Chapter I — The Language of Schemes

§4. Subpreschemes and Immersion Morphisms

Translation status. Translation skeleton with key definitions and theorem statements; full proofs reference .

4.1. Subpreschemes

Proposition (4.1.2). Let $(X,{\mathcal{O}}_X)$ be a prescheme and $\mathcal{J}\subset {\mathcal{O}}_X$ a quasi-coherent sheaf of ideals. The support of ${\mathcal{O}}_X/\mathcal{J}$ is closed in $X$; on this support, the pair $(Supp({\mathcal{O}}_X/\mathcal{J}),({\mathcal{O}}_X/\mathcal{J})|Supp)$ is a prescheme.

Definition (4.1.3). A closed subprescheme of $(X,{\mathcal{O}}_X)$ is a prescheme of the form $(Y,{\mathcal{O}}_X/\mathcal{J}|Y)$ with $\mathcal{J}\subset {\mathcal{O}}_X$ a quasi-coherent ideal sheaf and $Y=Supp({\mathcal{O}}_X/\mathcal{J})$. An open subprescheme is the prescheme $(U,{\mathcal{O}}_X|U)$ induced on an open $U\subset X$. A subprescheme is the closed subprescheme of an open subprescheme.

Proposition (4.1.5). A closed subprescheme is uniquely determined by its sheaf of ideals.

Proposition (4.1.6). For an affine scheme $X=\mathrm{Spec}(A)$: closed subpreschemes correspond bijectively to ideals $\mathfrak{J}\subset A$, the closed subprescheme being $\mathrm{Spec}(A/\mathfrak{J})$.

Proposition (4.1.9). Closed subpreschemes of a Noetherian prescheme satisfy the descending chain condition.

Corollary (4.1.10). Every prescheme is a union of its open affine subpreschemes (a closed cover of finite type, if Noetherian).

4.2. Immersion morphisms

Definition (4.2.1). A morphism $j:Y\to X$ of preschemes is an open immersion (resp. closed immersion, resp. immersion) if $j$ is a homeomorphism of $Y$ onto an open (resp. closed, resp. locally closed) subspace of $X$ and $j^{♯}:{\mathcal{O}}_X\to j_{\ast }({\mathcal{O}}_Y)$ is surjective onto the structure sheaf of the corresponding subprescheme. Equivalently, $j$ factors as $Y\stackrel{\sim }{\to }{Y}^{\prime }\hookrightarrow X$ with $Y^{\prime }$ an open (resp. closed, resp. locally closed) subprescheme.

Proposition (4.2.2). Composition of immersions is an immersion; the composite of two open (resp. closed) immersions is open (resp. closed).

Corollary (4.2.3). Immersions are monomorphisms of preschemes.

Corollary (4.2.4). An immersion is radicial.

Proposition (4.2.5). Immersions are preserved under base change.

4.3. Products of immersions

Proposition (4.3.1). If $j_1:Y_1\to X_1$ and $j_2:Y_2\to X_2$ are immersions of $S$-preschemes, then $j_1{\times }_S{j}_2:Y_1{\times }_S{Y}_2\to X_1{\times }_S{X}_2$ is an immersion (open or closed if both factors are).

Corollary (4.3.2). Open immersions and closed immersions are stable under fiber products.

4.4. Inverse images of subpreschemes

Proposition (4.4.1). For a morphism $f:X\to Y$ and a closed subprescheme $Y^{\prime }\subset Y$ with ideal sheaf $\mathcal{J}$, the inverse image $f^{-1}(Y^{\prime })=X{\times }_Y{Y}^{\prime }$ is a closed subprescheme of $X$ with ideal sheaf $f\ast (\mathcal{J})\cdot {\mathcal{O}}_X=\mathcal{J}{\mathcal{O}}_X$.

Corollary (4.4.2). Inverse images preserve closed (resp. open) subpreschemes.

Corollary (4.4.3). For composable morphisms, $(g\circ f{)}^{-1}(Z)=f^{-1}(g^{-1}(Z))$.

Corollary (4.4.4). Inverse images preserve immersions.

Proposition (4.4.5). For morphisms $X\to Y\to S$ and a subprescheme $Y^{\prime }\subset Y$, $X{\times }_Y{Y}^{\prime }=X{\times }_S{Y}^{\prime }$ (when $Y^{\prime }\hookrightarrow Y$ is an immersion).

4.5. Local immersions and local isomorphisms

Definition (4.5.1). A morphism $f:X\to Y$ is a local immersion at $x\in X$ if there is an open $U\ni x$ such that $f|U:U\to Y$ is an immersion. $f$ is a local immersion if it is so at every point.

Definition (4.5.2). A morphism $f:X\to Y$ is a local isomorphism at $x$ if there is an open $U\ni x$ and an open $V\supset f(U)$ such that $f|U\to V$ is an isomorphism. $f$ is a local isomorphism if it is so at every point.

Proposition (4.5.4). A local immersion is radicial (and so injective on geometric points stalkwise).

Proposition (4.5.5). A local isomorphism is open. A radicial local isomorphism is a local immersion.