n\rangle}}|P_{E}}\subset G$ has order
$|I_{P|P_{E}}|\cdot|I_{P_{C^{\langle n\rangle}}|P}|=2\times 2=4$; it is
generated by $\delta$. The same applies to $I_{Q_{C^{\langle
n\rangle}}|Q}=\langle\delta\rangle$.

###  7.3 Cohomology of a locally constant sheaf on $E$

Consider the locally constant sheaf $\mathcal{F}$ on $E$ trivialised by $C$,
with generic fibre $M=\Lambda^{3}$, defined by the representation:

$\begin{array}[]{rcl}\operatorname{Aut}(C|E)&\longrightarrow&\operatorname{GL}_{3}(\Lambda)\\\
\sigma&\longmapsto&\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\end{array}$

Computations using Magma yield
$\operatorname{H}^{0}(E,\mathcal{F})\simeq\Lambda^{2}$ and
$\operatorname{H}^{1}(E,\mathcal{F})\simeq\Lambda^{8}$. More precisely, the
1-cocycles $c_{1},\dots,c_{8}$ below form a basis of
$\operatorname{H}^{1}(G,M)$; the null-cohomologous cocycle $c^{\prime}$ is the
image of the 0-cocycle $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)\in M$.

Cocycle $c$ | $c(\gamma_{1})$ | $c(\gamma_{2})$ | $c(\gamma_{3})$ | $c(\gamma_{4})$ | $c(\gamma_{5})$ | $c(\delta)$
---|---|---|---|---|---|---
$c_{1}$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$
$c_{2}$ | $(0\leavevmode\nobreak\ 1\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$
$c_{3}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$
$c_{4}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$
$c_{5}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 1\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$
$c_{6}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$
$c_{7}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$
$c_{8}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 1\leavevmode\nobreak\ 0)$
$c^{\prime}$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(0\leavevmode\nobreak\ 0\leavevmode\nobreak\ 0)$ | $(1\leavevmode\nobreak\ 0\leavevmode\nobreak\ 1)$

The action of $\phi\in\operatorname{Gal}(\mathbb{F}_{121}|\mathbb{F}_{11})$ on
$\operatorname{Hom_{cr}}(G,M)$ only affects the element $c_{7}$ of this basis,
sending it to $\phi^{\star}c_{7}=c_{7}+c^{\prime}$. Its action on
$\operatorname{H}^{1}(G,M)$ is therefore trivial.

######  Remark 7.1.

We now know that the action of
$\operatorname{Gal}(\mathbb{F}_{121}|\mathbb{F}_{11})$ on
$\operatorname{H}^{1}(X,\mathcal{F})$ is trivial. Therefore, we could have
chosen a subcover of $C^{\langle n\rangle}$ by first computing a basis of
$\operatorname{H}^{1}(C,\mu_{2})(\mathbb{F}_{11})$ and then taking $n$-th
roots of the functions appearing in this basis. With the notations above, such
a basis is given by
$(D_{1},f_{1}),(D_{2},f_{2}),(D_{4},f_{4}),(D_{3}+\bar{D}_{5},f_{3}f_{5})$.
Set $k(C^{\prime})=k(C)(z_{1},z_{2},z_{4},t_{3})$ where $z_{i}^{2}=f_{i}$ and
$t_{3}^{2}=f_{3}f_{5}$. Using the previous notations $h_{3},h_{4}$, a preimage
of $\sigma\in\operatorname{Aut}(C|E)$ in $\operatorname{Aut}(C^{\prime}|E)$ is
the automorphism $\delta$ given by:

$x\mapsto-x,\quad y\mapsto y,\quad z_{1}\mapsto\frac{1}{z_{1}},\quad
z_{2}\mapsto\frac{1}{z_{2}},\quad z_{4}\mapsto h_{4}(x,y)z_{2}z_{4},\quad
t_{3}\mapsto a^{30}h_{3}(x,y)t_{3}.$

The group $\operatorname{Aut}(C^{\prime}|E)$ only has order 32, and one can
check that the map

$\operatorname{R\Gamma}(\operatorname{Aut}(C^{\prime}|E),M)\to\operatorname{R\Gamma}(\operatorname{Aut}(C^{\langle
n\rangle}|E),M)$

is a quasi-isomorphism.

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