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m aB PBollettino di Matematica pura e applicata

VIII

The «Bollettino di Matematica Pura ed Applicata (BmPa)» is the scientificjournal of the Section of Mathematics of the Dipartimento di Energia,ingegneria dell’Informazione e modelli Matematici (DEIM) of theUniversità di Palermo. It publishes original research papers and surveypapers in pure and applied mathematics, and it is open to contributions ofItalian and stranger researchers. The papers emphasize the advances ofknowledge in mathematics problems and new applications. All the papersare peer-reviewed.

The BmPa was created in 2008 mainly to give, once in a year, an overviewof the activities of the peoples working in the Dipartimento di Metodi eModelli Matematici (DMMM). The DMMM, which has been merged to the DIEETCAM of theUniversità di Palermo, was created thanks to the will of the all the profes-sors of the mathematical area of the Facoltà di Ingegneria. Both thescientific and didactic activity of the DMMM can be found in the web-site www.dmmm.unipa.it. From its constitution it has been involved inmany collaborations with Italian and foreign researchers. The aim of theBollettino is that of contribute to the diffusion of studies and researchesin all fields of pure and applied mathematics.

Editors Maria Stella Mongiovì ([email protected]), Michele Sciacca(michele.[email protected]), Salvatore Triolo ([email protected]).

Editorial Committee

Pietro Aiena (Palermo)S. Twareque Ali (Montreal)Fabio Bagarello (Palermo)Maria Letizia Bertotti (Bolzano)Luis Funar (Grenoble)Renata Grimaldi (Palermo)David Jou (Barcelona)Valentin Poenaru (Paris)Karl Strambach (Erlangen)

Bollettino di Matematicapura e applicata

Volume VIII

EditorsMaria Stella Mongiovì

Michele SciaccaSalvatore Triolo

Copyright © MMXIVARACNE editrice int.le S.r.l.

via Quarto Negroni, 1500040 Ariccia (RM)(06) 93781065

ISBN 978–88–548–9251–4

I diritti di traduzione, di memorizzazione elettronica,di riproduzione e di adattamento anche parziale,

con qualsiasi mezzo, sono riservati per tutti i Paesi.

Non sono assolutamente consentite le fotocopiesenza il permesso scritto dell’Editore.

I edizione: dicembre 2015

Contents

G. Failla, M. Lahyane, R. Utano, J. B. Frìas Medina: Geometry of Rings: An

Elementary Introduction...............................................................................................1

F. Conforto, S. Giambò, V. La Rosa: Non equilibrium relaxation models for two phase relativistic

flows.........................................................................................................................107

G. Lorenzoni: A method to numerically solve every differential analytical model...............................125

N. Mohammed: A spatially homogeneous mathematical model of immune-cancer competition...............155

D. E. Otera, C. Tanasi: Some recent results on “easy” topological representations of groups.................167

R. Matarese Palmieri, A. Scinelli: A study on vitamin C content in a sample of fresh horticultural plants,

consumed for a healthy feed. A statistical approach................................................................173

R. Matarese Palmieri, A. Scinelli: A statistical examination on crude fibre content in a sample

of horticultural plants, used in diets with benefit effects............................................................181

Boll. di mat. Pura ed appl. Vol. VIII (2015)

Geometry of Rings: An Elementary Introduction

Gioia Failla1 Mustapha Lahyane2 Rosanna Utano3

Juan Bosco Frías Medina2

1DIIES, University of Reggio Calabria, Via Graziella, Feo di Vito, Reggio Calabria. Italy2Instituto de Física y Matemáticas (IFM), Universidad Michoacana de San Nicolás de Hidalgo

(UMSNH). Edificio C-3, Ciudad Universitaria. C. P. 58040, Morelia, Michoacán. Mexico3Department of Mathematical and Computer Sciences, Physical and Earth Sciences, University of

Messina, Viale Ferdinando Stagno D’ Alcontres 31, 98166 Messina. Italy

E-mail(s): [email protected]; [email protected], [email protected], [email protected]

Abstract

The aim of this work is to give an elementary introduction to the geometryof any given ring. In particular, it is self-contained, and offers not only a gen-eralization of classical algebraic, differential, and analytic geometries but alsoan efficient way to be familiar, without much efforts, to one of the basic andnecessary tools used frequently in Algebraic Geometry.

Key words: Algebraic Geometry, Commutative Algebra, General Topology, SheafTheory.MSC:14XX, 13XX, 14F05PACS:02.10.Hh

Contents

1 Introduction 2

2 Sheaf Theory 32.1 Presheaves and their Fundamental Related Notions . . . . . . . . . . . 3

2.1.1 Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Very Usefull Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Restriction of a Sheaf to any Open Subset . . . . . . . . . . . . 142.2.2 Direct Image Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Associated Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Inverse Image Sheaf . . . . . . . . . . . . . . . . . . . . . . . . 242.2.5 Restriction of a Sheaf to any Subset . . . . . . . . . . . . . . . 312.2.6 Kernel of a Morphism . . . . . . . . . . . . . . . . . . . . . . . 34

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2.2.7 Image of a Morphism . . . . . . . . . . . . . . . . . . . . . . . 352.2.8 Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Exact Sequences of Sheaves . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Sheaves of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.5 Some Very Important Sheaves of Modules . . . . . . . . . . . . . . . . 57

3 Geometry of Rings 633.1 A Very Important Tool: Localization . . . . . . . . . . . . . . . . . . . 643.2 Zariski Topology on Spec(A) . . . . . . . . . . . . . . . . . . . . . . . 803.3 A Canonically Sheaf of Rings on Spec(A) . . . . . . . . . . . . . . . . 873.4 Classification of Morphisms between Spectra . . . . . . . . . . . . . . 95

1 Introduction

Given a ring R, one may naturally associate to it a geometric object which consistsof a pair (Spec(R),OSpec(R)) where Spec(R) is the set of all prime ideals of R andOSpec(R) is a sheaf of rings on Spec(R). Here, a ring means a commutative ring witha unit, and it is assumed that such ring has at least two elements, except in caseswhere a ring may be reduced to only one element, and when these cases occur, wemention them explicitly. Such geometric object is called the Spectrum of the ring,and is a particular case of what is generally known as an affine scheme.

At the end of the 50’s, Alexander Grothendieck and Jean-Alexandre Dieudonnéformulated the Scheme Theory which is the base for the modern Algebraic Geometry.In recent years, it has been proved the power of such theory not only for solvingproblems coming from Algebraic Geometry, but also for their use in other areas suchas Differential and Complex Geometry, Number Theory, Differential Equations, etc.Furthermore, there exist concrete applications of such theory to other areas such asCoding Theory, Cryptography, Mathematical Physics, Phylogenetics and Robotics tomention a few.

The Scheme Theory is based on the language of Commutative Algebra, and thetools coming from Sheaf Theory. However, it could be difficult for the beginners toenter in this area, because the classic texts of study Algebraic Geometry have a levelthat could be not adequate. The aim of this survey is to present in a self-contained,detailed and elementary introduction to Sheaf Theory, and also to introduce in thesame way, the heart notion of an affine scheme, which is one of the most basic objectsin Scheme Theory, and some of their fundamental properties. The contents of thissurvey is based on [2] regarding the localization of modules and rings, and on [1] and[3] regarding Sheaf Theory, and the geometric object associated to a ring.

In Section 2, we give a detailed presentation of Sheaf Theory. In the first sub-subsection, we introduce the notion of a sheaf and morphism of sheaves, and also westudy some of their properties. as well as. their local data. The subsubsection twois devoted to the construction of some important sheaves. The notion of short exactsequence of sheaves is treated in subsubsection three while in subsubsection four, we

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handle the sheaves of modules. Finally, Section 3 deal with the geometry of rings:after a detailed study of the localization of modules and rings at multiplicative sets,we start to define the topological space associated to any given ring and to give someproperties of such topological space. Next, we define a canonical sheaf of rings onthe topological space just obtained, and study in a detailed way its properties. Inparticular, we define the geometric object associated to any given ring as the spec-trum of such ring. Then, we study morphisms between spectra, and show that thesemorphisms have a purely algebraic character. At the end, we offer some importantproperties regarding the basic open subsets of a given spectrum.

2 Sheaf TheoryIn this section, we present an introduction to the theory of sheaves with full details.

We only assume the reader to be familiar with some fundamental, but elementary,notions of Commutative Algebra, General Topology, and Modern Algebra. This sectionis based also on the first chapter of the unpublished book [1].

2.1 Presheaves and their Fundamental Related NotionsWe review in this subsection some basic concepts, and notions from Sheaf Theory

that we need in order to understand the geometric object associated to a given ring.So, we will first deal with presheaves, sheaves, and stalks, as well as, some of theirstandard properties.

2.1.1 Presheaves and Sheaves

Definition 2.1. Let X be a topological space. A presheaf of abelian groups on Xis a pair (F , ρF ), where F assigns an abelian group F(U) to every open subset U ofX, and ρF assigns a homomorphism of abelian groups ρFU

V : F(U) → F(V ) to everyopen subsets U and V of X with V ⊆ U , furthermore these data should satisfy thefollowing requirements:

PS1. F(∅) = {0},PS2. ρFU

U is the identity map for every open subset U of X, and

PS3. ρFVW ◦ ρFU

V = ρFUW for every open subsets U , V and W of X with W ⊆ V ⊆ U .

Definition 2.2. Let X be a topological space. A sheaf of abelian groups on X is apresheaf (F , ρF ) of abelian groups on X such that for every open subset U of X andevery open covering

(Ui

)i∈I

of U , the following requirements hold:

S1. Let f be an element of F(U). If ρFUUi(f) = 0F(Ui) for all i ∈ I, then f = 0F(U).

S2. If fi ∈ F(Ui) for every i ∈ I such that ρFUi

Ui∩Uj(fi) = ρF

Uj

Ui∩Uj(fj) for every

i, j ∈ I, then there exists f ∈ F(U) satisfying ρFUUi(f) = fi for all i ∈ I.

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Here, we fix some notation, and terminologies: Let (F , ρF ) be a presheaf of abeliangroups on a topological space X.

1. We simply write “F is a presheaf on X” instead of writing “(F , ρF ) is a presheafof abelian groups on X”,

2. Let U be an open subset of X. Any element of F(U) is called a section of Fover U . We refer to a section of F over X as a global section of F .

3. Let U and V be open subsets of X such that V ⊆ U . The group homomorphismρFU

V is called the restriction of F from U to V , or only restriction wheneverthere is no need to emphasize on F , U and V .

4. Let U and V be open subsets of X such that V ⊆ U , and let h be a section ofF over U . h|V means ρFU

V (h).

Remark 2.3. Let F be a presheaf on a topological space X. One may be interestedin considering F(U) as a ring (respectively, a module, etc.) for every open subset Uof X, and the restrictions to be homomorphisms of rings (respectively, modules, etc).In such case, we mention that we have a presheaf of rings, (respectively, a presheaf ofmodules, etc). For example, we will handle the presheaves, and sheaves of modulesin Subsection 2.4.

Now, we consider the local data that one can infer immediately from a presheaf Fon a given topological space X. To do so, let p be an element of X. We firstly definethe following set:

Γp ={(U, s)

∣∣U is an open subset of X containing p and s is a section of F over U}.

Next, we define a relation ∼ on Γp as follows: Let (U, s) and (V, t) be elements of Γp.(U, s) ∼ (V, t) if there exists an open subset W of X containing p such that W ⊆ U∩Vand s|W = t|W .∼ is an equivalence relation. Indeed, let U , V and Z be open subsets of X containingp, and let s ∈ F(U), t ∈ F(V ) and r ∈ F(Z):

1. Reflexivity: (U, s) ∼ (U, s) follows immediately by taking W = U ,

2. Symmetry: If (U, s) ∼ (V, t), then there exists an open subset W of X containingp such that W ⊆ U ∩ V and s|W = t|W . Hence, W ensures also (V, t) ∼ (U, s),and

3. Transitivity: If (U, s) ∼ (V, t), and (V, t) ∼ (Z, r), then there exist open subsetsW1 and W2 of X both containing p such that W1 ⊆ U ∩ V , W2 ⊆ V ∩ Z,s|W1 = t|W1 and t|W2 = r|W2 . Therefore, W1 ∩W2 implies that (U, s) ∼ (Z, r).To see this, W1 ∩W2 is an open subset of X containing p, and is contained inU ∩ Z. Moreover,

s|W1∩W2

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= (s|W1)|W1∩W2

= (t|W1)|W1∩W2

= t|W1∩W2

= (t|W2)|W1∩W2

= (r|W2)|W1∩W2

= r|W1∩W2.

So, we are done.

So, we are able to construct the quotient setΓp

∼ of Γp by ∼ that we denotehereafter by Fp. Let (U, s) be an element of Γp, we denote its class in Fp by [(U, s)].It seems that Fp inherits an algebraic structure of an abelian group. For, we considerthe following operation +:

+ : Fp ×Fp → Fp([(U, s)], [(V, t)]

) → [(U ∩ V, s|U∩V + t|U∩V )].

We need to prove that this operation is binary, that is, + is well-defined: To do so,let [(U, s)], [(U , s)], [(V, t)], [(V , t)] be elements of Fp such that [(U, s)] = [(U , s)] and[(V, t)] = [(V , t)]. We have to check the equality [(U, s)] + [(V, t)] = [(U , s)] + [(V , t)],i.e., [(U∩V, s|U∩V +t|U∩V )] = [(U∩V , s|U∩V + t|U∩V )]. For, [(U, s)] = [(U , s)] (respec-tively, [(V, t)] = [(V , t)]) implies the existence of an open subset W1 (respectively,W2)of X containing p such that W1 ⊆ U ∩ U and s|W1

= s|W1(respectively, W2 ⊆ V ∩ V

and t|W2 = t|W2). Consider the open subset W3 = W1 ∩ W2 of X which certainelycontains p and W3 ⊆ (U ∩V )∩ (U ∩ V ). Furthermore, the fact that W3 ⊆ W1 (respec-tively, W3 ⊆ W2) ensures s|W3

= s|W3(respectively, t|W3

= t|W3). So, the equality

s|W3+ t|W3

= s|W3+ t|W3

holds, and we are done.Next, we show that the pair (Fp,+) is an abelian group. Indeed, let U , V and Z beopen subsets of X containing p, and let s ∈ F(U), t ∈ F(V ) and r ∈ F(Z).• Associativity:(

[(U, s)] + [(V, t)])+ [(Z, r)]

= [(U ∩ V, s|U∩V + t|U∩V )] + [(Z, r)]

= [((U ∩ V ) ∩ Z, (s|(U∩V )∩Z + t|(U∩V )∩Z) + r|(U∩V )∩Z)]

= [(U ∩ (V ∩ Z), s|U∩(V ∩Z) + (t|U∩(V ∩Z) + r|U∩(V ∩Z)))]

= [(U, s)] +([(V ∩ Z, t|V ∩Z + r|V ∩Z)]

)= [(U, s)] +

([(V, t)] + [(Z, r)]

).

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• Commutativity:

[(U, s)] + [(V, t)]

= [(U ∩ V, s|U∩V + t|U∩V )]

= [(V ∩ U, t|V ∩U + s|V ∩U )]

= [(V, t)] + [(U, s)].

• Identity element: [(X, 0F(X))] is the zero element for (Fp,+). In fact,

[(U, s)] + [(X, 0F(X))]

= [(U ∩X, s|U∩X + 0F(X)|U∩X)]

= [(U, s|U )]= [(U, s)].

• Every element has its inverse: The inverse element of [(U, s)] is [(U,−s)]. For,

[(U, s)] + [(U,−s)]

= [(U ∩ U, s|U∩U + (−s)|U∩U )]

= [(U, s− s)] = [(U, 0F(U))]

= [(X, 0F(X))].

Henceforth, Fp has naturaly a structure of an abelian group.

Definition 2.4. With notation as above. The stalk of F at p is the abelian group(Fp,+).

From now on, if U is an open subset of X containing p, and s is a section of F over U ,then we simply denote [(U, s)] by sp; and refer to sp as the germ of s at p. Moreover,one may observe that there exists a natural homomorphism of abelian groups betweenF(U) and Fp- In fact, we may consider the map given by:

γUp : F(U) → Fp

s → sp−Obviously, γU

p is a well-defined map. And, if s, t are sections of F over U , then

γUp (s+ t) = [(U, s+ t)] = [(U, s)] + [(U, t)] = γU

p (s) + γUp (t).

The following result gives one of the most important features of the concept ofstalk when dealing with sheaves.

Proposition 2.5. Let F be a sheaf on a topological space X, and let U be an opensubset of X. If s and t are sections of F over U , then the following statements areequivalent:

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1. s = t.

2. sp = tp, for every p ∈ U .

Proof. If U is empty, then there is nothing to prove. And, we need only to be surethat two implies one. Let p ∈ U . Since the germ [(U, s)] of s at p is equal to the germ[(U, t)] of t at p (by hypothesis), there exists an open subset Wp of X containing psuch that Wp ⊆ U and s|Wp

= t|Wp. So, (s− t)|Wp

= 0F(Wp). Therefore, we found anopen covering (Wp)p∈U of U such that the section (s−t) of F over U has the property(s− t)|Wp

= 0F(Wp) for every p ∈ U . From the fact that F is a sheaf, it follows thats− t = 0F(U), that is, s and t are equals. So, we are done.

Finally, we end this subsection with the following needed definition.

Definition 2.6. Let F be a presheaf on a topological space X. A subpresheaf of Fis a presheaf G on X such that G(U) is a subgroup of F(U) for every open subset Uof X, and the restrictions of G come from restrictions of F .

2.1.2 Morphisms

Here, we handle a way which decides when two presheaves (respectively, sheaves)can be considered as the same. This leads to the concept of morphisms, and theirinduced and related data.

Definition 2.7. Let F and G be presheaves on a topological space X. A morphismϕ : F → G of presheaves assigns a group homomorphism ϕU : F(U) → G(U) for everyopen subset U of X such that the following diagram commutes:

F(V )ϕV ��

ρFVW

��

G(V )

ρGVW

��F(W )

ϕW

�� G(W )

,

for every open subsets V and W of X with W ⊆ V .

Given a morphism ϕ : F → G of presheaves. It induces a group homomorphismsbetween the staks. For, let p be an element of X, we may define the following map:

ϕp : Fp → Gp

[(U, s)] → [(U,ϕU (s))].

Let us first prove that such map is well-defined. For, let U and V be open subsets ofX containing p, and let s ∈ F(U) and t ∈ F(V ) such that [(U, s)] = [(V, t)]. We need

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to be sure that ϕp([(U, s)]) = ϕp([(V, t)]), i.e., [(U,ϕU (s))] = [(V, ϕV (t))]. Indeed, byhypothesis, there exists an open subset W of X containing p with W ⊆ U ∩ V andρFU

W (s) = ρFVW (t). From tha fact that ϕ is a morphism, for every open subset Z of

X such that W ⊆ Z, the following diagram is commutative:

F(Z)ϕZ ��

ρFZW

��

G(Z)

ρGZW

��F(W )

ϕW

�� G(W )

.

Hence, ρGUW ◦ ϕU = ϕW ◦ ρFU

W (respectively, ρGVW ◦ ϕV = ϕW ◦ ρFV

W ), since U(respectively, V ) is an open subset ofX containingW . As a consequence, the followingequalities hold:

ρGUW ◦ ϕU (s) = ϕW ◦ ρFU

W (s) = ϕW ◦ ρFVW (t) = ρGV

W ◦ ϕV (t).

Therefore, W ensures the equality [(U,ϕU (s))] = [(V, ϕV (t))], and consequently, ϕp iswell-defined. Next, we show that ϕp is a group homomorphism. For, without loss ofgenerality, we may assume that the sections s and t of F are defined over the sameopen subset U of X and such that p belongs to U . So, we get the equalities below.

ϕp

([(U, s)] + [(U, t)]

)= ϕp

([(U, s+ t)]

)= [(U,ϕU (s+ t))]

= [(U,ϕU (s))] + [(U,ϕU (t))]

= ϕp

([(U, s)]

)+ ϕp

([(U, t)]

).

Thus, ϕp is a group homomorphism.

Remark 2.8. With notation as above.

1. Let U be an open subset of X containing p. The following diagram commutesby construction:

F(U)ϕU ��

γFUp

��

G(U)

γGUp

��Fp ϕp

�� Gp

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2. If G is a subpresheaf of F , then there is naturally a well-defined morphismj# : G → F given by the inclusion, that is, j#V : G(V ) → F(V ) is the inclusion forevery open subset V of X. Moreover, the induced homomorphism j#p : Gp → Fp

is injective for every p ∈ X. Consequently, Gp can be considered as a subgroupof Fp, for every p ∈ X.

Now, we deal with the concept of equality of presheaves.

Definition 2.9. Let ϕ : F → G and ψ : G → H be morphisms of presheaves on atopological space X.

1. The composition of ϕ and ψ is the morphism ψ ◦ϕ : F → H given by the data:(ψ ◦ ϕ)U = ψU ◦ ϕU , for every open subset U of X.

2. ϕ is an isomorphism if there exists a morphism ξ : G → F of presheaves onX such that ϕ ◦ ξ = idG and ξ ◦ ϕ = idF . In such case, F and G are calledisomorphic, and are denoted by F ∼= G. Here, idE : E → E is given by theidentity map idE(U) : E(U) → E(U) for every open set U of X, and for anypresheaf E on X.

Definition 2.10. Let F and G be sheaves on a topological space X. ϕ is injective(respectively, surjective) if ϕp is injective (respectively, surjective), for every p ∈ X.

Lemma 2.11. Let ϕ : F → G and ψ : G → H be morphisms of presheaves on atopological space X. Then, ψp ◦ ϕp = (ψ ◦ ϕ)p, for every p ∈ X.

Proof. Let U be an open subset of X containing p and s a section of F over U . Thefollowing equalities are straightforward:

ψp ◦ ϕp

([(U, s)]

)= ψp

([(U,ϕU (s))]

)= [(U,ψU ◦ ϕU (s))]

= [(U, (ψ ◦ ϕ)U (s))]= (ψ ◦ ϕ)p

([(U, s)]

).

The next result offers a local characterization of isomorphisms of sheaves.

Theorem 2.12. Let X be a topological space and ϕ : F → G a morphism of presheaveson X. The following statements are equivalent:

1. ϕ is an isomorphism.

2. ϕU : F(U) → G(U) is an isomorphism, for every open subset U of X.

Furthermore, if F and G are sheaves, then the above statements are equivalent to:

3. ϕp : Fp → Gp is an isomorphism, for every p ∈ X.

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Proof. The statements 1 and 2 are obviously equivalents using only the definitions.Now, let F and G be sheaves onX. Let us prove firstly that 2 implies 3. For, let p be anelement of X and U an open subset of X containing p. Let s be a section of F over Usuch that [(U, s)] ∈ Kerϕp. From the equality [(U,ϕU (s))] = [(X, 0G(X))], there existsan open subset W of X containing p with W ⊆ U , and such that ρGU

W ◦ ϕU (s) =0G(W ). Since ϕ is a morphism, the equality ϕW ◦ ρFU

W = ρGUW ◦ ϕU holds. Thus,

ϕW ◦ ρFUW (s) = ρGU

W ◦ ϕU (s) = 0G(W ). So, ρFUW (s) = 0F(W ) (since ϕW is injective).

Hence, [(U, s)] = [(W,ρFUW (s))] = [(W, 0F(W ))] = [(X, 0F(X))]. Next, we prove that

ϕp is surjective. Indeed, let U be an open subset of X containing p, and t a sectionof G over U . Since ϕU is surjective, there exists a section s of F over U such thatϕU (s) = t. Then, ϕp

([(U, s)]

)= [(U, t)]. So, we are done.

Here, we prove that 3 implies 2. Let U be a non-empty subset of X. Let us show thatϕU is injective. For, let s be a section of F over U such that s ∈ KerϕU and p ∈ U .From the fact that ϕU (s) = 0G(U), it follows that

(ϕU (s)

)p= [(U,ϕU (s))] = 0Gp

.And, since ϕp(sp) = [(U,ϕU (s))] = 0Gp

, we get sp = 0Fpby the injectivity of ϕp.

Thus, by Proposition 2.5, s = 0F(U). Hence, ϕU is injective as claimed.Let us show the surjectivity of ϕU . For, let t ∈ G(U) and p ∈ U . Since ϕp issurjective, there exists an open subset Up of X containing p and s(p) ∈ F(Up) suchthat ϕp

([(Up, s(p))]

)= tp, we may assume (without lost of generality) that Up is

contained in U . Thus, from the equality [(Up, ϕUp(s(p)))] = [(U, t)], it follows the

existence of an open subset Wp of X containing p such that Wp ⊆ Up and moreover,ρG

Up

Wp◦ϕUp

(s(p)) = ρGUWp

(t). Hence, since ϕ is a morphism, we get ϕWp◦ρFUp

Wp(s(p)) =

ρGUp

Wp◦ ϕUp(s(p)) = ρGU

Wp(t). So, we obtain an open covering (Wp)p∈U of U with a

family(ρF

Up

Wp(s(p))

)p∈U

of sections of F such that ρFUp

Wp(s(p)) ∈ F(Wp) for every

p ∈ U . Thus, what is left is to be sure that such family of sections can be extended toa section of F over U . Indeed, let p and q be elements of U . We would like to showthat the equality ρF

Wp

Wp∩Wq

(ρF

Up

Wp(s(p))

)= ρF

Wq

Wp∩Wq

(ρF

Uq

Wq(s(q))

)holds true. To do

so, it is worth noting that the following equalities occur:

ϕWp∩Wq

(ρF

Wp

Wp∩Wq

(ρF

Up

Wp(s(p))

))= ϕWp∩Wq

(ρF

Up

Wp∩Wq(s(p))

)= ρGU

Wp∩Wq(t)

= ϕWp∩Wq

(ρF

Uq

Wp∩Wq(s(q))

)= ϕWp∩Wq

(ρF

Wq

Wp∩Wq

(ρF

Uq

Wq(s(q))

)).

Hence, the equality ρFWp

Wp∩Wq

(ρF

Up

Wp(s(p))

)= ρF

Wq

Wp∩Wq

(ρF

Uq

Wq(s(q))

)follows from

the injectivity of ϕWp∩Wq. Now, since F is a sheaf, there exists s ∈ F(U) such that

ρFUWp

(s) = ρFUp

Wp(s(p)), for every p ∈ U . Finally, let us show that ϕU (s) = t. For,

since ϕ is a morphism, we get ρGUWp

◦ ϕU (s) = ϕWp◦ ρFU

Wp(s) = ρGU

Wp(t), for all

p ∈ U , i.e., ρGUWp

(ϕU (s)− t) = 0G(Wp). Therefore, from the facts that U =⋃

p∈U Wp

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and G is a sheaf, it follows that ϕU (s)− t = 0G(U), and consequently, we deduce thatϕU (s) = t, proving thus the surjectivity of ϕU .

Using the same technique given in the proof of the last theorem, we obtain thefollowing result whose proof is left to the reader:

Proposition 2.13. Let ϕ : F → G be a morphism of sheaves on a topological spaceX.

1. ϕ is injective, if and only if, ϕU is injective for every open subset U of X.

2. If ϕU is surjective for every open subset U of X, then ϕ is surjective.

3. ϕ is an isomorphism, if and only if, ϕ is injective and surjective.

Here comes a simple way for constructing morphisms of sheaves.

Proposition 2.14. Let F and G be sheaves on a topological space X, and let Bbe a basis for the topology of X. If for every B ∈ B, there is a homomorphismϕB : F(B) → G(B) of abelian groups such that for every B1, B2 ∈ B with B2 ⊆ B1,the following diagram commutes:

F(B1)ϕB1 ��

ρFB1B2

��

G(B1)

ρGB1B2

��F(B2) ϕB2

�� G(B2),

then, there exists a unique morphism ψ : F → G of sheaves on X such that ψB = ϕB,for all B ∈ B.

Proof. Let U be a non-empty open subset of X. Since B is a basis for the topologyof X, there exists a set IU such that U =

⋃i∈IU

Bi, where Bi ∈ B for every i ∈ IU .Let s be a section of F over U , and i ∈ IU . From the fact that ρFU

Bi(s) is a section of

F ove Bi, it follows that ϕBi(ρFU

Bi(s)) is a section of G over Bi. Thus, we construct

a family(ϕBi

(ρFUBi(s)))i∈IU

of sections of G such that ϕBi(ρFU

Bi(s)) ∈ G(Bi), for all

i ∈ IU . Now, we proceed to prove that such family can be extended to a section of Gover U . Indeed, Let i, j ∈ IU , we need to check that ρGBi

Bi∩Bj

(ϕBi

(ρFUBi(s)))is equal

to ρGBj

Bi∩Bj

(ϕBj

(ρFUBj

(s))). For, if Bi ∩ Bj = ∅, then we are done. Hence, we may

assune that Bi ∩ Bj = ∅, so let us take p ∈ Bi ∩Bj . Therefore, there exists Bkp∈ B

such that p ∈ Bkpand Bkp

⊆ Bi ∩Bj . Using the hypothesis on the homomorphisms

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for the open subsets of the basis, we infer the following equalities:

ρGBi∩Bj

Bkp◦ ρGBi

Bi∩Bj◦ ϕBi

◦ ρFUBi(s) = ρGBi

Bkp◦ ϕBi

◦ ρFUBi(s)

= ϕBkp◦ ρFBi

Bkp◦ ρFU

Bi(s)

= ϕBkp◦ ρFU

Bkp(s)

= ϕBkp◦ ρFBj

Bkp◦ ρFU

Bj(s)

= ρGBj

Bkp◦ ϕBj ◦ ρFU

Bj(s)

= ρGBi∩Bj

Bkp◦ ρGBj

Bi∩Bj◦ ϕBj

◦ ρFUBj

(s).

Thus, the last equality implies the next one:

ρGBi∩Bj

Bkp

(ρGBi

Bi∩Bj

(ϕBi

(ρFUBi(s)))− ρG

Bj

Bi∩Bj

(ϕBj

(ρFUBj

(s))))

= 0G(Bkp ).

On the other hand, since Bi∩Bj =⋃

p∈Bi∩BjBkp , and G is a sheaf, the last equation

gives the claimed equality. Moreover, from the facts that U =⋃

i∈IUBi and G is a

sheaf, there exists a section of G over U , denoted by ψU (s), such that ρGUBi(ψU (s))

is equal to ϕBi(ρFU

Bi(s)), for every i ∈ IU . Henceforth, we are able to define ψU as

follows.

ψU : F(U) → G(U)

s → ψU (s),

where ψU (s) is the section that we have just obtained, for every s ∈ F(U). This mapis well-defined. Indeed, let s, t ∈ F(U) such that s = t, i.e., s− t = 0F(U). It is worthnoting that for every i ∈ IU , the following equalities occur:

ρGUBi(ψU (s)− ψU (t)) = ϕBi(ρF

UBi(s))− ϕBi(ρF

UBi(t)) = ϕBi(ρF

UBi(s− t)) = 0G(Bi).

Thus, ψU (s) − ψU (t) = 0G(U) (since U =⋃

i∈IUBi, and G is a sheaf), that is, ψU (s)

is equal to ψU (t).Next, we will prove that ψU is a group homomorphism. For, let s and t be sectionsof F over U . For every i ∈ IU , we have:

ρGUBi(ψU (s+ t)− ψU (s)− ψU (t))

= ϕBi(ρFU

Bi(s+ t))− ϕBi

(ρFUBi(s))− ϕBi

(ρFUBi(t))

= ϕBi(ρFU

Bi(s+ t− s− t))

= 0G(Bi).

So, ψU (s + t) − ψU (s) − ψU (t) = 0G(U) (since U =⋃

i∈IUBi and G is a sheaf).

Consequently, ψU (s + t) = ψU (s) + ψU (t). Hence, ψU is a group homomorphism.Now, let us prove that the family (ψU ){U is an open subset ofX} is a morphism that we

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Boll. di mat. Pura ed appl. Vol. VIII (2015)

will denote by ψ. For, let U and V be open subsets of X such that V ⊆ U . Here, weshow the commutativity of the following diagram:

F(U)ψU ��

ρFUV

��

G(U)

ρGUV

��F(V )

ψV

�� G(V ),

that is, we need to check that the equality ρGUV ◦ ψU = ψV ◦ ρFU

V is true. Indeed, lets be a section of F over U . It is worth noting that for every i ∈ IV , the followingequalities are satisfied:

ρGVBi

(ρGU

V ◦ ψU (s)− ψV ◦ ρFUV (s))

= ρGVBi

◦ ρGUV ◦ ψU (s)− ρGV

Bi◦ ψV ◦ ρFU

V (s)

= ϕBi

(ρFU

Bi(s))− ϕBi

(ρFV

Bi◦ ρFU

V (s)).

= 0G(Bi).

Using the fact that V =⋃

i∈IVBi and G is a sheaf, we get the equality between

ρGUV ◦ψU (s)−ψV ◦ρFU

V (s) and 0G(U), i.e., ρGUV ◦ψU (s) is equal to ψV ◦ρFU

V (s). Thus,ψ is a morphism. Finally, we have to be sure that if B ∈ B, then ψB = ϕB . In fact,let B ∈ B, and let s be a section of F over B. By construction of ψB , the followingequalities hold:

ψB(s) = ρGBB(ψB(s)) = ϕB(ρFB

B(s)) = ϕB(s).

Therefore, ψB = ϕB . So, we are done.

Corollary 2.15. With notation and hypothesis as in the last proposition. If ϕB isan isomorphism, for every B ∈ B, then F and G are isomorphic.

Proof. By Proposition 2.14, there exists naturally a morphism ψ : F → G of sheaveson X. So, we need only to prove that ψ is an isomorphism. To do so, we prove thatthe induced homomorphism between the stalks ψp : Fp → Gp is an isomorphism, forevery p ∈ X. For, let p ∈ X.Firstly, we deal with the injectivity of ψp. So, let U be an open subset of X containingp, and s be a section of F over U such that [(U, s)] ∈ Kerψp. Since B is a basis for thetopology of X, there exists B1 ∈ B such that p ∈ B1, and B1 ⊆ U . From the equality[(B1, ψB1

(ρFUB1

(s)))] = [(X, 0G(X))], we deduce the existence of an open subset W

of X containing p, moreover is contained in B1, and ρGB1

W

(ψB1

(ρFUB1

(s)))= 0G(W ).

Using again the fact that B is a basis for the topology of X, we infer the existence ofB2 ∈ B such that p ∈ B2, and B2 ⊆ W . Now, the equality ρGB1

W

(ψB1

(ρFUB1

(s)))=

0G(W ) implies ρGB1

B2

(ψB1(ρF

UB1

(s)))= 0G(B2). And, since ψ is a morphism, we get

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ψB2

(ρFB1

B2◦ρFU

B1(s))= 0G(B2). Hence, ρFU

B2(s) = 0F(B2) (since ψB2 is injective), and

consequently we obtain:

[(U, s)] = [(B2, ρFUB2

(s))] = [(B2, 0F(B2))] = 0Fp.

Therefore, ψp is injective.Secondly, we take care of the surjectivity of ψp. Indeed, let U be an open subset ofX containing p, and let t be a section of G over U . We may take [(U, t)] of Gp. SinceB is a basis for the topology of X, there exists B ∈ B with p ∈ B, and B ⊆ U . Thus,from the surjectivity of ψB , we get the existence of a section s of F over B such thatψB(s) = ρGU

B(t). And, consequently, the following equalities:

ψp

([(B, s)]

)= [(B,ψB(s))] = [(B, ρGU

B(t))] = [(U, t)],

prove the surjectivity of ψp.Therefore, ψp is an isomorphism for every p ∈ X. Hence, ψ is an isomorphism ofsheaves.

2.2 Very Usefull Sheaves

In this subsection, we deal with some important sheaves, and study their proper-ties.

2.2.1 Restriction of a Sheaf to any Open Subset

Let F be a presheaf on a topological space X, and let U be a non-empty open setof X. U inherits a structure of a topological space that comes from the topology ofX, known as, the induced topology. Next, our task is to define naturally a presheafF|U on U . For, let V be an open subset of U , and let W be an open subset of U suchthat W ⊆ V , we define F|U (V ) = F(V ), and ρF|U

VW

= ρFVW . From the facts that U

is an open subset of X, and F is a presheaf on X, it follows obviously that F|U is apresheaf on U . Furthermore, if F is a sheaf, then F|U is also a sheaf. This justifiesthe following:

Definition 2.16. With notation as above, F|U is the restriction of the sheaf F to U .

Now, we are interested in the stalks of this sheaf. So, let p be a point of U . We mayconsider the following map:

ϕp : (F|U )p → Fp

[(V, s)] → [(V, s)].

It is easy to see that ϕp is a group homomorphism. Moreover, it is an isomorphism.Indeed,

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• ϕp is injective. For, let V be an open subset of U containing p, and let s bea section of F|U over V such that [(V, s)] ∈ Kerϕp. The equality [(V, s)] =[(X, 0F(X))] implies the existence of an open subset Z of X containing p suchthat Z ⊆ V , and ρFV

Z (s) = 0F(Z). We may observe that Z∩U is an open subsetof U containing p and is contained in V , and consequently, the following equali-ties hold: [(V, s)] = [(Z∩U, ρF|U

VZ∩U

(s))] = [(Z∩U, 0F|U (Z∩U))] = [(U, 0F|U (U))].Thus, ϕp is injective.

• ϕp is surjective. Indeed, let V be an open subset X containing p, and let s bea section of F over V . Since V ∩ U is an open subset of U containing p, wemay consider the element [(V ∩ U, ρF|U

VV ∩U

(s))] of (F|U )p. As a consequence,ϕp

([(V ∩ U, ρF|U

VV ∩U

(s))] and [(V, s)] are equals. Therefore, ϕp is surjective.

This study leads to:

Proposition 2.17. With notation as above. (F|U )p is isomorphic to Fp, for everyp ∈ U .

Using the restriction of sheaves, the following result offers a way to detect thesurjectivity of a given morphism of sheaves:

Lemma 2.18. Let ϕ : F → G be a morphism of sheaves on a topological space X. ϕis surjective, if and only if, the morphism ϕ|U : F|U → G|U is surjective, for everyopen subset U of X.

Proof. Let us assume that ϕ|U : F|U → G|U is surjective, for every open subset Uof X. The particular case where the open subset X is X itself, we get that themorphism ϕ|X : F|X → G|X is surjective, and from the fact that ϕ|X = ϕ, F|X = F ,and G|X = G, we deduce that ϕ : F → G is surjective.Conversely, let us assume that the morphism ϕ : F → G is surjective. Let U be anopen subset of X. If U = ∅, then there nothing to prove, since all are zeros. So, wemay consider U to be non-empty, and p ∈ X. It is worth nothing that the followingdiagram is commutative:

(F|U )p(ϕ|U )p ��

∼=

��

(G|U )p

∼=

��Fp ϕp

�� Gp

Hence, (ϕ|U )p is the composition of three homomorphisms that are all surjectives.Thus, (ϕ|U )p is surjective for every p in U . Consequently, ϕ|U is surjective.

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