Chapter 0 — Preliminaries

§3. Complements on Sheaves

3.1. Sheaves with values in a category

(3.1.1) Let $K$ be a category, let $(A_{\alpha }{)}_{\alpha \in I}$ and $(A_{\alpha \beta }{)}_{(\alpha ,\beta )\in I\times I}$ be two families of objects of $K$ with $A_{\beta \alpha }=A_{\alpha \beta }$, and let $({\rho }_{\alpha \beta }{)}_{(\alpha ,\beta )\in I\times I}$ be a family of morphisms ${\rho }_{\alpha \beta }:A_{\alpha }\to A_{\alpha \beta }$. We say that a pair consisting of an object $A$ of $K$ and a family of morphisms ${\rho }_{\alpha }:A\to A_{\alpha }$ is a solution of the universal problem defined by the data $(A_{\alpha })$, $(A_{\alpha \beta })$, $({\rho }_{\alpha \beta })$ if, for every object $B$ of $K$, the map sending $f\in \mathrm{Hom}(B,A)$ to $({\rho }_{\alpha }\circ f)\in {\prod }_{\alpha }\mathrm{Hom}(B,A_{\alpha })$ is a bijection of $\mathrm{Hom}(B,A)$ onto the set of $(f_{\alpha })$ such that ${\rho }_{\alpha \beta }\circ f_{\alpha }={\rho }_{\beta \alpha }\circ f_{\beta }$ for every pair of indices $(\alpha ,\beta )$. One sees at once that if such a solution exists, it is unique up to isomorphism.

(3.1.2) We shall not recall the definition of a presheaf $U\mapsto \mathcal{F}(U)$ on a topological space $X$ with values in a category $K$ (G, I, 1.9); we say that such a presheaf is a sheaf with values in $K$ if it satisfies the following axiom:

(F) For every cover $(U_{\alpha })$ of an open set $U\subset X$ by open sets $U_{\alpha }\subset U$, if ${\rho }_{\alpha }$ (resp. ${\rho }_{\alpha \beta }$) denotes the restriction morphism

ℱ(U) → ℱ(U_α)    (resp. ℱ(U_α) → ℱ(U_α ∩ U_β)),

then the pair $(\mathcal{F}(U),({\rho }_{\alpha }))$ is a solution of the universal problem for $(\mathcal{F}(U_{\alpha }))$, $(\mathcal{F}(U_{\alpha }\cap U_{\beta }))$, and $({\rho }_{\alpha \beta })$ (3.1.1).¹

Equivalently, for every object $T$ of $K$, the family $U\mapsto \mathrm{Hom}(T,\mathcal{F}(U))$ is a sheaf of sets.

(3.1.3) Suppose $K$ is the category defined by a "species of structure with morphisms" $\Sigma$ — the objects of $K$ being the sets endowed with structures of species $\Sigma$, and the morphisms those of $\Sigma$. Suppose moreover that $K$ satisfies:

(E) If $(A,({\rho }_{\alpha }))$ is a solution of a universal mapping problem in the category $K$ for families $(A_{\alpha })$, $(A_{\alpha \beta })$, $({\rho }_{\alpha \beta })$, then it is also a solution of the universal mapping problem for the same families in the category of sets — that is, when one regards $A$, the $A_{\alpha }$, and the $A_{\alpha \beta }$ as sets and ${\rho }_{\alpha }$, ${\rho }_{\alpha \beta }$ as maps.²

Under these conditions, axiom (F) implies that, viewed as a presheaf of sets, $U\mapsto \mathcal{F}(U)$ is a sheaf. Moreover, for a map $u:T\to \mathcal{F}(U)$ to be a morphism of $K$, it is necessary and sufficient, by (F), that each ${\rho }_{\alpha }\circ u$ be a morphism $T\to \mathcal{F}(U_{\alpha })$ — that is, that the structure of species $\Sigma$ on $\mathcal{F}(U)$ be the initial structure for the morphisms ${\rho }_{\alpha }$. Conversely, suppose that a presheaf $U\mapsto \mathcal{F}(U)$ on $X$ with values in $K$ is a sheaf of sets and satisfies the preceding condition; then it satisfies (F), hence is a sheaf with values in $K$.

(3.1.4) When $\Sigma$ is the species of group or ring structures, the fact that the presheaf $U\mapsto \mathcal{F}(U)$ with values in $K$ is a sheaf of sets implies ipso facto that it is a sheaf with values in $K$ — that is, a sheaf of groups or rings in the sense of (G).³ This is no longer the case when, for example, $K$ is the category of topological rings (with continuous representations as morphisms): a sheaf with values in $K$ is then a sheaf of sets $U\mapsto \mathcal{F}(U)$ such that for every open $U$ and every cover of $U$ by open $U_{\alpha }\subset U$, the topology of $\mathcal{F}(U)$ is the coarsest making the representations $\mathcal{F}(U)\to \mathcal{F}(U_{\alpha })$ continuous. In this case we say that $\mathcal{F}(U)$, viewed as a sheaf of rings (without topology), is underlying the sheaf of topological rings $U\mapsto \mathcal{F}(U)$. The morphisms $u_V:\mathcal{F}(V)\to \mathcal{G}(V)$ (for arbitrary open $V\subset X$) of sheaves of topological rings are thus homomorphisms of the underlying sheaves of rings such that $u_V$ is continuous for every open $V$; to distinguish them from arbitrary homomorphisms of the underlying sheaves of rings, we call them continuous homomorphisms of sheaves of topological rings. Analogous definitions and conventions hold for sheaves of topological spaces or topological groups.

(3.1.5) It is clear that for any category $K$, if $\mathcal{F}$ is a presheaf (resp. sheaf) on $X$ with values in $K$ and $U$ is an open subset of $X$, then the $\mathcal{F}(V)$ for $V\subset U$ form a presheaf (resp. sheaf) with values in $K$, called the presheaf (resp. sheaf) induced by $\mathcal{F}$ on $U$ and written $\mathcal{F}|U$.

For every morphism $u:\mathcal{F}\to \mathcal{G}$ of presheaves on $X$ with values in $K$, we shall write $u|U$ for the morphism $\mathcal{F}|U\to \mathcal{G}|U$ given by the $u_V$ for $V\subset U$.

(3.1.6) Suppose now that $K$ admits inductive limits [modern: direct limits] (T, 1.8); then, for every presheaf (and in particular every sheaf) $\mathcal{F}$ on $X$ with values in $K$ and every $x\in X$, one can define the stalk ${\mathcal{F}}_x$ as the object of $K$ given by the inductive limit of the filtered (by $\supset$) family of open neighborhoods $U$ of $x$ in $X$, with the restriction morphisms ${\rho }_U^V:\mathcal{F}(V)\to \mathcal{F}(U)$. If $u:\mathcal{F}\to \mathcal{G}$ is a morphism of presheaves with values in $K$, one defines for each $x\in X$ the morphism $u_x:{\mathcal{F}}_x\to {\mathcal{G}}_x$ as the inductive limit of the $u_U:\mathcal{F}(U)\to \mathcal{G}(U)$ over the system of open neighborhoods of $x$; this makes ${\mathcal{F}}_x$ a covariant functor in $\mathcal{F}$, with values in $K$, for each $x\in X$.

When in addition $K$ is defined by a species of structure with morphisms $\Sigma$, the elements of $\mathcal{F}(U)$ are still called sections over $U$ of the sheaf $\mathcal{F}$ with values in $K$, and we then write $\Gamma (U,\mathcal{F})$ in place of $\mathcal{F}(U)$; for $s\in \Gamma (U,\mathcal{F})$ and $V$ an open subset of $U$, we write $s|V$ in place of ${\rho }_V^U(s)$; for $x\in U$, the canonical image of $s$ in ${\mathcal{F}}_x$ is the germ of $s$ at $x$, written $s_x$. (We shall never use the notation $s(x)$ in this sense, since it is reserved for another notion concerning the particular sheaves considered in this treatise (5.5.1).)

If $u:\mathcal{F}\to \mathcal{G}$ is a morphism of sheaves with values in $K$, we shall write $u(s)$ in place of $u_V(s)$ for $s\in \Gamma (V,\mathcal{F})$.

If $\mathcal{F}$ is a sheaf of abelian groups, rings, or modules, the set of $x\in X$ such that ${\mathcal{F}}_x\ne 0$ is called the support of $\mathcal{F}$, written $Supp(\mathcal{F})$; this set is not necessarily closed in $X$.

When $K$ is defined by a species of structure with morphisms, we systematically refrain from introducing the "étalé-space" viewpoint for sheaves with values in $K$. In other words, we never view a sheaf as a topological space (nor even as the disjoint union of its stalks), and we shall not view a morphism $u:\mathcal{F}\to \mathcal{G}$ of such sheaves on $X$ as a continuous map of topological spaces.

3.2. Presheaves on a basis of open sets

(3.2.1) We restrict ourselves in what follows to categories $K$ admitting projective limits [modern: inverse limits] (in the generalized sense — corresponding to preordered sets that are not necessarily filtered; cf. (T, 1.8)). Let $X$ be a topological space and $\mathfrak{B}$ a basis of open sets for the topology of $X$. We call a presheaf on $\mathfrak{B}$ with values in $K$ a family of objects $\mathcal{F}(U)\in K$ attached to each $U\in \mathfrak{B}$, together with morphisms ${\rho }_U^V:\mathcal{F}(V)\to \mathcal{F}(U)$ defined for every pair $(U,V)$ of elements of $\mathfrak{B}$ with $U\subset V$, satisfying ${\rho }_U^U=identity$ and ${\rho }_U^W={\rho }_U^V\circ {\rho }_V^W$ whenever $U\subset V\subset W$ in $\mathfrak{B}$. To such a family we associate a presheaf with values in $K$, $U\mapsto {\mathcal{F}}^{\prime }(U)$ in the usual sense, by setting

ℱ′(U) = lim⃖_{V ∈ 𝔅, V ⊂ U} ℱ(V),

where $V$ ranges over the (in general non-filtered) ordered set of $V\in \mathfrak{B}$ with $V\subset U$; the $\mathcal{F}(V)$ form a projective system for the ${\rho }_V^W$ ($V\subset W\subset U$, $V,W\in \mathfrak{B}$). Indeed, if $U,U^{\prime }$ are open with $U\subset U^{\prime }$, define ${\rho }_U^{\prime U^{\prime }}$ as the projective limit (over $V\subset U$) of the canonical morphisms ${\mathcal{F}}^{\prime }(U^{\prime })\to \mathcal{F}(V)$ — that is, the unique morphism ${\mathcal{F}}^{\prime }(U^{\prime })\to {\mathcal{F}}^{\prime }(U)$ whose composition with each canonical ${\mathcal{F}}^{\prime }(U)\to \mathcal{F}(V)$ gives the canonical ${\mathcal{F}}^{\prime }(U^{\prime })\to \mathcal{F}(V)$. The transitivity of the ${\rho }_U^{\prime U^{\prime }}$ follows immediately. Moreover, for $U\in \mathfrak{B}$ the canonical morphism ${\mathcal{F}}^{\prime }(U)\to \mathcal{F}(U)$ is an isomorphism, allowing one to identify these two objects.⁴

(3.2.2) For the presheaf ${\mathcal{F}}^{\prime }$ defined above to be a sheaf, it is necessary and sufficient that the presheaf $\mathcal{F}$ on $\mathfrak{B}$ satisfy:

(F₀) For every cover $(U_{\alpha })$ of $U\in \mathfrak{B}$ by sets $U_{\alpha }\in \mathfrak{B}$ contained in $U$, and for every object $T\in K$, the map sending $f\in \mathrm{Hom}(T,\mathcal{F}(U))$ to $({\rho }_{U_{\alpha }}^U\circ f)\in {\prod }_{\alpha }\mathrm{Hom}(T,\mathcal{F}(U_{\alpha }))$ is a bijection of $\mathrm{Hom}(T,\mathcal{F}(U))$ onto the set of $(f_{\alpha })$ such that ${\rho }_V^{U_{\alpha }}\circ >f_{\alpha }={\rho }_V^{U_{\beta }}\circ f_{\beta }$ for every pair $(\alpha ,\beta )$ and every $V\in \mathfrak{B}$ with $V\subset U_{\alpha }\cap U_{\beta }$.⁵

The condition is plainly necessary. For sufficiency, consider a second basis ${\mathfrak{B}}^{\prime }$ for the topology of $X$ with ${\mathfrak{B}}^{\prime }\subset \mathfrak{B}$, and let ${\mathcal{F}}^{\prime \prime }$ denote the presheaf obtained from the subfamily $(\mathcal{F}(V){)}_{V\in {\mathfrak{B}}^{\prime }}$. Then ${\mathcal{F}}^{\prime \prime }$ is canonically isomorphic to ${\mathcal{F}}^{\prime }$: for any open $U$, the projective limit (over $V\in {\mathfrak{B}}^{\prime }$, $V\subset U$) of the canonical ${\mathcal{F}}^{\prime }(U)\to \mathcal{F}(V)$ is a morphism ${\mathcal{F}}^{\prime }(U)\to {\mathcal{F}}^{\prime \prime }(U)$. For $U\in \mathfrak{B}$ this morphism is an isomorphism, since by hypothesis each canonical ${\mathcal{F}}^{\prime }(U)\to \mathcal{F}(V)$ for $V\in {\mathfrak{B}}^{\prime }$, $V\subset U$, factors as ${\mathcal{F}}^{\prime }(U)\to {\mathcal{F}}^{\prime \prime }(U)\to \mathcal{F}(V)$, and one checks at once that the composites ${\mathcal{F}}^{\prime }(U)\to {\mathcal{F}}^{\prime \prime }(U)\to {\mathcal{F}}^{\prime }(U)$ and ${\mathcal{F}}^{\prime \prime }(U)\to {\mathcal{F}}^{\prime }(U)\to {\mathcal{F}}^{\prime \prime }(U)$ are identities. This being so, for every open $U$ the morphisms ${\mathcal{F}}^{\prime }(U)\to {\mathcal{F}}^{\prime \prime }(W)=\mathcal{F}(W)$ for $W\in \mathfrak{B}$, $W\subset U$, satisfy the conditions characterizing the projective limit of the $\mathcal{F}(W)$ ($W\in \mathfrak{B}$, $W\subset U$), which proves our assertion by uniqueness of projective limits up to isomorphism.

That said, let $U$ be an arbitrary open set of $X$, $(U_{\alpha })$ a cover of $U$ by open sets contained in $U$, and ${\mathfrak{B}}^{\prime }$ the subfamily of $\mathfrak{B}$ consisting of those elements of $\mathfrak{B}$ contained in some $U_{\alpha }$. Plainly ${\mathfrak{B}}^{\prime }$ is still a basis for the topology of $X$, so ${\mathcal{F}}^{\prime }(U)$ (resp. ${\mathcal{F}}^{\prime }(U_{\alpha })$) is the projective limit of the $\mathcal{F}(V)$ for $V\in {\mathfrak{B}}^{\prime }$, $V\subset U$ (resp. $V\subset U_{\alpha }$); axiom (F) is then verified at once, by the definition of projective limit.

When (F₀) holds, we shall say by abuse of language that the presheaf $\mathcal{F}$ on the basis $\mathfrak{B}$ is a sheaf.

(3.2.3) Let $\mathcal{F}$, $\mathcal{G}$ be two presheaves on the basis $\mathfrak{B}$ with values in $K$. A morphism $u:\mathcal{F}\to \mathcal{G}$ is a family $(u_V{)}_{V\in \mathfrak{B}}$ of morphisms $u_V:\mathcal{F}(V)\to \mathcal{G}(V)$ satisfying the usual compatibility with the restriction morphisms ${\rho }_V^W$. With the notation of (3.2.1), one obtains a morphism $u^{\prime }:{\mathcal{F}}^{\prime }\to {\mathcal{G}}^{\prime }$ of the corresponding ordinary presheaves by taking for $u_U^{\prime }$ the projective limit of the $u_V$ for $V\in \mathfrak{B}$, $V\subset U$; the compatibility with the ${\rho }_U^{\prime U^{\prime }}$ follows from the functoriality of projective limits.

(3.2.4) If $K$ admits inductive limits and $\mathcal{F}$ is a presheaf on $\mathfrak{B}$ with values in $K$, then for every $x\in X$ the neighborhoods of $x$ belonging to $\mathfrak{B}$ form a cofinal subset (for $\supset$) of the neighborhoods of $x$; hence if ${\mathcal{F}}^{\prime }$ is the ordinary presheaf associated with $\mathcal{F}$, the stalk ${\mathcal{F}}_x^{\prime }$ equals $\underrightarrow{\lim }\mathcal{F}(V)$ over $V\in \mathfrak{B}$, $x\in V$. If $u:\mathcal{F}\to \mathcal{G}$ is a morphism of presheaves on $\mathfrak{B}$ and $u^{\prime }:{\mathcal{F}}^{\prime }\to {\mathcal{G}}^{\prime }$ the corresponding morphism of ordinary presheaves, then $u_x^{\prime }$ is the inductive limit of the $u_V:\mathcal{F}(V)\to \mathcal{G}(V)$ for $V\in \mathfrak{B}$, $x\in V$.

(3.2.5) Returning to the general setting of (3.2.1): if $\mathcal{F}$ is an ordinary sheaf with values in $K$ and ${\mathcal{F}}_1$ is the sheaf on $\mathfrak{B}$ obtained by restriction of $\mathcal{F}$ to $\mathfrak{B}$, then the ordinary sheaf ${\mathcal{F}}_1^{\prime }$ obtained from ${\mathcal{F}}_1$ by the construction of (3.2.1) is canonically isomorphic to $\mathcal{F}$, by (F) and the uniqueness of projective limits. We shall ordinarily identify $\mathcal{F}$ and ${\mathcal{F}}_1^{\prime }$.

If $\mathcal{G}$ is a second ordinary sheaf on $X$ with values in $K$ and $u:\mathcal{F}\to \mathcal{G}$ a morphism, the preceding remark shows that giving the $u_V:\mathcal{F}(V)\to \mathcal{G}(V)$ for $V\in \mathfrak{B}$ alone determines $u$ completely. Conversely, given $u_V$ for $V\in \mathfrak{B}$ satisfying the commutativity diagrams with the restriction morphisms ${\rho }_V^W$ for $V,W\in \mathfrak{B}$ and $V\subset W$, there is a unique morphism $u^{\prime }:\mathcal{F}\to \mathcal{G}$ with $u_V^{\prime }=u_V$ for every $V\in \mathfrak{B}$ (3.2.3).

(3.2.6) Continue to assume that $K$ admits projective limits. Then the category of sheaves on $X$ with values in $K$ also admits projective limits: if $({\mathcal{F}}_{\lambda })$ is a projective system of sheaves on $X$ with values in $K$, then the $\mathcal{F}(U)={\underleftarrow{\lim }}_{\lambda }{\mathcal{F}}_{\lambda }(U)$ define a presheaf with values in $K$, and axiom (F) holds by the transitivity of projective limits; that $\mathcal{F}$ is then the projective limit of the ${\mathcal{F}}_{\lambda }$ is immediate.

When $K$ is the category of sets, then for every projective system $({\mathcal{H}}_{\lambda })$ with ${\mathcal{H}}_{\lambda }$ a subsheaf of ${\mathcal{F}}_{\lambda }$ for each $\lambda$, ${\underleftarrow{\lim }}_{\lambda }{\mathcal{H}}_{\lambda }$ is canonically identified with a subsheaf of ${\underleftarrow{\lim }}_{\lambda }{\mathcal{F}}_{\lambda }$. If $K$ is the category of abelian groups, the covariant functor ${\underleftarrow{\lim }}_{\lambda }{\mathcal{F}}_{\lambda }$ is additive and left exact.

3.3. Gluing of sheaves

(3.3.1) Continue to assume that $K$ admits (generalized) projective limits. Let $X$ be a topological space, $\mathfrak{U}=(U_{\lambda }{)}_{\lambda \in L}$ an open cover of $X$, and for each $\lambda \in L$ let ${\mathcal{F}}_{\lambda }$ be a sheaf on $U_{\lambda }$ with values in $K$. Suppose that for every pair $(\lambda ,\mu )$ we are given an isomorphism ${\theta }_{\lambda \mu }:{\mathcal{F}}_{\mu }|(U_{\lambda }\cap U_{\mu })\stackrel{\sim }{\to }{\mathcal{F}}_{\lambda }|(U_{\lambda }\cap U_{\mu })$, and that for every triple $(\lambda ,\mu ,\nu )$, writing ${\theta }_{\lambda \mu }^{\prime },{\theta }_{\mu \nu }^{\prime },{\theta }_{\lambda \nu }^{\prime }$ for the restrictions of ${\theta }_{\lambda \mu },{\theta }_{\mu \nu },{\theta }_{\lambda \nu }$ to $U_{\lambda }\cap U_{\mu }\cap U_{\nu }$, one has ${\theta }_{\lambda \nu }^{\prime }={\theta }_{\lambda \mu }^{\prime }\circ {\theta }_{\mu \nu }^{\prime }$ (the gluing condition for the ${\theta }_{\lambda \mu }$). Then there is a sheaf $\mathcal{F}$ on $X$ with values in $K$, and for each $\lambda$ an isomorphism ${\eta }_{\lambda }:\mathcal{F}|U_{\lambda }\stackrel{\sim }{\to }{\mathcal{F}}_{\lambda }$, such that for every $(\lambda ,\mu )$, writing ${\eta }_{\lambda }^{\prime }$ and ${\eta }_{\mu }^{\prime }$ for the restrictions of ${\eta }_{\lambda }$ and ${\eta }_{\mu }$ to $U_{\lambda }\cap U_{\mu }$, one has ${\theta }_{\lambda \mu }={\eta }_{\lambda }^{\prime }\circ {\eta }_{\mu }^{\prime -1}$. Moreover, $\mathcal{F}$ and the ${\eta }_{\lambda }$ are determined up to unique isomorphism by these conditions. Uniqueness follows at once from (3.2.5). For existence, let $\mathfrak{B}$ be the basis of open sets contained in some $U_{\lambda }$; choose (via Hilbert's $\tau$-function) one of the ${\mathcal{F}}_{\lambda }(U)$ for some $\lambda$ with $U\subset U_{\lambda }$; call this object $\mathcal{F}(U)$. The ${\rho }_U^V$ for $U\subset V$, $U,V\in \mathfrak{B}$, are defined in the obvious way (using the ${\theta }_{\lambda \mu }$), and transitivity follows from the gluing condition. Axiom (F₀) is then immediate, so the presheaf on $\mathfrak{B}$ so defined is a sheaf; from it the general construction (3.2.1) yields an ordinary sheaf, again written $\mathcal{F}$, with the required property. We say that $\mathcal{F}$ is obtained by gluing the ${\mathcal{F}}_{\lambda }$ via the ${\theta }_{\lambda \mu }$, and we ordinarily identify ${\mathcal{F}}_{\lambda }$ and $\mathcal{F}|U_{\lambda }$ via ${\eta }_{\lambda }$.

It is clear that every sheaf $\mathcal{F}$ on $X$ with values in $K$ may be viewed as obtained by gluing the sheaves ${\mathcal{F}}_{\lambda }=\mathcal{F}|U_{\lambda }$ (where $(U_{\lambda })$ is any open cover of $X$) via the identity isomorphisms ${\theta }_{\lambda \mu }$.

(3.3.2) With the same notation, let ${\mathcal{G}}_{\lambda }$ be a second sheaf on $U_{\lambda }$ (for each $\lambda \in L$) with values in $K$, and suppose given for each $(\lambda ,\mu )$ an isomorphism ${\omega }_{\lambda \mu }:{\mathcal{G}}_{\mu }|(U_{\lambda }\cap U_{\mu })\stackrel{\sim }{\to }{\mathcal{G}}_{\lambda }|(U_{\lambda }\cap U_{\mu })$ satisfying the gluing condition. Suppose finally given for each $\lambda$ a morphism $u_{\lambda }:{\mathcal{F}}_{\lambda }\to {\mathcal{G}}_{\lambda }$ such that the diagrams

(3.3.2.1)    ℱ_μ|(U_λ ∩ U_μ) ──u_μ──→ 𝒢_μ|(U_λ ∩ U_μ)
                   │                          │
                   ↓                          ↓
             ℱ_λ|(U_λ ∩ U_μ) ──u_λ──→ 𝒢_λ|(U_λ ∩ U_μ)

commute. Then if $\mathcal{G}$ is obtained by gluing the ${\mathcal{G}}_{\lambda }$ via the ${\omega }_{\lambda \mu }$, there is a unique morphism $u:\mathcal{F}\to \mathcal{G}$ such that the diagrams

ℱ|U_λ ──u|U_λ──→ 𝒢|U_λ
  │                 │
  ↓                 ↓
 ℱ_λ ────u_λ───→  𝒢_λ

commute; this follows at once from (3.2.3). The correspondence between the family $(u_{\lambda })$ and $u$ is a functorial bijection of the subset of ${\prod }_{\lambda }\mathrm{Hom}({\mathcal{F}}_{\lambda },{\mathcal{G}}_{\lambda })$ satisfying (3.3.2.1) onto $\mathrm{Hom}(\mathcal{F},\mathcal{G})$.

(3.3.3) With the notation of (3.3.1), let $V$ be an open subset of $X$. The restrictions of the ${\theta }_{\lambda \mu }$ to $V\cap U_{\lambda }\cap U_{\mu }$ plainly satisfy the gluing condition for the induced sheaves ${\mathcal{F}}_{\lambda }|(V\cap U_{\lambda })$, and the sheaf on $V$ obtained by gluing these is canonically identified with $\mathcal{F}|V$.

3.4. Direct images of presheaves

(3.4.1) Let $X$, $Y$ be topological spaces and $\psi :X\to Y$ a continuous map. Let $\mathcal{F}$ be a presheaf on $X$ with values in $K$; for every open $U\subset Y$, set $\mathcal{G}(U)=\mathcal{F}({\psi }^{-1}(U))$, and for $U\subset V$ open in $Y$ let ${\rho }_U^{\prime V}$ be the morphism $\mathcal{F}({\psi }^{-1}(V))\to \mathcal{F}({\psi }^{-1}(U))$. The $\mathcal{G}(U)$ and ${\rho }_U^{\prime V}$ define a presheaf on $Y$ with values in $K$, called the direct image of $\mathcal{F}$ by $\psi$ and written ${\psi }_{\ast }(\mathcal{F})$. If $\mathcal{F}$ is a sheaf, axiom (F) for $\mathcal{G}$ is immediate, so ${\psi }_{\ast }(\mathcal{F})$ is a sheaf.

(3.4.2) Let ${\mathcal{F}}_1$, ${\mathcal{F}}_2$ be presheaves on $X$ with values in $K$ and let $u:{\mathcal{F}}_1\to {\mathcal{F}}_2$ be a morphism. As $U$ ranges over the open subsets of $Y$, the family $u_{{\psi }^{-1}(U)}:{\mathcal{F}}_1({\psi }^{-1}(U))\to {\mathcal{F}}_2({\psi }^{-1}(U))$ satisfies the compatibility with restriction morphisms, defining ${\psi }_{\ast }(u):{\psi }_{\ast }({\mathcal{F}}_1)\to {\psi }_{\ast }({\mathcal{F}}_2)$. If $v:{\mathcal{F}}_2\to {\mathcal{F}}_3$ is a further morphism, ${\psi }_{\ast }(v\circ u)={\psi }_{\ast }(v)\circ {\psi }_{\ast }(u)$; that is, ${\psi }_{\ast }$ is a covariant functor from presheaves (resp. sheaves) on $X$ with values in $K$ to presheaves (resp. sheaves) on $Y$ with values in $K$.

(3.4.3) Let $Z$ be a third topological space, ${\psi }^{\prime }:Y\to Z$ continuous, and set ${\psi }^{\prime \prime }={\psi }^{\prime }\circ \psi$. One has ${\psi }_{\ast }^{\prime \prime }(\mathcal{F})={\psi }_{\ast }^{\prime }({\psi }_{\ast }(\mathcal{F}))$ for every presheaf $\mathcal{F}$ on $X$ with values in $K$; for every morphism $u:\mathcal{F}\to \mathcal{G}$, ${\psi }_{\ast }^{\prime \prime }(u)={\psi }_{\ast }^{\prime }({\psi }_{\ast }(u))$. In other words, ${\psi }_{\ast }^{\prime \prime }$ is the composite of ${\psi }_{\ast }^{\prime }$ and ${\psi }_{\ast }$:

(ψ′ ∘ ψ)_* = ψ′_* ∘ ψ_*.

Moreover, for every open $U\subset Y$, the direct image by the restriction $\psi |{\psi }^{-1}(U)$ of the induced presheaf $\mathcal{F}|{\psi }^{-1}(U)$ is none other than the induced presheaf ${\psi }_{\ast }(\mathcal{F})|U$.

(3.4.4) Suppose $K$ admits inductive limits and let $\mathcal{F}$ be a presheaf on $X$ with values in $K$. For each $x\in X$, the morphisms $\Gamma ({\psi }^{-1}(U),\mathcal{F})\to {\mathcal{F}}_x$ ($U$ an open neighborhood of $\psi (x)$ in $Y$) form an inductive system; passage to the limit gives a stalk morphism

$${\psi }_x:({\psi }_{\ast }(\mathcal{F}){)}_{\psi (x)}\to {\mathcal{F}}_x.$$

In general, ${\psi }_x$ is neither injective nor surjective. It is functorial: for $u:{\mathcal{F}}_1\to {\mathcal{F}}_2$, the diagram

(ψ_*(ℱ_1))_{ψ(x)} ──ψ_x──→ (ℱ_1)_x
        │                       │
(ψ_*(u))_{ψ(x)}                u_x
        │                       │
        ↓                       ↓
(ψ_*(ℱ_2))_{ψ(x)} ──ψ_x──→ (ℱ_2)_x

commutes. If $Z$ is a third topological space, ${\psi }^{\prime }:Y\to Z$ continuous, and ${\psi }^{\prime \prime }={\psi }^{\prime }\circ \psi$, then ${\psi }_x^{\prime \prime }={\psi }_x\circ {\psi }_{\psi (x)}^{\prime }$ for $x\in X$.

(3.4.5) Under the hypotheses of (3.4.4), suppose further that $\psi$ is a homeomorphism of $X$ onto the subspace $\psi (X)$ of $Y$. Then for every $x\in X$, ${\psi }_x$ is an isomorphism. This applies in particular to the canonical injection $j$ of a subspace $X\subset Y$ into $Y$.

(3.4.6) Suppose $K$ is the category of groups, of rings, etc. If $\mathcal{F}$ is a sheaf on $X$ with support $S$, and $y\notin \psi \bar{S}$, then by the definition of ${\psi }_{\ast }(\mathcal{F})$ one has $({\psi }_{\ast }(\mathcal{F}){)}_y=0$; that is, $Supp({\psi }_{\ast }(\mathcal{F}))\subset \psi \bar{S}$. The support is not, however, necessarily contained in $\psi (S)$. Under the same hypotheses, if $j$ is the canonical injection of a subspace $X\subset Y$, the sheaf $j_{\ast }(\mathcal{F})$ induces $\mathcal{F}$ on $X$; if moreover $X$ is closed in $Y$, then $j_{\ast }(\mathcal{F})$ is the sheaf on $Y$ inducing $\mathcal{F}$ on $X$ and 0 on $Y-X$ (G, II, 2.9.2); but the two sheaves are in general distinct when $X$ is locally closed but not closed.

3.5. Inverse images of presheaves

(3.5.1) Under the hypotheses of (3.4.1), if $\mathcal{F}$ (resp. $\mathcal{G}$) is a presheaf on $X$ (resp. $Y$) with values in $K$, every morphism $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$ of presheaves on $Y$ is called a ψ-morphism of $\mathcal{G}$ into $\mathcal{F}$, also written $\mathcal{G}\to \mathcal{F}$. We write ${\mathrm{Hom}}_{\psi }(\mathcal{G},\mathcal{F})$ for ${\mathrm{Hom}}_Y(\mathcal{G},{\psi }_{\ast }(\mathcal{F}))$. For every pair $(U,V)$ with $U$ open in $X$, $V$ open in $Y$, and $\psi (U)\subset V$, one has a morphism $u_{U,V}:\mathcal{G}(V)\to \mathcal{F}(U)$ obtained by composing the restriction $\mathcal{F}({\psi }^{-1}(V))\to \mathcal{F}(U)$ with $u_V:\mathcal{G}(V)\to {\psi }_{\ast }(\mathcal{F})(V)=\mathcal{F}({\psi }^{-1}(V))$. These morphisms make the diagrams

(3.5.1.1)    𝒢(V) ──u_{U,V}──→ ℱ(U)
               │                  │
               ↓                  ↓
             𝒢(V′) ──u_{U′,V′}──→ ℱ(U′)

(for $U^{\prime }\subset U$, $V^{\prime }\subset V$, $\psi (U^{\prime })\subset V^{\prime }$) commute. Conversely, a family $(u_{U,V})$ making (3.5.1.1) commute defines a $\psi$-morphism $u$: take $u_V=u_{{\psi }^{-1}(V),V}$.

If $K$ admits (generalized) projective limits and $\mathfrak{B}$, ${\mathfrak{B}}^{\prime }$ are bases for the topologies of $X$ and $Y$, then to define a $\psi$-morphism of sheaves it suffices to give $u_{U,V}$ for $U\in \mathfrak{B}$, $V\in {\mathfrak{B}}^{\prime }$, $\psi (U)\subset V$, satisfying (3.5.1.1) for $U,U^{\prime }\in \mathfrak{B}$ and $V,V^{\prime }\in {\mathfrak{B}}^{\prime }$; for an arbitrary open $W\subset Y$, define $u_W$ as the projective limit of the $u_{U,V}$ for $V\in {\mathfrak{B}}^{\prime }$, $V\subset W$, $U\in \mathfrak{B}$, $\psi (U)\subset V$.

When $K$ admits inductive limits, one obtains for each $x\in X$ a morphism $\mathcal{G}(V)\to \mathcal{F}({\psi }^{-1}(V))\to {\mathcal{F}}_x$ for every open neighborhood $V$ of $\psi (x)$ in $Y$; these form an inductive system whose limit is a stalk morphism ${\mathcal{G}}_{\psi (x)}\to {\mathcal{F}}_x$.

(3.5.2) Under the hypotheses of (3.4.3), let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be presheaves on $X$, $Y$, $Z$ with values in $K$, and let $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$, $v:\mathcal{H}\to {\psi }_{\ast }^{\prime }(\mathcal{G})$ be a $\psi$-morphism and a ${\psi }^{\prime }$-morphism. One obtains a ${\psi }^{\prime \prime }$-morphism

w : ℋ ──v──→ ψ′_*(𝒢) ──ψ′_*(u)──→ ψ′_*(ψ_*(ℱ)) = ψ″_*(ℱ),

called by definition the composite of $u$ and $v$. One may therefore regard pairs $(X,\mathcal{F})$ of a topological space $X$ and a presheaf $\mathcal{F}$ on $X$ (with values in $K$) as forming a category, with morphisms the pairs $(\psi ,\theta ):(X,\mathcal{F})\to (Y,\mathcal{G})$ consisting of a continuous map $\psi :X\to Y$ and a $\psi$-morphism $\theta :\mathcal{G}\to \mathcal{F}$.

(3.5.3) Let $\psi :X\to Y$ be a continuous map and $\mathcal{G}$ a presheaf on $Y$ with values in $K$. We call an inverse image of $\mathcal{G}$ by $\psi$ a pair $({\mathcal{G}}^{\prime },\rho )$ consisting of a sheaf ${\mathcal{G}}^{\prime }$ on $X$ with values in $K$ and a $\psi$-morphism $\rho :\mathcal{G}\to {\mathcal{G}}^{\prime }$ (equivalently, a homomorphism $\mathcal{G}\to {\psi }_{\ast }({\mathcal{G}}^{\prime })$) such that, for every sheaf $\mathcal{F}$ on $X$ with values in $K$, the map

(3.5.3.1)    Hom_X(𝒢′, ℱ) → Hom_ψ(𝒢, ℱ) = Hom_Y(𝒢, ψ_*(ℱ))

sending $v$ to ${\psi }_{\ast }(v)\circ \rho$ is a bijection; this map being functorial in $\mathcal{F}$, it then defines an isomorphism of functors in $\mathcal{F}$. As a solution of a universal problem, the pair $({\mathcal{G}}^{\prime },\rho )$, when it exists, is determined up to unique isomorphism. We then write ${\mathcal{G}}^{\prime }=\psi \ast (\mathcal{G})$, $\rho ={\rho }_{\mathcal{G}}$, and by abuse of language we call $\psi \ast (\mathcal{G})$ the inverse image sheaf of $\mathcal{G}$ by $\psi$, with the understanding that $\psi \ast (\mathcal{G})$ is taken together with the canonical $\psi$-morphism ${\rho }_{\mathcal{G}}:\mathcal{G}\to {\psi }_{\ast }(\psi \ast (\mathcal{G}))$, i.e. with the canonical homomorphism of presheaves on $Y$

$$(\mathrm{3.5.3.2}){\rho }_{\mathcal{G}}:\mathcal{G}\to {\psi }_{\ast }(\psi \ast (\mathcal{G})).$$

For any homomorphism $v:\psi \ast (\mathcal{G})\to \mathcal{F}$ (where $\mathcal{F}$ is a sheaf on $X$ with values in $K$), set $v^{♭}={\psi }_{\ast }(v)\circ {\rho }_{\mathcal{G}}:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$. By definition, every morphism $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$ of presheaves is of the form $v^{♭}$ for a unique $v$, which we write $u^{♯}$. In other words, every such $u$ factors uniquely as

(3.5.3.3)    u : 𝒢 ──ρ_𝒢──→ ψ_*(ψ*(𝒢)) ──ψ_*(u^♯)──→ ψ_*(ℱ).

(3.5.4) Suppose now that the category $K$ is such⁶ that every presheaf $\mathcal{G}$ on $Y$ with values in $K$ admits an inverse image by $\psi$, denoted $\psi \ast (\mathcal{G})$.

We shall see that $\psi \ast (\mathcal{G})$ may be defined as a covariant functor in $\mathcal{G}$, from presheaves on $Y$ to sheaves on $X$, in such a way that the isomorphism $v\mapsto v^{♭}$ is an isomorphism of bifunctors

(3.5.4.1)    Hom_X(ψ*(𝒢), ℱ) ⥲ Hom_Y(𝒢, ψ_*(ℱ))

in $\mathcal{G}$ and $\mathcal{F}$.

Indeed, for every morphism $w:{\mathcal{G}}_1\to {\mathcal{G}}_2$ of presheaves on $Y$, consider the composite ${\mathcal{G}}_1\stackrel{w}{\to }{\mathcal{G}}_2\stackrel{{\rho }_{{\mathcal{G}}_2}}{\to }{\psi }_{\ast }(\psi \ast ({\mathcal{G}}_2))$; to it corresponds a morphism $({\rho }_{{\mathcal{G}}_2}\circ w{)}^{♯}:\psi \ast ({\mathcal{G}}_1)\to \psi \ast ({\mathcal{G}}_2)$, which we write $\psi \ast (w)$. By (3.5.3.3),

(3.5.4.2)    ψ_*(ψ*(w)) ∘ ρ_{𝒢_1} = ρ_{𝒢_2} ∘ w.

For every morphism $u:{\mathcal{G}}_2\to {\psi }_{\ast }(\mathcal{F})$ (with $\mathcal{F}$ a sheaf on $X$ with values in $K$), by (3.5.3.3), (3.5.4.2), and the definition of $u^{♭}$,

(u^♯ ∘ ψ*(w))^♭ = ψ_*(u^♯) ∘ ψ_*(ψ*(w)) ∘ ρ_{𝒢_1} = ψ_*(u^♯) ∘ ρ_{𝒢_2} ∘ w = u ∘ w,

that is,

(3.5.4.3)    (u ∘ w)^♯ = u^♯ ∘ ψ*(w).

Taking in particular $u={\rho }_{{\mathcal{G}}_3}\circ w^{\prime }$ for a morphism $w^{\prime }:{\mathcal{G}}_2\to {\mathcal{G}}_3$, one gets $\psi \ast (w^{\prime }\circ w)=({\rho }_{{\mathcal{G}}_3}\circ w^{\prime }\circ w{)}^{♯}=({\rho }_{{\mathcal{G}}_3}\circ w^{\prime }{)}^{♯}\circ \psi \ast (w)=\psi \ast (w^{\prime })\circ \psi \ast (w)$, proving functoriality.

Finally, for a sheaf $\mathcal{F}$ on $X$ with values in $K$, let $i_{\mathcal{F}}$ be the identity of ${\psi }_{\ast }(\mathcal{F})$ and set

σ_ℱ = (i_ℱ)^♯ : ψ*(ψ_*(ℱ)) → ℱ;

(3.5.4.3) then gives the factorization

(3.5.4.4)    u^♯ : ψ*(𝒢) ──ψ*(u)──→ ψ*(ψ_*(ℱ)) ──σ_ℱ──→ ℱ

for every morphism $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$. We call ${\sigma }_{\mathcal{F}}$ the canonical morphism.

(3.5.5) Let ${\psi }^{\prime }:Y\to Z$ be continuous and suppose every presheaf $\mathcal{H}$ on $Z$ with values in $K$ admits an inverse image ${\psi }^{\prime }\ast (\mathcal{H})$. Then (under the hypotheses of (3.5.4)) every presheaf $\mathcal{H}$ on $Z$ admits an inverse image by ${\psi }^{\prime \prime }={\psi }^{\prime }\circ \psi$, and there is a canonical functorial isomorphism

$$(\mathrm{3.5.5.1}){\psi }^{\prime \prime }\ast (\mathcal{H})\stackrel{\sim }{\to }\psi \ast ({\psi }^{\prime }\ast (\mathcal{H})).$$

This follows at once from the definitions, given that ${\psi }_{\ast }^{\prime \prime }={\psi }_{\ast }^{\prime }\circ {\psi }_{\ast }$. Moreover, if $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$ is a $\psi$-morphism, $v:\mathcal{H}\to {\psi }_{\ast }^{\prime }(\mathcal{G})$ a ${\psi }^{\prime }$-morphism, and $w={\psi }_{\ast }^{\prime }(u)\circ v$ their composite (3.5.2), one checks at once that $w^{♯}$ is the composite

w^♯ : ψ*(ψ′*(ℋ)) ──ψ*(v^♯)──→ ψ*(𝒢) ──u^♯──→ ℱ.

(3.5.6) Take in particular $\psi =1_X:X\to X$. If an inverse image by $\psi$ of a presheaf $\mathcal{F}$ on $X$ exists, this inverse image is called the sheaf associated with the presheaf $\mathcal{F}$. Every morphism $u:\mathcal{F}\to {\mathcal{F}}^{\prime }$ from $\mathcal{F}$ to a sheaf ${\mathcal{F}}^{\prime }$ with values in $K$ factors uniquely as

ℱ ──ρ_ℱ──→ 1_X*(ℱ) ──u^♯──→ ℱ′.

3.6. Simple and locally simple sheaves

(3.6.1) A presheaf $\mathcal{F}$ on $X$ with values in $K$ is called constant if the canonical morphisms $\mathcal{F}(X)\to \mathcal{F}(U)$ are isomorphisms for every nonempty open $U\subset X$; note that $\mathcal{F}$ is not necessarily a sheaf. A sheaf $\mathcal{F}$ is called simple if it is associated (3.5.6) with a constant presheaf. A sheaf $\mathcal{F}$ is called locally simple if every $x\in X$ admits an open neighborhood $U$ such that $\mathcal{F}|U$ is simple.

(3.6.2) Suppose $X$ is irreducible (2.1.1). The following are equivalent:

a) $\mathcal{F}$ is a constant presheaf on $X$; b) $\mathcal{F}$ is a simple sheaf on $X$; c) $\mathcal{F}$ is a locally simple sheaf on $X$.

Indeed, let $\mathcal{F}$ be a constant presheaf on $X$. If U, V are nonempty open subsets, then $U\cap V$ is nonempty; so $\mathcal{F}(X)\to \mathcal{F}(U)\to \mathcal{F}(U\cap V)$ and $\mathcal{F}(X)\to \mathcal{F}(V)\to \mathcal{F}(U\cap V)$ are isomorphisms, whence so are $\mathcal{F}(U)\to \mathcal{F}(U\cap V)$ and $\mathcal{F}(V)\to \mathcal{F}(U\cap V)$. One concludes at once that axiom (F) of (3.1.2) holds, so $\mathcal{F}$ is isomorphic to its associated sheaf, proving a) ⇒ b).

Now let $(U_{\alpha })$ be an open cover of $X$ by nonempty open sets and $\mathcal{F}$ a sheaf on $X$ with $\mathcal{F}|U_{\alpha }$ simple for every $\alpha$. Since $U_{\alpha }$ is irreducible, $\mathcal{F}|U_{\alpha }$ is a constant presheaf by the above. Since $U_{\alpha }\cap U_{\beta }\ne \varnothing$, both $\mathcal{F}(U_{\alpha })\to \mathcal{F}(U_{\alpha }\cap U_{\beta })$ and $\mathcal{F}(U_{\beta })\to \mathcal{F}(U_{\alpha }\cap U_{\beta })$ are isomorphisms, giving a canonical isomorphism ${\theta }_{\alpha \beta }:\mathcal{F}(U_{\alpha })\to \mathcal{F}(U_{\beta })$ for every pair. Applying (F) with $U=X$, one sees that for every index ${\alpha }_0$, the pair $(\mathcal{F}(U_{{\alpha }_0}),({\theta }_{\alpha {\alpha }_0}))$ is a solution of the universal problem; by uniqueness, $\mathcal{F}(X)\to \mathcal{F}(U_{\alpha })$ is an isomorphism, proving c) ⇒ a).

3.7. Inverse images of presheaves of groups or rings

(3.7.1) We show that when $K$ is the category of sets, the inverse image by $\psi$ of every presheaf $\mathcal{G}$ always exists (notation and hypotheses on $X$, $Y$, $\psi$ as in (3.5.3)). For each open $U\subset X$, define ${\mathcal{G}}^{\prime }(U)$ as follows: an element $s^{\prime }$ of ${\mathcal{G}}^{\prime }(U)$ is a family $(s_z^{\prime }{)}_{z\in U}$ with $s_z^{\prime }\in {\mathcal{G}}_{\psi (z)}$ such that for every $x\in U$, the following holds: there is an open neighborhood $V$ of $\psi (x)$ in $Y$, a neighborhood $W\subset {\psi }^{-1}(V)\cap U$ of $x$, and an element $s\in \mathcal{G}(V)$ with $s_z^{\prime }=s_{\psi (z)}$ for $z\in W$. One checks at once that $U\mapsto {\mathcal{G}}^{\prime }(U)$ is a sheaf.

Now let $\mathcal{F}$ be a sheaf of sets on $X$, and let $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$, $v:{\mathcal{G}}^{\prime }\to \mathcal{F}$ be morphisms. Define $u^{♯}$ and $v^{♭}$ as follows: if $s^{\prime }$ is a section of ${\mathcal{G}}^{\prime }$ over a neighborhood $U$ of $x\in X$ and $V$ is an open neighborhood of $\psi (x)$ with $s_z^{\prime }=s_{\psi (z)}$ for $z$ in a neighborhood of $x$ contained in ${\psi }^{-1}(V)\cap U$, set $u_x^{♯}(s_x^{\prime })=u_{\psi (x)}(s_{\psi (x)})$. If $s\in \mathcal{G}(V)$ for $V$ open in $Y$, let $v^{♭}(s)$ be the section of $\mathcal{F}$ over ${\psi }^{-1}(V)$, image under $v$ of the section $s^{\prime }$ of ${\mathcal{G}}^{\prime }$ with $s_z^{\prime }=s_{\psi (z)}$ for $z\in {\psi }^{-1}(V)$. The canonical homomorphism (3.5.3) $\rho :\mathcal{G}\to {\psi }_{\ast }(\psi \ast (\mathcal{G}))$ is defined by: for $V\subset Y$ open and $s\in \Gamma (V,\mathcal{G})$, $\rho (s)$ is the section $(s_{\psi (z)}{)}_{z\in {\psi }^{-1}(V)}$ of $\psi \ast (\mathcal{G})$ over ${\psi }^{-1}(V)$. The relations $(u^{♭}{)}^{♯}=u$, $(v^{♯}{)}^{♭}=v$, and $v^{♭}={\psi }_{\ast }(v)\circ \rho$ are immediate.

One checks that if $w:{\mathcal{G}}_1\to {\mathcal{G}}_2$ is a morphism of presheaves of sets on $Y$, then $\psi \ast (w)$ is given explicitly by: if $s^{\prime }=(s_z^{\prime }{)}_{z\in U}$ is a section of $\psi \ast ({\mathcal{G}}_1)$ over an open $U\subset X$, then $(\psi \ast (w))(s^{\prime })=(w_{\psi (z)}(s_z^{\prime }){)}_{z\in U}$. For every open $V\subset Y$, the inverse image of $\mathcal{G}|V$ by $\psi |{\psi }^{-1}(V)$ is identical with $\psi \ast (\mathcal{G})|{\psi }^{-1}(V)$.

When $\psi =1_X$ is the identity, the construction of ${\mathcal{G}}^{\prime }$ recovers the usual sheaf of sets associated with a presheaf (G, II, 1.2). The above goes through unchanged when $K$ is the category of groups or rings (not necessarily commutative).

When $X$ is an arbitrary subspace of a topological space $Y$ and $j:X\to Y$ is the canonical injection, the inverse image $j\ast (\mathcal{G})$, when it exists, is called the sheaf induced on $X$ by $\mathcal{G}$; for sheaves of sets (or groups, or rings) this recovers the usual definition (G, II, 1.5).

(3.7.2) Keeping the notation and hypotheses of (3.5.3), suppose $\mathcal{G}$ is a sheaf of groups (resp. rings) on $Y$. The construction of sections of $\psi \ast (\mathcal{G})$ (3.7.1) shows, in view of (3.4.4), that the stalk homomorphism ${\psi }_x\circ {\rho }_{\psi (x)}:{\mathcal{G}}_{\psi (x)}\to (\psi \ast (\mathcal{G}){)}_x$ is a functorial isomorphism in $\mathcal{G}$, which allows one to identify these two stalks. Under this identification, $u_x^{♯}$ is identical with the homomorphism defined in (3.5.1), and in particular

$$Supp(\psi \ast (\mathcal{G}))={\psi }^{-1}(Supp(\mathcal{G})).$$

An immediate consequence is that the functor $\psi \ast (\mathcal{G})$ is exact in $\mathcal{G}$ in the abelian category of sheaves of abelian groups.

3.8. Sheaves of pseudo-discrete spaces

(3.8.1) Let $X$ be a topological space whose topology admits a basis $\mathfrak{B}$ of quasi-compact open sets. Let $\mathcal{F}$ be a sheaf of sets on $X$. Endowing each $\mathcal{F}(U)$ with the discrete topology makes $U\mapsto \mathcal{F}(U)$ a presheaf of topological spaces. We shall see that there exists a sheaf of topological spaces ${\mathcal{F}}^{\prime }$ associated with $\mathcal{F}$ (3.5.6) such that $\Gamma (U,{\mathcal{F}}^{\prime })$ is the discrete space $\mathcal{F}(U)$ for every quasi-compact open $U$. For this, it suffices to show that the presheaf $U\mapsto \mathcal{F}(U)$ of topological spaces on $\mathfrak{B}$ satisfies (F₀) of (3.2.2), and more generally that if $U$ is a quasi-compact open set and $(U_{\alpha })$ is a cover of $U$ by elements of $\mathfrak{B}$, then the coarsest topology $\mathcal{T}$ on $\Gamma (U,\mathcal{F})$ making the maps $\Gamma (U,\mathcal{F})\to \Gamma (U_{\alpha },\mathcal{F})$ continuous is the discrete topology. There is a finite set of indices ${\alpha }_i$ with $U={\bigcup }_i{U}_{{\alpha }_i}$. Let $s\in \Gamma (U,\mathcal{F})$ and let $s_i$ be its image in $\Gamma (U_{{\alpha }_i},\mathcal{F})$; the intersection of the inverse images of $s_i$ is by definition a $\mathcal{T}$-neighborhood of $s$. But since $\mathcal{F}$ is a sheaf of sets and the $U_{{\alpha }_i}$ cover $U$, this intersection reduces to $s$, whence the assertion.

Note that if $U$ is a non-quasi-compact open subset of $X$, the topological space $\Gamma (U,{\mathcal{F}}^{\prime })$ still has $\Gamma (U,\mathcal{F})$ as underlying set, but its topology is in general not discrete: it is the coarsest making the maps $\Gamma (U,\mathcal{F})\to \Gamma (V,\mathcal{F})$ continuous, for $V\in \mathfrak{B}$, $V\subset U$ (the $\Gamma (V,\mathcal{F})$ being discrete).

The above applies without change to sheaves of groups or rings (not necessarily commutative), associating with them sheaves of topological groups or topological rings respectively. For brevity, we say that ${\mathcal{F}}^{\prime }$ is the sheaf of pseudo-discrete spaces (resp. groups, rings) associated with the sheaf of sets (resp. groups, rings) $\mathcal{F}$.

(3.8.2) Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of sets (resp. groups, rings) on $X$ and $u:\mathcal{F}\to \mathcal{G}$ a homomorphism. Then $u$ is also a continuous homomorphism ${\mathcal{F}}^{\prime }\to {\mathcal{G}}^{\prime }$, where ${\mathcal{F}}^{\prime }$, ${\mathcal{G}}^{\prime }$ are the associated pseudo-discrete sheaves; this follows from (3.2.5).

(3.8.3) Let $\mathcal{F}$ be a sheaf of sets, $\mathcal{H}$ a subsheaf of $\mathcal{F}$, and ${\mathcal{F}}^{\prime }$, ${\mathcal{H}}^{\prime }$ the associated pseudo-discrete sheaves. For every open $U\subset X$, $\Gamma (U,{\mathcal{H}}^{\prime })$ is closed in $\Gamma (U,{\mathcal{F}}^{\prime })$: indeed, it is the intersection of the inverse images of $\Gamma (V,\mathcal{H})$ (for $V\in \mathfrak{B}$, $V\subset U$) under the continuous maps $\Gamma (U,\mathcal{F})\to \Gamma (V,\mathcal{F})$, and $\Gamma (V,\mathcal{H})$ is closed in the discrete space $\Gamma (V,\mathcal{F})$.