Chapter I — The Language of Schemes

§9. Complements on Quasi-coherent Sheaves

Translation status. Skeleton with definitions and principal statements; full proofs reference .

9.1. Tensor product of quasi-coherent sheaves

Proposition (9.1.1). For quasi-coherent ${\mathcal{O}}_X$-modules $\mathcal{F}$, $\mathcal{G}$ on a prescheme $X$, $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$ is quasi-coherent.

Definition (9.1.2). For preschemes $X_1,X_2$ and quasi-coherent ${\mathcal{O}}_{X_i}$-modules ${\mathcal{F}}_i$ ($i=1,2$), the tensor product on distinct preschemes ${\mathcal{F}}_1⊠{\mathcal{F}}_2=p_1^{\ast }({\mathcal{F}}_1){\otimes }_{{\mathcal{O}}_{X_1{\times }_S{X}_2}}{p}_2^{\ast }({\mathcal{F}}_2)$ on $X_1{\times }_S{X}_2$ (where $p_i$ is the projection).

Proposition (9.1.3). External tensor product is quasi-coherent; bifunctorial in ${\mathcal{F}}_1,{\mathcal{F}}_2$.

Proposition (9.1.4). Restriction of scalars and base change preserve quasi-coherence: for $f:X\to Y$ and $\mathcal{F}$ quasi-coherent on $Y$, $f\ast (\mathcal{F})$ is quasi-coherent on $X$.

Corollaries (9.1.5)–(9.1.7). Behavior under sub- and quotient-modules, kernels, cokernels, inductive limits.

Proposition (9.1.9)–(9.1.12). Exactness properties: $f\ast$ is right exact; if $f$ is flat, $f\ast$ is exact.

Corollary (9.1.13). Stalks of pullback: $(f\ast (\mathcal{F}){)}_x={\mathcal{F}}_{f(x)}{\otimes }_{{\mathcal{O}}_{f(x)}}{\mathcal{O}}_x$.

9.2. Direct image of a quasi-coherent sheaf

Proposition (9.2.1). For $f:X\to Y$ a quasi-compact, separated morphism, and $\mathcal{F}$ quasi-coherent on $X$, the direct image $f_{\ast }(\mathcal{F})$ is quasi-coherent on $Y$.

Corollary (9.2.2). For an affine morphism $f:X\to Y$ (i.e., the preimage of every affine open is affine), $f_{\ast }$ preserves quasi-coherence.

9.3. Extension of sections

Theorem (9.3.1). Let $X$ be a Noetherian prescheme, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-module, $U\subset X$ open. Every section $s\in \Gamma (U,\mathcal{F})$ extends to a section of $\mathcal{F}\otimes {\mathcal{O}}_X(D)$ for some divisor $D$ supported on $X\setminus U$ (after multiplying by powers of local equations).

Corollary (9.3.2)–(9.3.3). Sections over $U=X_f$ extend to global sections after multiplying by a power of $f$.

Proposition (9.3.4). Behavior of section extension under flat base change.

Corollary (9.3.5). For Noetherian $X$ and quasi-compact $U$, $\Gamma (U,\mathcal{F})$ is a localization of $\Gamma (X,\mathcal{F})$ in a precise functorial sense.

9.4. Extension of quasi-coherent sheaves

(9.4.1) For $X$ a Noetherian prescheme, $U\subset X$ open, and $\mathcal{F}$ quasi-coherent on $U$, an extension of $\mathcal{F}$ to $X$ is a quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ with $\mathcal{F}|U\cong \mathcal{F}$.

Proposition (9.4.2). Existence of extension: every quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$.

Corollary (9.4.3). Coherent extensions: a coherent sheaf on $U$ extends to a coherent sheaf on $X$ whose restriction to $U$ is the given one.

Corollary (9.4.5). Quasi-coherent sub-Modules of a coherent sheaf extend to quasi-coherent sub-Modules.

Lemma (9.4.6). Bounded-from-below $\mathcal{J}$-adic filtrations: for a quasi-coherent ideal $\mathcal{J}$ and coherent $\mathcal{F}$, the canonical filtration ${\mathcal{J}}^n\mathcal{F}$ decreases to a quasi-coherent submodule.

Theorem (9.4.7). Theorem of extension of subsheaves: every quasi-coherent subsheaf of $\mathcal{F}|U$ extends to a quasi-coherent subsheaf of $\mathcal{F}$ on $X$.

Corollaries (9.4.8)–(9.4.10). Consequences for closed subpreschemes, base change, and morphisms.

9.5. Closed image of a prescheme; closure of a subprescheme

Proposition (9.5.1). For $f:X\to Y$ with $X$ Noetherian (or $f$ quasi-compact), the kernel $\mathcal{J}=Ker({\mathcal{O}}_Y\to f_{\ast }({\mathcal{O}}_X))$ is a quasi-coherent ideal sheaf, defining the closed image $f\bar{X}$ (with reduced subprescheme structure) — the smallest closed subprescheme of $Y$ through which $f$ factors.

Corollary (9.5.2). The closed image is the closure of the set-theoretic image in $Y$ with its reduced subprescheme structure when $Y$ is reduced.

Definition (9.5.3). The closure of a subprescheme $Y^{\prime }\hookrightarrow Y$ (or adherence of a subprescheme) is the closed image of $Y^{\prime }\to Y$.

Propositions (9.5.4)–(9.5.6). Functoriality of the closure under base change and composition.

Propositions (9.5.8)–(9.5.10). Closures and irreducible components: the closure of $Y^{\prime }$ decomposes by the irreducible components of $Y\overline{{}^{\prime }}$.

Corollary (9.5.11). Closure of a subprescheme: $Y^{\prime }\hookrightarrow Y$ extends uniquely (up to isomorphism) to a closed subprescheme of $Y$ containing $Y^{\prime }$ as a dense open subprescheme.

9.6. Quasi-coherent sheaves of algebras; change of structure sheaf

Proposition (9.6.1). For a quasi-coherent ${\mathcal{O}}_X$-algebra $\mathcal{A}$, the ringed space $(X,\mathcal{A})$ is not in general a prescheme, but its "Spec" is — see (9.6.5).

Proposition (9.6.3). Quasi-coherence is preserved under change of structure sheaf for affine morphisms.

Corollary (9.6.4). Tensor products of quasi-coherent algebras are quasi-coherent.

Proposition (9.6.5) (Spec of a quasi-coherent algebra). For $(X,{\mathcal{O}}_X)$ a prescheme and $\mathcal{A}$ a quasi-coherent ${\mathcal{O}}_X$-algebra, there is an affine morphism $f:\mathrm{Spec}(\mathcal{A})\to X$ such that $f_{\ast }({\mathcal{O}}_{\mathrm{Spec}(\mathcal{A})})=\mathcal{A}$ and $\mathrm{Spec}(\mathcal{A})$ is universal among $X$-preschemes whose direct image yields $\mathcal{A}$.

Proposition (9.6.6). Properties of $\mathrm{Spec}(\mathcal{A})$: it is separated; coherent if $\mathcal{A}$ is locally finitely presented; functorial in $\mathcal{A}$.