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@ Appl. Gen. Topol. 18, no. 2 (2017), 219-230doi:10.4995/agt.2017.3371
c© AGT, UPV, 2017

Sheaf Cohomology on network codings:

Maxflow-Mincut Theorem

Miradain Atontsa Nguemo
a
and Calvin Tcheka

b

a
IRMP, Université Catholique de Louvain, Chemin du Cyclotron 2 , Louvain-La-Neuve, Bel-

gium (miradain.atontsa@uclouvain.be)
b

Department of Mathematics, University of Dschang, PO Box: 67, Dschang, Cameroon

(jtcheka@yahoo.fr)

Communicated by M. Sanchis

Abstract

Surveying briefly a novel algebraic topological application sheaf theory

into directed network coding problems, we obtain the weak duality in

multiple source scenario by using the idea of modified graph. Further-

more,we establish the maxflow-mincut theorem with network coding

sheaves in the case of multiple source.

2010 MSC: 18F20; 18G25; 68M10.

Keywords: network information flow; network coding sheaves; topological

cut; relative sheaf cohomology.

1. Introduction

A sheaf is a mathematical tool for storing local information over a global
structure. In our case, it assigns vector spaces to each open set. Sheaf theory
was invented in the mid 1940s as a branch of algebraic topology to deal with the
collation of local data on topological spaces. However, in spite of its generality
dealing with local to global transitions, applications to other sciences have
not well been established so far except for logic and semantics in computer
science. The purpose of this paper is to show the usefulness of sheaf theory
and sheaf cohomology on a network, which as a tool, might also be extendable
on higher dimensional space such as simplicial complexes [6], [2]. The first

Received 07 November 2014 – Accepted 01 August 2017

http://dx.doi.org/10.4995/agt.2017.3371


M. Atontsa and C. Tcheka

step in this direction had been taken by Robert Ghrist and Yasuaki Hiraoka
[4] who introduced in 2011 a class of sheaves designed to model the flow of
information over graphs. They proved the maxflow bound inequality using a
lot of homological algebra tools such as relative homology and exact sequences.
In this paper, we extend this approach proved on the single source context to
the multi source case, and we solve the optimization problem with this class
through the Maxflow-Mincut theorem.

From a network coding X with multiple sources, we associate a modified
graph X by duplicating the edges which conduct information from multiple
sources. The information flow of X is very related to the original flow as we
use it to construct an upper bound for the maxflow.

Theorem A: If D is an open of the graph X which include some receiver nodes,
and not the sources, then D defines a cut and we have this inequality:

maxflow(X) = max
F

dimH0(X;F) ≤ mincut(X) = min
D

dim H0(D;F)

The proof uses Lemma 1 and Lemma 2 which give a concrete description be-
tween H0(X;F) and H0(X;F).

To complete the proof of our theorem we prove that the maxflow is an upper
bound of the mincut.

Theorem B: If X is a graph with multiple source nodes and some receivers,

maxflow(X) := max
F

dimH0(X;F) ≥ mincut(X) =: min
D

dim H0(D;F)

The technical idea of the proof is this result uses a dual version of theorem
A. Namely, from a network X, we associate a new one X1 by adding nodes to
edges outcoming from sources in X −S. The information on the two graphs are
equivalent(Lemma 3). Moreover the rank-nullity theorem on the differential
δ0 tells that H0(X;F) ∼= H1(X;F), so we rather use H1, still unused, which
correlates more the two graphs.

This paper has 2 sections. We start with the preliminaries which remind the
basics in network coding theory, and sheaf theory on graphs. The second section
is more technical and contains all our constructions to prove the maxflow-
mincut theorem. The first three subsections aim to prove Theorem A, and the
last subsection is based on the proof of Theorem B.

Throughout the paper, we refer to [3] and [1] for general discussion on sheaf
theory.

2. Preliminaries

This section is a survey of the work of Ghrist-Hiraoka who pioneered this
application of algebraic topology in network coding theory. Therefore we refer
to [4] for additional comprehension to the topic.

We also fix in the notation to understand our main results in the next section.
Throughout the paper, K if a fix field, and the networks viewed as directed
graphs are not allowed to have loops.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 220



Sheaf Cohomology on network codings: Maxflow-Mincut Theorem

2.1. Network Coding. A network coding is a directed graph G = (V,E),
where V and E are finite sets of nodes and edges1, respectively, satisfying the
following properties:

(i) there exist a subset S = {s1,s2, ...,sk} ⊂ V of nodes called sources
which transmit elements of Knsi for each si, i ∈ {1,2, ...,k}

(ii) there exists a subset R = {r1,r2, ...,rl} ⊂ V of nodes called receivers
which require each information from some sources.

(iii) there exists a capacity map cap: E → N which assign any edge e with
a capacity cap(e).

(iv) For each edge e = |vw|, there exists local coding maps φwv
2, which

are linear maps given by

φwv : K
nv ⊕ Klv → Kcap(e)

with

{
lv =

∑
e∈In(v) cap(e)

nv = 0 if v /∈ S

and In(v) (resp. Out(v)) the subset of edges having v as a head (resp.
tail).

Given a network coding G = (V,E), one build an augmented network X =

(V, Ẽ), where Ẽ is obtained from the set E by adding the edges e = |rjsi|,
whenever the node rj receives information from si. One assign to these edge
the capacities cap(e) = nsi. In the next sections we will rather work with
augmented graphs since they encode decoding on the network.

2.2. Network Coding Sheaves. To see X as a topological space, we consider
its geometric realization. The following definition is not as general as possible
and will be restricted to the setting of sheaves over a graph. We will first assign
sections to special open sets.

Definition 2.1 (Local sections).

1.: For a connected open set U contained in an edge e ∈ X,F(U) :=
Kcap(e).

2.: For a connected open set U which contains only one node v,F(U) :=
Knv ⊕ Klv

Under these definitions of local sections, we define the following local maps.

Definition 2.2 (Local restriction maps).

1.: For connected open sets V ⊂ U ⊂ e, for some edge e ∈ X,̺V U :=
IdF U : F(U) → F(V ).

2.: For connected open sets V ⊂ U, where U contains only one node v,
and V is located in one edge e ∈ In(v),̺V U : F(U) → F(V ) is a natural
projection of Knv ⊕ Klv on Kcap(e).

1If e = |vw| ∈ E, head and tail are two applications defined as: head(e) := w and
tail(e) := v

2φwv is also denoted φe

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M. Atontsa and C. Tcheka

3.: For connected open sets V ⊂ U, where U contains only one node v,
and V is located in one edge e = |vw| ∈ Out(v),̺V U := φwv : F(U) →
F(V )

These local sections and local restriction maps are used to construct F(U)
for arbitrary open sets U ⊂ X. The process used is called sheafification which
is an universal tool for tuning any pre-sheaf into a sheaf.

Definition 2.3. For an open set U ⊂ X,F(U) is the set of equivalent classes
σ = [(σi,Ui)i∈I], where a representative (σi,Ui)i∈I with a covering U = ∪

i∈I
Ui

is given by a family of local sections σi ∈ F(Ui) satisfying σi|Ui∩Uj = σj|Ui∩Uj,
and the equivalent relation is defined by:

(σi,Ui)i∈I ∼ (τj,Vj)j∈J ⇐⇒ σi|Ui∩Vj = τj|Ui∩Vj for i ∈ I,j ∈ J.

For arbitrary open sets V ⊂ U ⊂ X The restriction map ̺V U : F(U) →
F(V ) is induced by the local restriction maps on a representative. The sheaf
defined by the sheafification process from the local sections and local coding
maps is called network coding sheaf of the network X.

2.3. C̆ech cohomology. There are six operations on sheaves that are impor-
tant in the general theory, but only one of them (Cohomology) plays an impor-
tant role in this note. Suppose F is a sheaf on X, and that U = {U1,U2, ...}
is a cover of X. We define the Čech cochain spaces Čn(U;F) to be the direct
sum of sections over each n−wise intersection of elements in U. i.e Čn(U;F) =

Π
{i0,i1,...,in}⊂{1,2,...}

F(Ui0 ∩ ... ∩ Uin).

We define a sequence of linear maps δn : Čn(U;F) → Čn+1(U;F) by

δn(σ)(Ui1, ...,Uin+1) =

n+1∑

j=0

(−1)jσ(Ui1 ∩...∩Ŭij ∩...∩Uin+1)|Ui1 ∩...∩Uij ∩...∩Uin+1

where the hat means that an element has been omitted from the list. The
standard computation shows that δn+1 ◦ δn = 0. We obtain then the following
C̆ech complex: (Č∗(U;F),δ∗), and the C̆ech cohomology Ȟ∗(U;F). We define
the n−cohomology group of F on X to be the direct limit of the Ȟn(U;F) as
U becomes finer.

Ȟn(X;F) = lim
−→

Ȟn(U;F).

In practice, this direct limit is not easy to work with. Instead, what is used
is the fact that under some condition on the cover U, the equalities Ȟn(X;F) =
Ȟn(U;F) hold for all n. This condition is that U be acyclic for F, in the sense
that Ȟn(Ui1 ∩ ... ∩ Uil) = 0, for n > 0 and any i1, i2, ..., il.

2.4. Sheaf cohomology. We basically define cohomology of sheaves using the
derived functors of the global section functor (see [9]), but Robin has proven

in [7] that sheaf cohomology and C̆ech cohomology coincides on graphs.
Let consider the open covering X = ∪

v∈V
Uv, where Uv is the largest con-

nected open of X which contains only the node v. For any edge e = |vw|,

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 222



Sheaf Cohomology on network codings: Maxflow-Mincut Theorem

the intersection Uv ∩ Uw := Ue is the biggest open contained in e. Then
C̆0(X;F) = C̆0(U;F) = Π

v∈V
F(Uv), C̆

1(X;F) = C̆1(U;F) = Π
e∈E

F(Ue) and

the C̆ech complex is reduced to be :

0 −→ C̆0(X;F)
δ
0

−→ C̆1(X;F) −→ 0.

The sheaf cohomology is the cohomology associated to the C̆ech complex de-
fined as follow:

H0(X;F) := ker(δ0), and H1(X;F) := C̆1(X;F)/Im(δ0).

2.5. Information flow.

Definition 2.4. The information flow on a network X for a family of trans-
mitted data

Z = (Zs1,Zs2, ...,Zsk ),

where Zsi ∈ K
nsi , is an assignment ψ(e) ∈ Kcap(e) for each edge e, which

satisfies the following so-called flow conditions:
For e = |vw| and assuming that In(v) = {e1,e2, ...,en},

(i) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = ψ(e), for v /∈ (S ∪ R).
(ii) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = ψ(e), for v = si ∈ S \ R.
(iii) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = Zsi, for v = rj ∈ R \ S and w = si.
(iv) φwv(ψ(e1),ψ(e2), ...,ψ(en)) = ψ(e), for v = rj ∈ R \ S and w /∈ S(rj)

Let σ = {σv} ∈ H
0(X;F), where for each i,σsi = (Zsi, σ̃si) ∈ K

nsi ⊕ Klsi .
One defines the map ψ : E −→ ⊕

e∈E
Kcap(e) as follows: ∀(e = |vw|) ∈ E,ψ(e) :=

σv|Ue = σw|Ue.
It is clear that the family {ψ(e)} defines an information flow on the network

for the data Z = (Zs1,Zs2, ...,Zsk ).
This construction makes it possible to apply homological algebra tools to

network coding problems as we state in the next theorem.

Theorem 2.5 (Information theoretical meaning; [4]). For any network coding
sheaf F of a graph X fitted with ψ, H0(X;F) is equivalent to the information
flows on the network.

Namely this theorem tells that H0(X;F) carries all the data that can be
transmitted on the network.

2.6. Topological Cut. α : S → 2R is the function which assigns to each
source si the set α(si) of all receiver nodes receiving information from si,

Definition 2.6 (Cut-set and cut value in network theory).

(1) A cut-set C on a graph X is a set of edges whose removal breaks
the connection between a source and some receivers. Namely, if s is a
source and D is an open set of a graph X which includes some receivers
r1,r2, ...,rj ∈ α(s) but the node s, then the set of incoming edges into
D define a cut-set C between s and r1,r2, ...,rj.

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M. Atontsa and C. Tcheka

(2) The value of a cut-set C, denoted Val(C), is the sum of the capacities
of its edges.

Remark 2.7. The topological cut values of the graph X which satisfy the sheaf
F are presented by the cohomology classes of H0(D;F).

The following theorem from which we recover the physical definition of cut
value shows that H0(D;F) is independent of the sheaf F .

Proposition 2.8 ([4]). The value of the cut-set is equivalent to topological cut
values. More precisely

Val (C) = dim(H0(D;F)).

3. Optimization of the network: Maxflow-Mincut

3.1. Duplication of networks. In this subsection, we construct from a net-
work X, a modified network X which still keep some information from the
original one. Let γ(v,w) be the reunion of all paths in X connecting v to w,
and γ(si) := ∪

r∈α(si)
γ(si,r). The modified graph X is obtained by duplicat-

ing edges which conduct information from multiple sources. More precisely,
X := {(x,i),x ∈ γ(si), i = 1,2, ...,k}, and hence X is the disjoint union of the
subgraphs γ(si). Heuristically for us, the purpose for this construction is to
apply the results of the single source case [4] to each subgraphs γ(si).

As an illustration, Fig.1. and Fig.2 show how we construct the graph X from
X. The adding edges e = |rjsi| are not represented on purpose to simplify the
construction.

Fig.1. Graph X

s1

v1

e4

v

e1

s2

e2

w

e

r1
e5

e6

v2

e3

r2

e8

e7

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Sheaf Cohomology on network codings: Maxflow-Mincut Theorem

(s1,1)

(v,1)

(w,1)

(v1,1)
(r1,1)

(r2,1)

(s2,2)

(v,2)
(w,2)

(r1,2)

(v2,2)

(r2,2)

(e4,1)

(e1,1) (e,1)

(e5,1)

(e6,1)

(e7,1)

(e6,2)

(e7,2)

(e8,2)

(e,2)

(e2,2)

(e3,2)

Fig.2. Graph X = γ(s1) ∪ γ(s2)

To make the difference with the sheaf F of X, we denote by F : X −→ V ect,
the network coding sheaf on X = ∪

i∈{1,...,k}
∪

v∈V∩γ(si)
(Uv, i).

The natural projection j : X −→ X,(x,i) 7→ x, induces an injective map:
j∗ : H0(X;F) −→ H0(X;F). Therefore dimH0(X;F) ≤ dimH0(X;F), and
in terms of information theory, any information flow on the network X can be
extended to an information flow on X.

3.2. Relative Sheaf Cohomology. let D ⊂ X be an open of a graph X.
That inclusion induces a surjective map p∗ : C∗(X;F) → C∗(D;F). Therefore
we have the short exact sequence defined as follow:

0 → C∗(X,D;F)
i
∗

→ C∗(X;F)
p
∗

→ C∗(D;F) → 0

where C∗(X,D;F) ∼=
C

∗(D;F )
C∗(X;F )

The short exact sequence induces the long exact sequence below:

0 → H0(X,D;F)
i
0

→ H0(X;F)
p
0

→ H0(D;F)
δ
0

→ H1(X,D;F)
i
1

→ ...

It has been proven in [4] that if the graph X contains only one source s and
the open D includes some receivers, but does not include the source node s,
then for any network coding sheaf F,H0(X,D;F) = 0. Thus the long exact
sequence theorem applied to the above short exact sequence shows that

dimH0(X;F) ≤ dimH0(D;F).

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 225



M. Atontsa and C. Tcheka

In the sequel we consider that the graph X has multiple sources along with
multiple receivers. Using the preliminary result on the single source case to
the graph X introduced in Section 6, we obtain the following inequalities on
its subgraphs γ(si) = ∪

r∈α(si)
γ(si,r), for all i ∈ {1,2, ...,k}:

dimH0i (X;F) ≤ dimH
0(Di;F),

for any open Di of the subgraph γ(si) = ∪
r∈α(si)

γ(si,r) which contains some

receiver nodes and does not include the source node si, and H
∗
i (X;F) :=

H0(γ(si);F). If B := ∪
i
Di, this leads to the following inequality:

∑
i∈{1,2,...,k}

dimH0i (X;F) ≤
∑

i∈{1,2,...,k}

dim H0(Di;
−

F)

‖ ‖
dimH0(X;F) ≤ dim H0(B;F)

3.3. Maxflow bound. The following theorem generally known as the weak
duality generalizes the upper bound theorem proved in [4] for the single-source
scenario.

Theorem A: If D is an open of the graph X which include some receiver nodes,
and not the sources, then D defines a cut and we have this inequality:

maxflow(X) = max
F

dimH0(X;F) ≤ mincut(X) = min
D

dim H0(D;F)

To prove this theorem, we will need the next two lemmas.
Let p : X −→ X the natural surjection, I =

⋃
si,sj∈S

p(γ(si)) ∩ p(γ(sj)), and

if D denotes a cut on the network X, let J = {e = |vw| ⊂ I,e cutting edge}.

Lemma 3.1. There exists a finite vector space M of dimension dimM =∑
e=|vw|⊂I

cap(e) such that

H0(X;F) ∼= H0(X;F) ⊕ M,

Proof. The surjection p : X −→ X induces the injection: p∗ : H0(X;F) −→
H0(X;F). Therefore we have H0(X;F) = p(H0(X;F)) ⊕ M, where dim M ≤
dimH0(X;F). Let σ = {σv,i} ∈ H

0(X;F).
σ = σ1 + σ2, with σ1 ∈ H

0(X;F) and σ2 ∈ M. σ * p(H
0(X;F)) ⇔ σ2 6= 0.

However,σ * p(H0(X;F)) ⇔ ∃(i,j) ∈ {1,2, ...,k}2, i 6= j, and e = |vw| ∈
p(γ(si))∩p(γ(sj)) such that σw,i|(Ue,i) 6= σw,j|(Ue,j) ⇐⇒ σw,i|(Ue,i)−σw,j|(Ue,j) 6=
0.

i.e (σw,i|(Ue,i),σw,j|(Ue,j)) ∈ Imf, where
f : Kcap(e) × Kcap(e) −→ Kcap(e)

(x,y) 7−→ x − y
.

It is clear that σ2 consists of such couple (σw,i|(Ue,i),σw,j|(Ue,j)) ∈ Imf and
dim Imf = cap(e).

Hence dimM =
∑

e=|vw|⊂I

cap(e) �

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Sheaf Cohomology on network codings: Maxflow-Mincut Theorem

Lemma 3.2. There exists a finite vector space N of dimension dimN =∑
e=|vw|⊂J

cap(e) such that

H0(p−1(D);F) ∼= H0(D;F) ⊕ N

Proof. The surjection p : p−1(D) −→ D implies the injection: p∗ : H0(D;F) −→
H0(p−1(D);F). Therefore H0(p−1(D);F) = p∗(H(D;F))⊕N, where dimN ≤
dimH0(p−1(D);F). Let p−1(D) = ∪

i∈{1,2,...,k}
Di. H

0(p−1(D);F)

= H0( ∪
i∈{1,2,...,k}

Di;F) = ⊕
i∈{1,2,...,k}

H0(Di;F).

dimH0(p−1(D);F) =
∑
i

dimH0(Di;F) =
∑
i

val(Ci),

where Ci denotes the cut created by Di on the subgraph γ(si).
On the other hand, dimH0(D;F) = val(C), where C is the cut created by

the open D on the graph X. val(C) =
∑

e is cutting edge

cap(e). It is however clear

that,
∑
i

val(Ci) =
∑

e is cutting edge on X by D

cap(e) +
∑

e⊂p(γ(si))∩p(γ(sj)),e is cutting edge

cap(e)

this leads to:

dimH0(p−1(D);F) = dimH0(D;F) +
∑

e⊂p(γ(si))∩p(γ(sj)),e is cutting edge

cap(e).

Hence

dim N =
∑
e⊂J

cap(e).

�

Proof of Theorem A.
By using the inequality: dimH0(X;F) ≤ dimH0(p−1(D);F) with lemma
3.1. and lemma 3.2., we have this:

dimH0(X;F) + dimM ≤ dimH0(D;F) + dimN ⇐⇒ dimH0(X;F) +
dimM − dimN ≤ dimH0(D;F)

However, it is clear that dim M − dimN ≥ 0, hence dimH0(X;F) ≤
dimH0(D;F), for all open D and network coding sheaf F on the graph X.
Therefore we have the following result:

max
F

dimH0(X;F) ≤ min
D

dim H0(D;F)

3.4. Mincut bound. As we consider in this paper that the graphs wear the de-
coding maps, each node is expressed by the incoming edges and one have natu-
rally the isomorphism C0(X;F) ∼= C1(X;F). It turns out using the rank-nullity
theorem on the differential δ0, which is linear in our case, that H0(X;F) ∼=
H1(X;F). We will use this identification in proving the next result.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 227



M. Atontsa and C. Tcheka

Theorem B : If X is a graph with multiple source nodes and some receivers,

maxflow(X) := max
F

dimH0(X;F) ≥ mincut(X) =: min
D

dim H0(D;F)

The proof of this salient theorem uses the following construction on the
network.

Let us consider now a graph X and U an open which contains all the nodes
but the source nodes si. We admit to add some virtual nodes on the cutting
edges such that they are include in the open set U, and we then obtain the
modified graph X1. The following figures are an illustration of that concept
where the nodes G and H have been added.

Fig.4. Graph X with an open U

S

B

e1

C

e2
J

e6

Ee4 r1e5

F
e3

e7

e8

r2e9

U

Fig.5. Graph X1

S

B

e1

C

e2
J

e6

Ee4 r1e5

F
e3

e7

e8

r2e9

U

G

H

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Sheaf Cohomology on network codings: Maxflow-Mincut Theorem

We consider in particular that the network X1 have the following charac-
teristics: for a cutting edge e = |sw| of X, where we have added a node v, Uv
will still be the largest connected open of the graph X1 which contains only
the node v. Moreover we define the local coding maps φvs := φws and φwv :=
id

Kcap(e)
1Imφws, where 1Imφws means the characteristic function on the vectorial

space Imφws. The network coding sheaf on the network (X1,(φba)e=|ab|∈X1)
defined above is denoted F1.

Heuristically for us, the purpose of this modification is to understand more
the properties of the cut created by the open set U without changing the value
of the cut-set. To make sure that we have not significantly changed something,
we have the following lemma:

Lemma 3.3. For any graph X fitted with ψ,H0(X1,F1) is equivalent to the
information on the graph X.

Proof. We denote X1 = (V1,E1).
Let now σ = {σv}v∈V1 ∈ H

0(X1,F1). If e = |sw| ∈ E is a cutting edge by
the open set U which have been added a node v as follows:

s

w

e

s

w

e1

v

e2

We now have the following proposition which first stresses a link between
the two graphs.

Proposition 3.4. For any open set U satisfying the above conditions, we have
the following equality:

H1(U,F1) ∼= H
1(X,F).

Proof. Using the above notation, if e = |sw| is a cutting edge and v is the node
which is added and include in the cut U, we denote e2 = |vw| and e1 = |sv|.
It is clear from the definition that F1(Ue2) = F(Ue), and a direct computation
shows that Imφws ∼= Imφwv. Therefore F1(Ue2)/Imφwv

∼= F(Ue)/Imφsw, and we
obtain the isomorphism. �

Proof of Theorem B.
Let U ⊂ X an open set which satisfies the condition stated earlier in Proposition
3.4.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 229



M. Atontsa and C. Tcheka

We have the following square:

H1(U;F1)
(1)
−→
∼=

H1(X;F)

∼= ↓(2) 	 ∼= ↓(3)

H0(U;F1)
(4)
−→
∼=

H0(X;F)

‖
H0(U;F),

where (1) is the isomorphism of Proposition 3.4, (2) and (3) are isomorphisms
that results from comments at the beginning of this subsection. Then (4) is
the unique morphism which makes the diagram to commute. We obtain from
the square the equality:

dimH0(U;F) = dimH0(X;F)

Thus, min
D

dimH0(D;F) ≤ dimH0(U,F) = dimH0(X;F) ≤ max
F

dimH0(X;F).

Then, {
σs|Ue1 = σv|Ue1
σv|Ue2 = σw|Ue2

⇐⇒ {
φvs(σs) = σv
φwv(σv) = σw|Ue2

(1).

But φvs := φws on the network X, and φwv(σv) = φwv(φvs(σs)) := φvs(σs).
Therefore

(1) ⇐⇒ φvs(σs) = σw|Ue2 .

Hence any cocycle σ = {σv}v∈V1 ∈ H
0(X1,F1) is equivalent to the induced

element
∼
σ = {σv}v∈V ∈ H

0(X,F). We then have H0(X1,F1) ∼= H
0(X,F). �

Acknowledgements. The authors would like to thank Tchatchiem Kamche

Hermann for useful discussions during this work

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