Chapter 0 — Preliminaries

§4. Ringed Spaces

4.1. Ringed spaces; $\mathcal{A}$-Modules; $\mathcal{A}$-Algebras

(4.1.1) A ringed space (resp. topologically ringed space) is a pair $(X,\mathcal{A})$ consisting of a topological space $X$ and a sheaf of rings (not necessarily commutative) (resp. a sheaf of topological rings) $\mathcal{A}$ on $X$; we call $X$ the topological space underlying the ringed space $(X,\mathcal{A})$, and $\mathcal{A}$ the structure sheaf. The latter will be written ${\mathcal{O}}_X$, and its stalk at $x\in X$ will be written ${\mathcal{O}}_{X,x}$ or simply ${\mathcal{O}}_x$ when no confusion can arise.

We shall write 1 or $e$ for the unit section of ${\mathcal{O}}_X$ over $X$ (the unit element of $\Gamma (X,{\mathcal{O}}_X)$).

Since in this treatise we shall mainly consider sheaves of commutative rings, when we speak of a ringed space $(X,\mathcal{A})$ without further qualification it will be understood that $\mathcal{A}$ is a sheaf of commutative rings.

Ringed spaces with structure sheaf not necessarily commutative (resp. topologically ringed spaces) form a category, with morphisms $(X,\mathcal{A})\to (Y,\mathcal{B})$ defined as pairs $(\psi ,\theta )=\Psi$ consisting of a continuous map $\psi :X\to Y$ and a ψ-morphism $\theta :\mathcal{B}\to \mathcal{A}$ (3.5.1) of sheaves of rings (resp. sheaves of topological rings). The composite of a second morphism ${\Psi }^{\prime }=({\psi }^{\prime },{\theta }^{\prime }):(Y,\mathcal{B})\to (Z,\mathcal{C})$ with $\Psi$, written ${\Psi }^{\prime \prime }={\Psi }^{\prime }\circ \Psi$, is $({\psi }^{\prime \prime },{\theta }^{\prime \prime })$ with ${\psi }^{\prime \prime }={\psi }^{\prime }\circ \psi$ and ${\theta }^{\prime \prime }$ the composite of $\theta$ and ${\theta }^{\prime }$ (equal to ${\psi }_{\ast }^{\prime }(\theta )\circ {\theta }^{\prime }$; cf. 3.5.2). For ringed spaces, recall that ${\theta }^{\prime \prime ♯}={\theta }^{♯}\circ \psi \ast ({\theta }^{\prime ♯})$ (3.5.5); hence if ${\theta }^{♯}$ and ${\theta }^{\prime ♯}$ are injective (resp. surjective) homomorphisms, so is ${\theta }^{\prime \prime ♯}$, using that ${\psi }_x\circ {\rho }_{\psi (x)}$ is an isomorphism for every $x\in X$ (3.7.2). One checks at once that when $\psi$ is injective and ${\theta }^{♯}$ is surjective, the morphism $(\psi ,\theta )$ is a monomorphism (T, 1.1) in the category of ringed spaces.

By abuse of language, we shall often replace $\psi$ by $\Psi$ in the notation — for instance writing ${\Psi }^{-1}(U)$ in place of ${\psi }^{-1}(U)$ for $U\subset Y$ — when no confusion can arise.

(4.1.2) For every subset $M\subset X$, the pair $(M,\mathcal{A}|M)$ is plainly a ringed space, called induced on $M$ by $(X,\mathcal{A})$ (or the restriction of $(X,\mathcal{A})$ to $M$). If $j:M\to X$ is the canonical injection and $\omega$ the identity of $\mathcal{A}|M$, then $(j,{\omega }^{♭})$ is a monomorphism $(M,\mathcal{A}|M)\to (X,\mathcal{A})$ of ringed spaces, called the canonical injection. The composite of a morphism $\Psi :(X,\mathcal{A})\to (Y,\mathcal{B})$ with this injection is called the restriction of $\Psi$ to $M$.

(4.1.3) We shall not return to the definition of $\mathcal{A}$-Modules or algebraic sheaves on a ringed space $(X,\mathcal{A})$ (G, II, 2.2); when $\mathcal{A}$ is a sheaf of rings not necessarily commutative, by $\mathcal{A}$-Module we shall always mean "left $\mathcal{A}$-Module" unless explicit mention is made to the contrary. The $\mathcal{A}$-submodules of $\mathcal{A}$ will be called sheaves of (left, right, or two-sided) ideals in $\mathcal{A}$, or $\mathcal{A}$-Ideals.

When $\mathcal{A}$ is a sheaf of commutative rings, and the module structure is replaced everywhere by an algebra structure in the definition of $\mathcal{A}$-Modules, one obtains the definition of an $\mathcal{A}$-Algebra on $X$. Equivalently, an $\mathcal{A}$-Algebra (not necessarily commutative) is an $\mathcal{A}$-Module $\mathcal{C}$ together with an $\mathcal{A}$-Module homomorphism $\varphi :\mathcal{C}{\otimes }_{\mathcal{A}}\mathcal{C}\to \mathcal{C}$ and a section $e$ over $X$ such that:

1° the diagram

𝒞 ⊗_𝒜 𝒞 ⊗_𝒜 𝒞 ──φ⊗1──→ 𝒞 ⊗_𝒜 𝒞
        │                         │
      1⊗φ                         φ
        │                         │
        ↓                         ↓
   𝒞 ⊗_𝒜 𝒞 ─────φ────→         𝒞

commutes;

2° for every open $U\subset X$ and every section $s\in \Gamma (U,\mathcal{C})$, $\varphi ((e|U)\otimes s)=\varphi (s\otimes (e|U))=s$.

To say that $\mathcal{C}$ is a commutative $\mathcal{A}$-Algebra is to say in addition that the diagram

𝒞 ⊗_𝒜 𝒞 ──σ──→ 𝒞 ⊗_𝒜 𝒞
       ↘φ      ↙φ
          𝒞

commutes, where $\sigma$ is the canonical symmetry of the tensor product $\mathcal{C}{\otimes }_{\mathcal{A}}\mathcal{C}$.

Homomorphisms of $\mathcal{A}$-Algebras are defined as homomorphisms of $\mathcal{A}$-Modules as in (G, II, 2.2), but of course no longer form an abelian group.

If $\mathcal{M}$ is an $\mathcal{A}$-submodule of an $\mathcal{A}$-Algebra $\mathcal{C}$, the sub-$\mathcal{A}$-Algebra of $\mathcal{C}$ generated by $\mathcal{M}$ is the sum of the images of the homomorphisms ${\otimes }^n\mathcal{M}\to \mathcal{C}$ (for $n\ge 0$). It is also the sheaf associated with the presheaf of algebras $U\mapsto \mathcal{B}(U)$, where $\mathcal{B}(U)$ is the subalgebra of $\Gamma (U,\mathcal{C})$ generated by the submodule $\Gamma (U,\mathcal{M})$.

(4.1.4) A sheaf of rings $\mathcal{A}$ on a topological space $X$ is said to be reduced at a point $x\in X$ if the stalk ${\mathcal{A}}_x$ is a reduced ring (1.1.1); $\mathcal{A}$ is reduced if it is reduced at every point. Recall that a ring $A$ is said to be regular if each local ring $A_{\mathfrak{p}}$ ($\mathfrak{p}$ ranging over the prime ideals of $A$) is a regular local ring; we say that a sheaf of rings $\mathcal{A}$ on $X$ is regular at a point $x$ (resp. regular) if the stalk ${\mathcal{A}}_x$ is a regular ring (resp. if $\mathcal{A}$ is regular at every point). Finally, we say that $\mathcal{A}$ is normal at a point $x$ (resp. normal) if ${\mathcal{A}}_x$ is integral and integrally closed (resp. if $\mathcal{A}$ is normal at every point). A ringed space $(X,\mathcal{A})$ has one of these properties if $\mathcal{A}$ does.

A sheaf of graded rings $\mathcal{A}$ is by definition a sheaf of rings that is the direct sum (G, II, 2.7) of a family $({\mathcal{A}}_n{)}_{n\in \mathbb{Z}}$ of sheaves of abelian groups with ${\mathcal{A}}_m\cdot {\mathcal{A}}_n\subset {\mathcal{A}}_{m+n}$. A graded $\mathcal{A}$-Module is an $\mathcal{A}$-Module $\mathcal{F}$ that is a direct sum of a family $({\mathcal{F}}_n{)}_{n\in \mathbb{Z}}$ of sheaves of abelian groups satisfying ${\mathcal{A}}_m\cdot {\mathcal{F}}_n\subset {\mathcal{F}}_{m+n}$. Equivalently, $({\mathcal{A}}_m{)}_x({\mathcal{A}}_n{)}_x\subset ({\mathcal{A}}_{m+n}{)}_x$ (resp. $({\mathcal{A}}_m{)}_x({\mathcal{F}}_n{)}_x\subset ({\mathcal{F}}_{m+n}{)}_x$) at every point $x$.

(4.1.5) For a ringed space $(X,\mathcal{A})$ (not necessarily commutative), we shall not recall the definitions of the bifunctors $\mathcal{F}{\otimes }_{\mathcal{A}}\mathcal{G}$, $\mathcal{H}o{m}_{\mathcal{A}}(\mathcal{F},\mathcal{G})$, and ${\mathrm{Hom}}_{\mathcal{A}}(\mathcal{F},\mathcal{G})$ (G, II, 2.8 and 2.2) on the categories of left or right $\mathcal{A}$-Modules (as the case may be), with values in the category of sheaves of abelian groups (or more generally $\mathcal{C}$-Modules, where $\mathcal{C}$ is the center of $\mathcal{A}$). The stalk $(\mathcal{F}{\otimes }_{\mathcal{A}}\mathcal{G}{)}_x$ at any $x\in X$ is canonically identified with ${\mathcal{F}}_x{\otimes }_{{\mathcal{A}}_x}{\mathcal{G}}_x$, and there is a canonical functorial homomorphism $(\mathcal{H}o{m}_{\mathcal{A}}(\mathcal{F},\mathcal{G}){)}_x\to {\mathrm{Hom}}_{{\mathcal{A}}_x}({\mathcal{F}}_x,{\mathcal{G}}_x)$ which in general is neither injective nor surjective. The bifunctors above are additive and in particular commute with finite direct sums; $\mathcal{F}{\otimes }_{\mathcal{A}}\mathcal{G}$ is right exact in $\mathcal{F}$ and in $\mathcal{G}$, commutes with inductive limits, and $\mathcal{A}\otimes \mathcal{G}$ (resp. $\mathcal{F}{\otimes }_{\mathcal{A}}\mathcal{A}$) is canonically identified with $\mathcal{G}$ (resp. $\mathcal{F}$). The functors $\mathcal{H}o{m}_{\mathcal{A}}(\mathcal{F},\mathcal{G})$ and ${\mathrm{Hom}}_{\mathcal{A}}(\mathcal{F},\mathcal{G})$ are left exact in $\mathcal{F}$ and $\mathcal{G}$; precisely, given an exact sequence $0\to {\mathcal{G}}^{\prime }\to \mathcal{G}\to {\mathcal{G}}^{\prime \prime }$, the sequence

0 → ℋom_𝒜(ℱ, 𝒢′) → ℋom_𝒜(ℱ, 𝒢) → ℋom_𝒜(ℱ, 𝒢″)

is exact; given an exact sequence ${\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$, the sequence

0 → ℋom_𝒜(ℱ″, 𝒢) → ℋom_𝒜(ℱ, 𝒢) → ℋom_𝒜(ℱ′, 𝒢)

is exact; analogous properties hold for Hom. Moreover, $\mathcal{H}o{m}_{\mathcal{A}}(\mathcal{A},\mathcal{G})$ is canonically identified with $\mathcal{G}$; finally, for every open $U\subset X$,

Γ(U, ℋom_𝒜(ℱ, 𝒢)) = Hom_{𝒜|U}(ℱ|U, 𝒢|U).

For any left (resp. right) $\mathcal{A}$-Module $\mathcal{F}$, the dual of $\mathcal{F}$, written $\check{\mathcal{F}}$, is the right (resp. left) $\mathcal{A}$-Module $\mathcal{H}o{m}_{\mathcal{A}}(\mathcal{F},\mathcal{A})$.

Finally, if $\mathcal{A}$ is a sheaf of commutative rings and $\mathcal{F}$ is an $\mathcal{A}$-Module, then $U\mapsto {\bigwedge }^p\Gamma (U,\mathcal{F})$ is a presheaf whose associated sheaf is an $\mathcal{A}$-Module written ${\bigwedge }^p\mathcal{F}$, called the $p$-th exterior power of $\mathcal{F}$. One checks easily that the canonical map of the presheaf $U\mapsto {\bigwedge }^p\Gamma (U,\mathcal{F})$ into ${\bigwedge }^p\mathcal{F}$ is injective, and that for every $x\in X$, $({\bigwedge }^p\mathcal{F}{)}_x={\bigwedge }^p({\mathcal{F}}_x)$. Plainly ${\bigwedge }^p\mathcal{F}$ is a covariant functor in $\mathcal{F}$.

(4.1.6) Suppose $\mathcal{A}$ is a sheaf of rings not necessarily commutative, $\mathcal{J}$ a sheaf of left ideals of $\mathcal{A}$, and $\mathcal{F}$ a left $\mathcal{A}$-Module. We write $\mathcal{J}\mathcal{F}$ for the sub-$\mathcal{A}$-Module of $\mathcal{F}$ that is the image of $\mathcal{J}{\otimes }_{\mathbb{Z}}\mathcal{F}$ (where $\mathbb{Z}$ is the sheaf associated with the constant presheaf $U\mapsto \mathbb{Z}$) under the canonical map $\mathcal{J}{\otimes }_{\mathbb{Z}}\mathcal{F}\to \mathcal{F}$; plainly $(\mathcal{J}\mathcal{F}{)}_x={\mathcal{J}}_x{\mathcal{F}}_x$ for every $x\in X$. When $\mathcal{A}$ is commutative, $\mathcal{J}\mathcal{F}$ is also the canonical image of $\mathcal{J}{\otimes }_{\mathcal{A}}\mathcal{F}\to \mathcal{F}$. It is immediate that $\mathcal{J}\mathcal{F}$ is also the $\mathcal{A}$-Module associated with the presheaf $U\mapsto \Gamma (U,\mathcal{J})\Gamma (U,\mathcal{F})$. If ${\mathcal{J}}_1,{\mathcal{J}}_2$ are two sheaves of left ideals, then ${\mathcal{J}}_1({\mathcal{J}}_2\mathcal{F})=({\mathcal{J}}_1{\mathcal{J}}_2)\mathcal{F}$.

(4.1.7) Let $(X_{\lambda },{\mathcal{A}}_{\lambda }{)}_{\lambda \in L}$ be a family of ringed spaces; for each pair $(\lambda ,\mu )$, suppose given an open subset $V_{\mu \lambda }\subset X_{\lambda }$ and an isomorphism of ringed spaces ${\varphi }_{\lambda \mu }:(V_{\mu \lambda },{\mathcal{A}}_{\mu }|V_{\mu \lambda })\stackrel{\sim }{\to }(V_{\lambda \mu },{\mathcal{A}}_{\lambda }|V_{\lambda \mu })$, with $V_{\lambda \lambda }=X_{\lambda }$ and ${\varphi }_{\lambda \lambda }$ the identity. Suppose moreover that for every triple $(\lambda ,\mu ,\nu )$, writing ${\varphi }_{\lambda \mu }^{\prime }$ for the restriction of ${\varphi }_{\lambda \mu }$ to $V_{\mu \lambda }\cap V_{\mu \nu }$, the map ${\varphi }_{\lambda \mu }^{\prime }$ is an isomorphism of $(V_{\mu \lambda }\cap V_{\mu \nu },{\mathcal{A}}_{\mu }|(V_{\mu \lambda }\cap V_{\mu \nu }))$ onto $(V_{\lambda \mu }\cap V_{\lambda \nu },{\mathcal{A}}_{\lambda }|(V_{\lambda \mu }\cap V_{\lambda \nu }))$, and that ${\varphi }_{\lambda \nu }^{\prime }={\varphi }_{\lambda \mu }^{\prime }\circ {\varphi }_{\mu \nu }^{\prime }$ (gluing condition for the ${\varphi }_{\lambda \mu }$). One first forms the topological space obtained by gluing the $X_{\lambda }$ along the $V_{\lambda \mu }$ via the ${\varphi }_{\lambda \mu }$; identifying $X_{\lambda }$ with the open subset $X_{\lambda }^{\prime }$ of $X$, the hypotheses imply that the three subsets $V_{\lambda \mu }\cap V_{\lambda \nu }$, $V_{\mu \nu }\cap V_{\mu \lambda }$, $V_{\nu \lambda }\cap V_{\nu \mu }$ are identified with $X_{\lambda }^{\prime }\cap X_{\mu }^{\prime }\cap X_{\nu }^{\prime }$. One then transports the ringed-space structure of $X_{\lambda }$ to $X_{\lambda }^{\prime }$; the resulting ${\mathcal{A}}_{\lambda }^{\prime }$ satisfy the gluing condition (3.3.1) and hence define a sheaf of rings $\mathcal{A}$ on $X$. One says that $(X,\mathcal{A})$ is the ringed space obtained by gluing the $(X_{\lambda },{\mathcal{A}}_{\lambda })$ along the $V_{\lambda \mu }$ via the ${\varphi }_{\lambda \mu }$.

4.2. Direct image of an $\mathcal{A}$-Module

(4.2.1) Let $(X,\mathcal{A})$, $(Y,\mathcal{B})$ be two ringed spaces and $\Psi =(\psi ,\theta )$ a morphism $(X,\mathcal{A})\to (Y,\mathcal{B})$. Then ${\psi }_{\ast }(\mathcal{A})$ is a sheaf of rings on $Y$ and $\theta :\mathcal{B}\to {\psi }_{\ast }(\mathcal{A})$ is a sheaf-of-rings homomorphism. Let $\mathcal{F}$ be an $\mathcal{A}$-Module; its direct image ${\psi }_{\ast }(\mathcal{F})$ is a sheaf of abelian groups on $Y$. Moreover, for every open $U\subset Y$,

Γ(U, ψ_*(ℱ)) = Γ(ψ⁻¹(U), ℱ)

is equipped with a module structure over the ring $\Gamma (U,{\psi }_{\ast }(\mathcal{A}))=\Gamma ({\psi }^{-1}(U),\mathcal{A})$; the bilinear maps defining these structures are compatible with restriction, giving ${\psi }_{\ast }(\mathcal{F})$ a structure of ${\psi }_{\ast }(\mathcal{A})$-Module. The homomorphism $\theta :\mathcal{B}\to {\psi }_{\ast }(\mathcal{A})$ then equips ${\psi }_{\ast }(\mathcal{F})$ with a $\mathcal{B}$-Module structure as well; this $\mathcal{B}$-Module is called the direct image of $\mathcal{F}$ by $\Psi$ and is written ${\Psi }_{\ast }(\mathcal{F})$. If ${\mathcal{F}}_1,{\mathcal{F}}_2$ are two $\mathcal{A}$-Modules on $X$ and $u:{\mathcal{F}}_1\to {\mathcal{F}}_2$ is an $\mathcal{A}$-homomorphism, it is immediate (considering sections over open subsets of $Y$) that ${\psi }_{\ast }(u)$ is a ${\psi }_{\ast }(\mathcal{A})$-homomorphism ${\psi }_{\ast }({\mathcal{F}}_1)\to {\psi }_{\ast }({\mathcal{F}}_2)$, and a fortiori a $\mathcal{B}$-homomorphism ${\Psi }_{\ast }({\mathcal{F}}_1)\to {\Psi }_{\ast }({\mathcal{F}}_2)$; as such, it is written ${\Psi }_{\ast }(u)$. Thus ${\Psi }_{\ast }$ is a covariant functor from $\mathcal{A}$-Modules to $\mathcal{B}$-Modules; it is left exact (G, II, 2.12).

On ${\psi }_{\ast }(\mathcal{A})$, the $\mathcal{B}$-Module structure together with the sheaf-of-rings structure gives a structure of $\mathcal{B}$-Algebra; this $\mathcal{B}$-Algebra is written ${\Psi }_{\ast }(\mathcal{A})$.

(4.2.2) Let $\mathcal{M}$, $\mathcal{N}$ be two $\mathcal{A}$-Modules. For every open $U\subset Y$, there is a canonical map

Γ(ψ⁻¹(U), ℳ) × Γ(ψ⁻¹(U), 𝒩) → Γ(ψ⁻¹(U), ℳ ⊗_𝒜 𝒩)

which is bilinear over $\Gamma ({\psi }^{-1}(U),\mathcal{A})=\Gamma (U,{\psi }_{\ast }(\mathcal{A}))$, and a fortiori over $\Gamma (U,\mathcal{B})$; it defines a homomorphism

Γ(U, Ψ_*(ℳ)) ⊗_{Γ(U, ℬ)} Γ(U, Ψ_*(𝒩)) → Γ(U, Ψ_*(ℳ ⊗_𝒜 𝒩))

which is compatible with restriction; the result is a canonical functorial homomorphism of $\mathcal{B}$-Modules

(4.2.2.1)    Ψ_*(ℳ) ⊗_ℬ Ψ_*(𝒩) → Ψ_*(ℳ ⊗_𝒜 𝒩),

in general neither injective nor surjective. If $\mathcal{P}$ is a third $\mathcal{A}$-Module, the diagram

(4.2.2.2)    Ψ_*(ℳ) ⊗_ℬ Ψ_*(𝒩) ⊗_ℬ Ψ_*(𝒫) → Ψ_*(ℳ ⊗_𝒜 𝒩) ⊗_ℬ Ψ_*(𝒫)
                        │                                    │
                        ↓                                    ↓
             Ψ_*(ℳ) ⊗_ℬ Ψ_*(𝒩 ⊗_𝒜 𝒫) ─────────────→  Ψ_*(ℳ ⊗_𝒜 𝒩 ⊗_𝒜 𝒫)

commutes.

(4.2.3) Let $\mathcal{M}$, $\mathcal{N}$ be two $\mathcal{A}$-Modules. For every open $U\subset Y$, by definition $\Gamma ({\psi }^{-1}(U),\mathcal{H}o{m}_{\mathcal{A}}(\mathcal{M},\mathcal{N}))={\mathrm{Hom}}_{\mathcal{A}|V}(\mathcal{M}|V,\mathcal{N}|V)$ with $V={\psi }^{-1}(U)$. The map $u\mapsto {\psi }_{\ast }(u)$ is a homomorphism

Hom_{𝒜|V}(ℳ|V, 𝒩|V) → Hom_{ℬ|U}(Ψ_*(ℳ)|U, Ψ_*(𝒩)|U)

for the $\Gamma (U,\mathcal{B})$-module structures. These homomorphisms are compatible with restriction, so they define a canonical functorial homomorphism of $\mathcal{B}$-Modules

(4.2.3.1)    Ψ_*(ℋom_𝒜(ℳ, 𝒩)) → ℋom_ℬ(Ψ_*(ℳ), Ψ_*(𝒩)).

(4.2.4) If $\mathcal{C}$ is an $\mathcal{A}$-Algebra, the composite

Ψ_*(𝒞) ⊗_ℬ Ψ_*(𝒞) → Ψ_*(𝒞 ⊗_𝒜 𝒞) → Ψ_*(𝒞)

equips ${\Psi }_{\ast }(\mathcal{C})$ with a $\mathcal{B}$-Algebra structure, by (4.2.2.2). Similarly, if $\mathcal{M}$ is a $\mathcal{C}$-Module, ${\Psi }_{\ast }(\mathcal{M})$ is canonically equipped with a ${\Psi }_{\ast }(\mathcal{C})$-Module structure.

(4.2.5) Consider the special case in which $X$ is a closed subspace of $Y$ and $\psi$ is the canonical injection $j:X\to Y$. Let ${\mathcal{B}}^{\prime }=\mathcal{B}|X=j\ast (\mathcal{B})$ be the restriction of the sheaf of rings $\mathcal{B}$ to $X$; an $\mathcal{A}$-Module $\mathcal{M}$ may be viewed as a ${\mathcal{B}}^{\prime }$-Module via ${\theta }^{♯}:{\mathcal{B}}^{\prime }\to \mathcal{A}$. Then ${\Psi }_{\ast }(\mathcal{M})$ is the $\mathcal{B}$-Module that induces $\mathcal{M}$ on $X$ and 0 elsewhere. If $\mathcal{N}$ is a second $\mathcal{A}$-Module, ${\Psi }_{\ast }(\mathcal{M}){\otimes }_{\mathcal{B}}{\Psi }_{\ast }(\mathcal{N})$ is identified with ${\Psi }_{\ast }(\mathcal{M}{\otimes }_{{\mathcal{B}}^{\prime }}\mathcal{N})$, and $\mathcal{H}o{m}_{\mathcal{B}}({\Psi }_{\ast }(\mathcal{M}),{\Psi }_{\ast }(\mathcal{N}))$ with ${\Psi }_{\ast }(\mathcal{H}o{m}_{{\mathcal{B}}^{\prime }}(\mathcal{M},\mathcal{N}))$.

(4.2.6) Let $(Z,\mathcal{C})$ be a third ringed space and ${\Psi }^{\prime }=({\psi }^{\prime },{\theta }^{\prime }):(Y,\mathcal{B})\to (Z,\mathcal{C})$ a morphism; if ${\Psi }^{\prime \prime }={\Psi }^{\prime }\circ \Psi$, then plainly ${\Psi }_{\ast }^{\prime \prime }={\Psi }_{\ast }^{\prime }\circ {\Psi }_{\ast }$.

4.3. Inverse image of a $\mathcal{B}$-Module

(4.3.1) Under the hypotheses and notation of (4.2.1), let $\mathcal{G}$ be a $\mathcal{B}$-Module and $\psi \ast (\mathcal{G})$ its inverse image (3.7.1) — a sheaf of abelian groups on $X$. The construction of sections of $\psi \ast (\mathcal{G})$ and $\psi \ast (\mathcal{B})$ (3.7.1) shows that $\psi \ast (\mathcal{G})$ is canonically a $\psi \ast (\mathcal{B})$-Module. The homomorphism ${\theta }^{♯}:\psi \ast (\mathcal{B})\to \mathcal{A}$ makes $\mathcal{A}$ a $\psi \ast (\mathcal{B})$-Module, written ${\mathcal{A}}_{[\theta ]}$ when confusion threatens; the tensor product $\psi \ast (\mathcal{G}){\otimes }_{\psi \ast (\mathcal{B})}{\mathcal{A}}_{[\theta ]}$ is then an $\mathcal{A}$-Module. We call this $\mathcal{A}$-Module the inverse image of $\mathcal{G}$ by $\Psi$ and write it $\Psi \ast (\mathcal{G})$. If ${\mathcal{G}}_1,{\mathcal{G}}_2$ are $\mathcal{B}$-Modules and $v:{\mathcal{G}}_1\to {\mathcal{G}}_2$ is a $\mathcal{B}$-homomorphism, one checks at once that $v$ is a $\psi \ast (\mathcal{B})$-homomorphism $\psi \ast ({\mathcal{G}}_1)\to \psi \ast ({\mathcal{G}}_2)$; hence $\psi \ast (v)\otimes 1$ is an $\mathcal{A}$-homomorphism $\Psi \ast ({\mathcal{G}}_1)\to \Psi \ast ({\mathcal{G}}_2)$, written $\Psi \ast (v)$. Thus $\Psi \ast$ is a covariant functor from $\mathcal{B}$-Modules to $\mathcal{A}$-Modules. Unlike $\psi \ast$, this functor is not exact in general but only right exact, tensoring with $\mathcal{A}$ being a right-exact functor on $\psi \ast (\mathcal{B})$-Modules.

For every $x\in X$, by (3.7.2),

(Ψ*(𝒢))_x = 𝒢_{ψ(x)} ⊗_{ℬ_{ψ(x)}} 𝒜_x.

The support of $\Psi \ast (\mathcal{G})$ is therefore contained in ${\psi }^{-1}(Supp(\mathcal{G}))$.

(4.3.2) Let $({\mathcal{G}}_{\lambda })$ be an inductive system of $\mathcal{B}$-Modules with $\mathcal{G}=\underrightarrow{\lim }{\mathcal{G}}_{\lambda }$. The canonical homomorphisms ${\mathcal{G}}_{\lambda }\to \mathcal{G}$ give $\psi \ast (\mathcal{B})$-homomorphisms $\psi \ast ({\mathcal{G}}_{\lambda })\to \psi \ast (\mathcal{G})$, and so a canonical homomorphism $\underrightarrow{\lim }\psi \ast ({\mathcal{G}}_{\lambda })\to \psi \ast (\mathcal{G})$. Since the stalk of an inductive limit of sheaves is the inductive limit of stalks (G, II, 1.11), this map is bijective (3.7.2). Tensor product also commutes with inductive limits, so there is a canonical functorial isomorphism of $\mathcal{A}$-Modules

lim⃗ Ψ*(𝒢_λ) ⥲ Ψ*(lim⃗ 𝒢_λ).

For a finite direct sum ${\oplus }_i{\mathcal{G}}_i$ of $\mathcal{B}$-Modules, plainly $\psi \ast ({\oplus }_i{\mathcal{G}}_i)={\oplus }_i\psi \ast ({\mathcal{G}}_i)$, so tensoring with ${\mathcal{A}}_{[\theta ]}$ gives

(4.3.2.1)    Ψ*(⊕_i 𝒢_i) = ⊕_i Ψ*(𝒢_i).

By passage to inductive limit, the equality extends to arbitrary direct sums.

(4.3.3) Let ${\mathcal{G}}_1,{\mathcal{G}}_2$ be $\mathcal{B}$-Modules. From the construction of inverse images of sheaves of abelian groups (3.7.1) one obtains at once a canonical homomorphism

ψ*(𝒢_1) ⊗_{ψ*(ℬ)} ψ*(𝒢_2) → ψ*(𝒢_1 ⊗_ℬ 𝒢_2)

of $\psi \ast (\mathcal{B})$-Modules; since the stalk of a tensor product of sheaves is the tensor product of stalks (G, II, 2.8), (3.7.2) shows that this map is an isomorphism. Tensoring with $\mathcal{A}$ gives a canonical functorial isomorphism

(4.3.3.1)    Ψ*(𝒢_1) ⊗_𝒜 Ψ*(𝒢_2) ⥲ Ψ*(𝒢_1 ⊗_ℬ 𝒢_2).

(4.3.4) Let $\mathcal{C}$ be a $\mathcal{B}$-Algebra; giving $\mathcal{C}$ an algebra structure amounts to giving a $\mathcal{B}$-homomorphism $\mathcal{C}{\otimes }_{\mathcal{B}}\mathcal{C}\to \mathcal{C}$ satisfying associativity and commutativity (verified stalkwise). The isomorphism above transports this into an $\mathcal{A}$-Module homomorphism $\Psi \ast (\mathcal{C}){\otimes }_{\mathcal{A}}\Psi \ast (\mathcal{C})\to \Psi \ast (\mathcal{C})$ with the same properties, equipping $\Psi \ast (\mathcal{C})$ with an $\mathcal{A}$-Algebra structure. In particular, the $\mathcal{A}$-Algebra $\Psi \ast (\mathcal{B})$ is equal to $\mathcal{A}$ (up to canonical isomorphism).

Similarly, if $\mathcal{M}$ is a $\mathcal{C}$-Module, giving the module structure amounts to a $\mathcal{B}$-homomorphism $\mathcal{C}{\otimes }_{\mathcal{B}}\mathcal{M}\to \mathcal{M}$ satisfying associativity; transport gives a $\Psi \ast (\mathcal{C})$-Module structure on $\Psi \ast (\mathcal{M})$.

(4.3.5) Let $\mathcal{J}$ be a sheaf of ideals of $\mathcal{B}$; since $\psi \ast$ is exact, the $\psi \ast (\mathcal{B})$-Module $\psi \ast (\mathcal{J})$ is canonically identified with a sheaf of ideals of $\psi \ast (\mathcal{B})$. The canonical injection $\psi \ast (\mathcal{J})\to \psi \ast (\mathcal{B})$ then gives an $\mathcal{A}$-Module homomorphism $\Psi \ast (\mathcal{J})=\psi \ast (\mathcal{J}){\otimes }_{\psi \ast (\mathcal{B})}{\mathcal{A}}_{[\theta ]}\to \mathcal{A}$; we write $\Psi \ast (\mathcal{J})\cdot \mathcal{A}$, or $\mathcal{J}\mathcal{A}$ when no confusion threatens, for the image of $\Psi \ast (\mathcal{J})$ under this map. Thus by definition $\mathcal{J}\mathcal{A}={\theta }^{♯}(\psi \ast (\mathcal{J}))\cdot \mathcal{A}$, and in particular $(\mathcal{J}\mathcal{A}{)}_x={\theta }_x^{♯}({\mathcal{J}}_{\psi (x)})\cdot {\mathcal{A}}_x$ using the canonical identification of stalks of $\psi \ast (\mathcal{J})$ with those of $\mathcal{J}$ (3.7.2). If ${\mathcal{J}}_1,{\mathcal{J}}_2$ are two ideals of $\mathcal{B}$, $({\mathcal{J}}_1{\mathcal{J}}_2)\mathcal{A}={\mathcal{J}}_1({\mathcal{J}}_2\mathcal{A})=({\mathcal{J}}_1\mathcal{A})({\mathcal{J}}_2\mathcal{A})$. For an $\mathcal{A}$-Module $\mathcal{F}$, set $\mathcal{J}\mathcal{F}=(\mathcal{J}\mathcal{A})\mathcal{F}$.

(4.3.6) Let $(Z,\mathcal{C})$ be a third ringed space and ${\Psi }^{\prime }=({\psi }^{\prime },{\theta }^{\prime }):(Y,\mathcal{B})\to (Z,\mathcal{C})$ a morphism. If ${\Psi }^{\prime \prime }={\Psi }^{\prime }\circ \Psi$, then by the definition (4.3.1) and (4.3.3.1), ${\Psi }^{\prime \prime }\ast =\Psi \ast \circ {\Psi }^{\prime }\ast$.

4.4. Relations between direct and inverse images

(4.4.1) Under the hypotheses and notation of (4.2.1), let $\mathcal{G}$ be a $\mathcal{B}$-Module. A $\mathcal{B}$-Module homomorphism $u:\mathcal{G}\to {\Psi }_{\ast }(\mathcal{F})$ is also called a $\Psi$-homomorphism of $\mathcal{G}$ into $\mathcal{F}$ — or simply a homomorphism of $\mathcal{G}$ into $\mathcal{F}$, written $u:\mathcal{G}\to \mathcal{F}$, when no confusion threatens. To give such a homomorphism is to give, for every pair $(U,V)$ with $U\subset X$ open, $V\subset Y$ open, $\psi (U)\subset V$, a homomorphism $u_{U,V}:\Gamma (V,\mathcal{G})\to \Gamma (U,\mathcal{F})$ of $\Gamma (V,\mathcal{B})$-modules — $\Gamma (U,\mathcal{F})$ viewed as a $\Gamma (V,\mathcal{B})$-module via ${\theta }_{U,V}:\Gamma (V,\mathcal{B})\to \Gamma (U,\mathcal{A})$ — with the $u_{U,V}$ making the diagrams (3.5.1.1) commute. It suffices to give the $u_{U,V}$ for $U$ (resp. $V$) ranging over a basis $\mathfrak{B}$ of the topology of $X$ (resp. ${\mathfrak{B}}^{\prime }$ of $Y$), provided one checks (3.5.1.1) for those restrictions.

(4.4.2) Under the hypotheses of (4.2.1) and (4.2.6), let $\mathcal{H}$ be a $\mathcal{C}$-Module and $v:\mathcal{H}\to {\Psi }_{\ast }^{\prime }(\mathcal{G})$ a ${\Psi }^{\prime }$-morphism; then

w : ℋ ──v──→ Ψ′_*(𝒢) ──Ψ′_*(u)──→ Ψ′_*(Ψ_*(ℱ))

is a ${\Psi }^{\prime \prime }$-morphism, called the composite of $u$ and $v$.

(4.4.3) We now show that there is a canonical isomorphism of bifunctors in $\mathcal{F}$ and $\mathcal{G}$

(4.4.3.1)    Hom_𝒜(Ψ*(𝒢), ℱ) ⥲ Hom_ℬ(𝒢, Ψ_*(ℱ)),

written $v\mapsto v^{♭}$ (or simply $v\mapsto v^{♭}$); the inverse is written $u\mapsto u^{♯}$. The definition is as follows: composing $v:\Psi \ast (\mathcal{G})\to \mathcal{F}$ with the canonical map $\psi \ast (\mathcal{G})\to \Psi \ast (\mathcal{G})$ gives a sheaf-of-groups homomorphism $v^{\prime }:\psi \ast (\mathcal{G})\to \mathcal{F}$, which is also a $\psi \ast (\mathcal{B})$-module homomorphism. From it (3.7.1) one deduces a homomorphism $v^{\prime ♭}:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})={\Psi }_{\ast }(\mathcal{F})$, also a $\mathcal{B}$-Module homomorphism (as one checks); set $v^{♭}=v^{\prime ♭}$. Similarly, from $u:\mathcal{G}\to {\Psi }_{\ast }(\mathcal{F})$ one deduces (3.7.1) a $\psi \ast (\mathcal{B})$-Module homomorphism $u^{♯}:\psi \ast (\mathcal{G})\to \mathcal{F}$, and by tensoring with $\mathcal{A}$ an $\mathcal{A}$-Module homomorphism $\Psi \ast (\mathcal{G})\to \mathcal{F}$, which we write $u^{♯}$. One verifies at once that $(u^{♯}{)}^{♭}=u$ and $(v^{♭}{)}^{♯}=v$, and the functoriality in $\mathcal{F}$ of $v\mapsto v^{♭}$. The functoriality in $\mathcal{G}$ of $u\mapsto u^{♯}$ then follows formally as in (3.5.4) (this reasoning would also give the functoriality of $\Psi \ast$ established directly in (4.3.1)).

Taking $v$ to be the identity of $\Psi \ast (\mathcal{G})$, $v^{♭}$ is a homomorphism

$$(\mathrm{4.4.3.2}){\rho }_{\mathcal{G}}:\mathcal{G}\to {\Psi }_{\ast }(\Psi \ast (\mathcal{G}));$$

taking $u$ to be the identity of ${\Psi }_{\ast }(\mathcal{F})$, $u^{♯}$ is a homomorphism

$$(\mathrm{4.4.3.3}){\sigma }_{\mathcal{F}}:\Psi \ast ({\Psi }_{\ast }(\mathcal{F}))\to \mathcal{F};$$

these will be called canonical. In general they are neither injective nor surjective. Canonical factorizations analogous to (3.5.3.3) and (3.5.4.4) hold.

Note that if $s$ is a section of $\mathcal{G}$ over an open $V\subset Y$, then ${\rho }_{\mathcal{G}}(s)$ is the section $s^{\prime }\otimes 1$ of $\Psi \ast (\mathcal{G})$ over ${\psi }^{-1}(V)$, where $s_x^{\prime }=s_{\psi (x)}$ for $x\in {\psi }^{-1}(V)$. Note also that any homomorphism $u:\mathcal{G}\to {\psi }_{\ast }(\mathcal{F})$ defines for each $x\in X$ a stalk homomorphism $u_x:{\mathcal{G}}_{\psi (x)}\to {\mathcal{F}}_x$, factoring through the canonical map $s_x\mapsto s_x\otimes 1$ of ${\mathcal{G}}_{\psi (x)}$ into $(\Psi \ast (\mathcal{G}){)}_x={\mathcal{G}}_{\psi (x)}{\otimes }_{{\mathcal{B}}_{\psi (x)}}{\mathcal{A}}_x$. The map $u_x$ is also the inductive limit of $\Gamma (V,\mathcal{G})\stackrel{u}{\to }\Gamma ({\psi }^{-1}(V),\mathcal{F})\to {\mathcal{F}}_x$ over neighborhoods $V$ of $\psi (x)$.

(4.4.4) Let ${\mathcal{F}}_1,{\mathcal{F}}_2$ be $\mathcal{A}$-Modules and ${\mathcal{G}}_1,{\mathcal{G}}_2$ be $\mathcal{B}$-Modules, with $u_i:{\mathcal{G}}_i\to {\mathcal{F}}_i$ a homomorphism ($i=1,2$). We write $u_1\otimes u_2$ for the homomorphism $u:{\mathcal{G}}_1{\otimes }_{\mathcal{B}}{\mathcal{G}}_2\to {\mathcal{F}}_1{\otimes }_{\mathcal{A}}{\mathcal{F}}_2$ with $u^{♯}=(u_1{)}^{♯}\otimes (u_2{)}^{♯}$ (using (4.3.3.1)); one checks that $u$ is also the composite

𝒢_1 ⊗_ℬ 𝒢_2 → Ψ_*(ℱ_1) ⊗_ℬ Ψ_*(ℱ_2) → Ψ_*(ℱ_1 ⊗_𝒜 ℱ_2),

where the first arrow is the ordinary tensor product $u_1\otimes u_2$ and the second is the canonical map (4.2.2.1).

(4.4.5) Let $({\mathcal{G}}_{\lambda }{)}_{\lambda \in L}$ be an inductive system of $\mathcal{B}$-Modules, and for each $\lambda$ let $u_{\lambda }:{\mathcal{G}}_{\lambda }\to {\Psi }_{\ast }(\mathcal{F})$ be a homomorphism, forming an inductive system. Set $\mathcal{G}=\underrightarrow{\lim }{\mathcal{G}}_{\lambda }$ and $u=\underrightarrow{\lim }{u}_{\lambda }$; then the $(u_{\lambda }{)}^{♯}$ form an inductive system of homomorphisms $\Psi \ast ({\mathcal{G}}_{\lambda })\to \mathcal{F}$, whose inductive limit is $u^{♯}$.

(4.4.6) Let $\mathcal{M}$, $\mathcal{N}$ be $\mathcal{B}$-Modules, $V\subset Y$ open, $U={\psi }^{-1}(V)$. The map $v\mapsto \Psi \ast (v)$ is a homomorphism

Hom_{ℬ|V}(ℳ|V, 𝒩|V) → Hom_{𝒜|U}(Ψ*(ℳ)|U, Ψ*(𝒩)|U)

for the $\Gamma (V,\mathcal{B})$-module structures (${\mathrm{Hom}}_{\mathcal{A}|U}(\Psi \ast (\mathcal{M})|U,\Psi \ast (\mathcal{N})|U)$ is naturally a $\Gamma (U,\psi \ast (\mathcal{B}))$-module, hence a $\Gamma (V,\mathcal{B})$-module via the canonical homomorphism $\Gamma (V,\mathcal{B})\to \Gamma (U,\psi \ast (\mathcal{B}))$ of (3.7.2)). These homomorphisms are compatible with restriction, so they define a canonical functorial homomorphism

γ : ℋom_ℬ(ℳ, 𝒩) → Ψ_*(ℋom_𝒜(Ψ*(ℳ), Ψ*(𝒩))),

corresponding to a homomorphism

γ^♯ : Ψ*(ℋom_ℬ(ℳ, 𝒩)) → ℋom_𝒜(Ψ*(ℳ), Ψ*(𝒩)),

both functorial in $\mathcal{M}$ and $\mathcal{N}$.

(4.4.7) Suppose $\mathcal{F}$ (resp. $\mathcal{G}$) is an $\mathcal{A}$-Algebra (resp. a $\mathcal{B}$-Algebra). If $u:\mathcal{G}\to {\Psi }_{\ast }(\mathcal{F})$ is a $\mathcal{B}$-Algebra homomorphism, then $u^{♯}:\Psi \ast (\mathcal{G})\to \mathcal{F}$ is an $\mathcal{A}$-Algebra homomorphism; this follows from the commutativity of

𝒢 ⊗_ℬ 𝒢 ────────────→ 𝒢
    │                    │
    ↓                    ↓ u
Ψ_*(ℱ ⊗_𝒜 ℱ) ────→ Ψ_*(ℱ)

and (4.4.4). Similarly, if $v:\Psi \ast (\mathcal{G})\to \mathcal{F}$ is an $\mathcal{A}$-Algebra homomorphism, then $v^{♭}:\mathcal{G}\to {\Psi }_{\ast }(\mathcal{F})$ is a $\mathcal{B}$-Algebra homomorphism.

(4.4.8) Let $(Z,\mathcal{C})$ be a third ringed space, ${\Psi }^{\prime }=({\psi }^{\prime },{\theta }^{\prime }):(Y,\mathcal{B})\to (Z,\mathcal{C})$ a morphism, and ${\Psi }^{\prime \prime }={\Psi }^{\prime }\circ \Psi :(X,\mathcal{A})\to (Z,\mathcal{C})$. Let $\mathcal{H}$ be a $\mathcal{C}$-Module and $v^{\prime }:\mathcal{H}\to {\Psi }_{\ast }^{\prime }(\mathcal{G})$ a homomorphism. The composite $v^{\prime \prime }=v\circ v^{\prime }$ is defined as the homomorphism

ℋ ──v′──→ Ψ′_*(𝒢) ──Ψ′_*(v)──→ Ψ′_*(Ψ_*(ℱ));

one checks that $v^{\prime \prime ♯}$ is

Ψ*(Ψ′*(ℋ)) ──Ψ*(v′^♯)──→ Ψ*(𝒢) ──v^♯──→ ℱ.