§4. Subpreschemes and Immersion Morphisms

4.1. Subpreschemes

(4.1.1) Since the notion of quasi-coherent sheaf (0, 5.1.3) is local, a quasi-coherent ${\mathcal{O}}_X$-Module $\mathcal{F}$ on a prescheme $X$ may be defined by the condition that, for every affine open $V$ of $X$, $\mathcal{F}|V$ is isomorphic to the sheaf associated to a $\Gamma (V,{\mathcal{O}}_X)$-module (1.4.1). On a prescheme $X$ the structure sheaf ${\mathcal{O}}_X$ is quasi-coherent, and kernels, cokernels, and images of homomorphisms of quasi-coherent ${\mathcal{O}}_X$-Modules, as well as inductive limits and direct sums of quasi-coherent ${\mathcal{O}}_X$-Modules, are quasi-coherent (1.3.7 and 1.3.9).

Proposition (4.1.2). Let $X$ be a prescheme and $\mathcal{J}$ a quasi-coherent sheaf of ideals in ${\mathcal{O}}_X$. The support $Y$ of the sheaf ${\mathcal{O}}_X/\mathcal{J}$ is closed, and if ${\mathcal{O}}_Y$ denotes the restriction of ${\mathcal{O}}_X/\mathcal{J}$ to $Y$, then $(Y,{\mathcal{O}}_Y)$ is a prescheme.

Proof. By (2.1.3) it suffices to treat the case where $X=\mathrm{Spec}(A)$ is affine, and to show that there $Y$ is closed in $X$ and is an affine scheme. Indeed, ${\mathcal{O}}_X=\tilde{A}$ and $\mathcal{J}=\tilde{\mathfrak{J}}$, where $\mathfrak{J}$ is an ideal of $A$ (1.4.1); then $Y=V(\mathfrak{J})$ and is identified with the prime spectrum of $B=A/\mathfrak{J}$ (1.1.11). Moreover, if $\phi :A\to B=A/\mathfrak{J}$ is the canonical homomorphism, the direct image ${\phi }_{\ast }(\tilde{B})$ is canonically identified with the sheaf $\widetilde{A/\mathfrak{J}}={\mathcal{O}}_Y$ (1.6.3 and 1.3.9), which completes the proof.

We say $(Y,{\mathcal{O}}_Y)$ is the subprescheme of $(X,{\mathcal{O}}_X)$ defined by the sheaf of ideals $\mathcal{J}$; it is a special case of the general notion of subprescheme.

Definition (4.1.3). A ringed space $(Y,{\mathcal{O}}_Y)$ is a subprescheme of a prescheme $(X,{\mathcal{O}}_X)$ if: 1° $Y$ is a locally closed subspace of $X$; 2° if $U$ denotes the largest open of $X$ containing $Y$ such that $Y$ is closed in $U$ (i.e. the complement in $X$ of the frontier of $Y$ relative to $Y$), then $(Y,{\mathcal{O}}_Y)$ is a subprescheme of $(U,{\mathcal{O}}_X|U)$ defined by a quasi-coherent sheaf of ideals of ${\mathcal{O}}_X|U$. The subprescheme $(Y,{\mathcal{O}}_Y)$ is called closed if $Y$ is closed in $X$ (in which case $U=X$).

It follows at once from this definition and (4.1.2) that the closed subpreschemes of $X$ are in canonical bijective correspondence with the quasi-coherent sheaves of ideals $\mathcal{J}$ of ${\mathcal{O}}_X$: if two such sheaves $\mathcal{J}$, ${\mathcal{J}}^{\prime }$ have the same (closed) support $Y$ and the restrictions of ${\mathcal{O}}_X/\mathcal{J}$ and ${\mathcal{O}}_X/{\mathcal{J}}^{\prime }$ to $Y$ agree, then ${\mathcal{J}}^{\prime }=\mathcal{J}$.

(4.1.4) Let $(Y,{\mathcal{O}}_Y)$ be a subprescheme of $X$, $U$ the largest open of $X$ containing $Y$ in which $Y$ is closed, and $V$ an open of $X$ contained in $U$; then $V\cap Y$ is closed in $V$. If moreover $Y$ is defined by the quasi-coherent sheaf of ideals $\mathcal{J}$ of ${\mathcal{O}}_X|U$, then $\mathcal{J}|V$ is a quasi-coherent sheaf of ideals of ${\mathcal{O}}_X|V$, and the prescheme induced by $Y$ on $Y\cap V$ is the closed subprescheme of $V$ defined by $\mathcal{J}|V$. Conversely:

Proposition (4.1.5). Let $(Y,{\mathcal{O}}_Y)$ be a ringed space such that $Y$ is a subspace of $X$ and there is a cover $(V_{\alpha })$ of $Y$ by opens of $X$ such that, for each $\alpha$, $Y\cap V_{\alpha }$ is closed in $V_{\alpha }$ and the ringed space $(Y\cap V_{\alpha },{\mathcal{O}}_Y|(Y\cap V_{\alpha }))$ is a closed subprescheme of the prescheme induced on $V_{\alpha }$ by $X$. Then $(Y,{\mathcal{O}}_Y)$ is a subprescheme of $X$.

Proof. The hypothesis implies $Y$ is locally closed in $X$ and the largest open $U$ containing $Y$ in which $Y$ is closed contains all the $V_{\alpha }$; we may thus reduce to $U=X$ and $Y$ closed in $X$. Define a quasi-coherent sheaf of ideals $\mathcal{J}$ of ${\mathcal{O}}_X$ by taking for $\mathcal{J}|V_{\alpha }$ the sheaf of ideals of ${\mathcal{O}}_X|V_{\alpha }$ defining the closed subprescheme $(Y\cap V_{\alpha },{\mathcal{O}}_Y|(Y\cap V_{\alpha }))$, and $\mathcal{J}|W={\mathcal{O}}_X|W$ for every open $W$ of $X$ not meeting $Y$. By (4.1.3) and (4.1.4) there is a unique sheaf of ideals $\mathcal{J}$ satisfying these conditions, and it defines the closed subprescheme $(Y,{\mathcal{O}}_Y)$. In particular, the prescheme induced by $X$ on an open of $X$ is a subprescheme of $X$.

Proposition (4.1.6). A subprescheme (resp. closed subprescheme) of a subprescheme (resp. closed subprescheme) of $X$ is canonically identified with a subprescheme (resp. closed subprescheme) of $X$.

Proof. A locally closed subset of a locally closed subspace of $X$ is a locally closed subspace of $X$, so by (4.1.5) the question is local and we may assume $X$ affine; the proposition then follows from the canonical identification of $A/{\mathfrak{J}}^{\prime }$ with $(A/\mathfrak{J})/({\mathfrak{J}}^{\prime }/\mathfrak{J})$ when $\mathfrak{J}\subset {\mathfrak{J}}^{\prime }$ are two ideals of a ring $A$. We shall always make this identification in the sequel.

(4.1.7) Let $Y$ be a subprescheme of a prescheme $X$, and $\psi :Y\to X$ the canonical injection of underlying spaces; the inverse image ${\psi }^{\ast }({\mathcal{O}}_X)$ is the restriction ${\mathcal{O}}_X|Y$ (0, 3.7.1). For each $y\in Y$, let ${\omega }_y$ be the canonical homomorphism $({\mathcal{O}}_X{)}_y\to ({\mathcal{O}}_Y{)}_y$; these are the stalk restrictions of a surjective homomorphism $\omega :{\mathcal{O}}_X|Y\to {\mathcal{O}}_Y$ of sheaves of rings (it suffices to check locally, assuming $X$ affine and $Y$ closed: if $\mathcal{J}$ defines $Y$, the ${\omega }_y$ are the stalk restrictions of ${\mathcal{O}}_X\to ({\mathcal{O}}_X/\mathcal{J})|Y$). We have thus defined a monomorphism of ringed spaces (0, 4.1.1) $j=(\psi ,\omega )$, which is evidently a morphism $Y\to X$ of preschemes (2.2.1), the canonical injection morphism. If $f:X\to Z$ is a morphism, we call the composite $Y\to X\to Z$ the restriction of $f$ to the subprescheme $Y$.

(4.1.8) In accordance with the general definitions (T, I, 1.1), we say that a morphism of preschemes $f:Z\to X$ is dominated by the injection morphism $j:Y\to X$ of a subprescheme $Y$ of $X$ if $f$ factors as $Z\stackrel{g}{\to }Y\stackrel{j}{\to }X$, where $g$ is a morphism of preschemes; $g$ is necessarily unique since $j$ is a monomorphism.

Proposition (4.1.9). For a morphism $f:Z\to X$ to be dominated by an injection morphism $j:Y\to X$, it is necessary and sufficient that $f(Z)\subset Y$ and that, for every $z\in Z$, setting $y=f(z)$, the homomorphism $({\mathcal{O}}_X{)}_y\to {\mathcal{O}}_z$ corresponding to $f$ factor through $({\mathcal{O}}_X{)}_y\to ({\mathcal{O}}_Y{)}_y\to {\mathcal{O}}_z$ (equivalently, that the kernel of $({\mathcal{O}}_X{)}_y\to {\mathcal{O}}_z$ contain that of $({\mathcal{O}}_X{)}_y\to ({\mathcal{O}}_Y{)}_y$).

Proof. The conditions are clearly necessary. To see they are sufficient, we may reduce to the case where $Y$ is a closed subprescheme of $X$, replacing $X$ if necessary by an open $U$ in which $Y$ is closed (4.1.3); $Y$ is then defined by a quasi-coherent sheaf $\mathcal{J}$ of ideals of ${\mathcal{O}}_X$. Write $f=(\psi ,\theta )$, and let $\mathcal{N}$ be the kernel sheaf of $\theta :{\psi }^{\ast }({\mathcal{O}}_X)\to {\mathcal{O}}_Z$; by the properties of the functor ${\psi }^{\ast }$ (0, 3.7.2), the hypothesis gives ${\psi }^{\ast }(\mathcal{J})\subset \mathcal{N}$. Hence $\theta$ factors as ${\psi }^{\ast }({\mathcal{O}}_X)\to {\psi }^{\ast }({\mathcal{O}}_X/\mathcal{J})\to {\mathcal{O}}_Z$, the first arrow being canonical. Letting ${\psi }^{\prime }$ be the continuous map $Z\to Y$ coinciding with $\psi$, one has ${\psi }^{\prime \ast }({\mathcal{O}}_Y)$ on the left, and the second arrow $\omega$ is a local homomorphism, so $g=({\psi }^{\prime },\omega )$ is a morphism $Z\to Y$ of preschemes (2.2.1) with $f=j\circ g$, whence the proposition.

Corollary (4.1.10). For an injection morphism $Z\to X$ to be dominated by the injection morphism $Y\to X$, it is necessary and sufficient that $Z$ be a subprescheme of $Y$.

We then write $Z\le Y$, and this is an order relation on the set of subpreschemes of $X$.

4.2. Immersion morphisms

Definition (4.2.1). A morphism $f:Y\to X$ is called an immersion (resp. closed immersion, resp. open immersion) if it factors as $Y\stackrel{g}{\to }Z\stackrel{j}{\to }X$, where $g$ is an isomorphism, $Z$ a subprescheme of $X$ (resp. a closed subprescheme, resp. a prescheme induced on an open), and $j$ the injection morphism.

The subprescheme $Z$ and the isomorphism $g$ are then uniquely determined: if $Z^{\prime }$ is a second subprescheme with injection $j^{\prime }$ and $g^{\prime }$ an isomorphism $Y\to Z^{\prime }$ with $j\circ g=j^{\prime }\circ g^{\prime }$, then $j^{\prime }=j\circ g\circ g^{\prime -1}$, whence $Z^{\prime }\le Z$ (4.1.10), and likewise $Z\le Z^{\prime }$, so $Z^{\prime }=Z$; since $j$ is a monomorphism, $g^{\prime }=g$. One calls $f=j\circ g$ the canonical factorization of the immersion $f$, and $Z$, $g$ the subprescheme and isomorphism associated to $f$. An immersion is a monomorphism of preschemes (4.1.7) and a fortiori a radicial morphism (3.5.4).

Proposition (4.2.2). a) For a morphism $f=(\psi ,\theta ):Y\to X$ to be an open immersion, it is necessary and sufficient that $\psi$ be a homeomorphism of $Y$ onto an open subset of $X$, and that for every $y\in Y$ the homomorphism ${\theta }_y^{♯}:({\mathcal{O}}_X{)}_{\psi (y)}\to {\mathcal{O}}_y$ be bijective. b) For $f=(\psi ,\theta ):Y\to X$ to be an immersion (resp. a closed immersion), it is necessary and sufficient that $\psi$ be a homeomorphism of $Y$ onto a locally closed (resp. closed) subset of $X$, and that for every $y\in Y$ the homomorphism ${\theta }_y^{♯}$ be surjective.

Proof. a) The conditions are clearly necessary. Conversely, if they hold, $\theta$ is an isomorphism of ${\mathcal{O}}_Y$ onto ${\psi }^{\ast }({\mathcal{O}}_X)$, and ${\psi }^{\ast }({\mathcal{O}}_X)$ is the sheaf obtained by transport of structure from ${\mathcal{O}}_X|\psi (Y)$ via ${\psi }^{-1}$, whence the conclusion.

b) The conditions being clearly necessary, we prove sufficiency. First suppose $X$ affine and $Z=\psi (Y)$ closed in $X$. By (0, 3.4.6), ${\psi }_{\ast }({\mathcal{O}}_Y)$ has support $Z$, and its restriction ${\mathcal{O}}_Z^{\prime }$ to $Z$ recovers $(Y,{\mathcal{O}}_Y)$ by transport of structure via $\psi$. We show ${\psi }_{\ast }({\mathcal{O}}_Y)$ is a quasi-coherent ${\mathcal{O}}_X$-Module. For $x\notin Z$ it vanishes near $x$; for $x=\psi (y)\in Z$, take an affine open neighborhood $V$ of $y$ in $Y$; then $\psi (V)$ is the trace on $Z$ of an open $U$ of $X$, and the restriction of ${\psi }_{\ast }({\mathcal{O}}_Y)$ to $U$ agrees with that of the direct image $({\psi }_V{)}_{\ast }({\mathcal{O}}_Y|V)$. The restriction of $(\psi ,\theta )$ to $(V,{\mathcal{O}}_Y|V)$ is of the form $({}^a\phi ,\tilde{\phi })$, where $\phi :A=\Gamma (X,{\mathcal{O}}_X)\to \Gamma (V,{\mathcal{O}}_Y)$ is a homomorphism (1.7.3), so $({\psi }_V{)}_{\ast }({\mathcal{O}}_Y|V)$ is quasi-coherent (1.6.3), proving the claim by the local nature of quasi-coherence. Since $\psi$ is a homeomorphism, (0, 3.4.5) gives that ${\theta }_y^{♯}$ is identified with a stalk of the canonical map ${\mathcal{O}}_X\to {\psi }_{\ast }({\mathcal{O}}_Y)$ via isomorphisms, so the surjectivity of ${\theta }_y^{♯}$ gives that of this map. As ${\psi }_{\ast }({\mathcal{O}}_Y)$ has support $Z$, the canonical homomorphism of ${\mathcal{O}}_X=\tilde{A}$ onto the quasi-coherent Module ${\psi }_{\ast }({\mathcal{O}}_Y)$ is surjective; hence there is a unique isomorphism $\omega$ of a quotient $\widetilde{A/\mathfrak{J}}$ ($\mathfrak{J}$ an ideal of $A$) onto ${\psi }_{\ast }({\mathcal{O}}_Y)$ inducing $\theta$ (1.3.8). With ${\mathcal{O}}_Z^{\prime }$ the restriction of $\widetilde{A/\mathfrak{J}}$ to $Z$, $(Z,{\mathcal{O}}_Z^{\prime })$ is a subprescheme of $X$, and $f$ factors as the canonical injection of this subprescheme and the isomorphism $({\psi }_0,{\omega }_0)$. For the general case, take an affine open $U$ of $X$ with $U\cap \psi (Y)$ closed and nonempty; restricting $f$ to ${\psi }^{-1}(U)$ reduces to the first case, giving a closed immersion ${\psi }^{-1}(U)\to U$ with canonical factorization $j_U\circ g_U$. For a second affine open $V\subset U$, uniqueness of the canonical factorization (4.2.1) forces compatibility on overlaps, so by (4.1.5) there is a subprescheme $Z$ of $X$ with underlying space $\psi (Y)$, and the $g_U$ glue to an isomorphism $g:Y\to Z$ with $f=j\circ g$.

Corollary (4.2.3). Let $X$ be an affine scheme. For a morphism $f=(\psi ,\theta ):Y\to X$ to be a closed immersion, it is necessary and sufficient that $Y$ be an affine scheme and the homomorphism $\Gamma (\theta ):\Gamma ({\mathcal{O}}_X)\to \Gamma ({\mathcal{O}}_Y)$ be surjective.

Corollary (4.2.4). a) Let $f:Y\to X$ be a morphism and $(V_{\lambda })$ a cover of $f(Y)$ by opens of $X$. For $f$ to be an immersion (resp. an open immersion), it is necessary and sufficient that its restriction to each induced prescheme $f^{-1}(V_{\lambda })$ be an immersion (resp. an open immersion) into $V_{\lambda }$. b) Let $f:Y\to X$ be a morphism and $(V_{\lambda })$ an open cover of $X$. For $f$ to be a closed immersion, it is necessary and sufficient that its restriction to each $f^{-1}(V_{\lambda })$ be a closed immersion into $V_{\lambda }$.

Proof. Write $f=(\psi ,\theta )$; in case a), ${\theta }_y^{♯}$ is surjective (resp. bijective) for all $y$, and in case b) surjective for all $y$, so it suffices to check the topological condition on $\psi$. Now $\psi$ is injective and carries neighborhoods to neighborhoods within $\psi (Y)$; in case a), $\psi (Y)\cap V_{\lambda }$ is locally closed (resp. open) in $V_{\lambda }$, so $\psi (Y)$ is locally closed (resp. open) in $\bigcup V_{\lambda }$, a fortiori in $X$; in case b), $\psi (Y)\cap V_{\lambda }$ is closed in $V_{\lambda }$, so $\psi (Y)$ is closed in $X=\bigcup V_{\lambda }$.

Proposition (4.2.5). The composite of two immersions (resp. two open immersions, two closed immersions) is an immersion (resp. an open immersion, a closed immersion). This follows trivially from (4.1.6).

4.3. Products of immersions

Proposition (4.3.1). Let $\alpha :X^{\prime }\to X$, $\beta :Y^{\prime }\to Y$ be two $S$-morphisms. If $\alpha$ and $\beta$ are immersions (resp. open immersions, resp. closed immersions), then $\alpha {\times }_S\beta :X^{\prime }{\times }_S{Y}^{\prime }\to X{\times }_{S}Y$ is an immersion (resp. open, resp. closed). Moreover, if $\alpha$ (resp. $\beta$) identifies $X^{\prime }$ (resp. $Y^{\prime }$) with a subprescheme $X^{\prime \prime }$ (resp. $Y^{\prime \prime }$) of $X$ (resp. $Y$), then $\alpha {\times }_S\beta$ identifies the underlying space of $X^{\prime }{\times }_S{Y}^{\prime }$ with the subspace $p^{-1}(X^{\prime \prime })\cap q^{-1}(Y^{\prime \prime })$ of $X{\times }_{S}Y$, where $p$, $q$ are the projections.

Proof. By (4.2.1) we may assume $X^{\prime }$, $Y^{\prime }$ are subpreschemes and $\alpha$, $\beta$ injection morphisms. The proposition is already known for subpreschemes induced on opens (3.2.7); since every subprescheme is a closed subprescheme of a prescheme induced on an open (4.1.3), we reduce to $X^{\prime }$, $Y^{\prime }$ closed subpreschemes. We may assume $S$ affine: covering $S$ by affine opens $S_{\mu }$ and using (3.2.5), (3.2.6.4), (3.2.7), and (4.2.4), the general case follows from the affine one. Likewise we may assume $X$, $Y$ affine: covering by affine opens and using (3.2.7), the relation $p^{\prime -1}(X_i^{\prime })\cap q^{\prime -1}(Y_j^{\prime })={\gamma }^{-1}(U_i{\times }_S{V}_j)$ (where $\gamma =\alpha {\times }_S\beta$) reduces to the affine case. So assume $X$, $Y$, $S$ affine with rings $B$, $C$, $A$; then $B$, $C$ are $A$-algebras, $X^{\prime }$, $Y^{\prime }$ affine with quotient rings $B^{\prime }$, $C^{\prime }$, and $\alpha =({}^a\rho ,\dots )$, $\beta =({}^a\sigma ,\dots )$ with $\rho :B\to B^{\prime }$, $\sigma :C\to C^{\prime }$ the canonical surjections (1.7.3). Now $X{\times }_{S}Y$ (resp. $X^{\prime }{\times }_S{Y}^{\prime }$) is affine with ring $B{\otimes }_{A}C$ (resp. $B^{\prime }{\otimes }_A{C}^{\prime }$), and $\alpha {\times }_S\beta =({}^a\tau ,\dots )$ with $\tau =\rho \otimes \sigma :B{\otimes }_{A}C\to B^{\prime }{\otimes }_A{C}^{\prime }$ (3.2.2, 3.2.3); since $\tau$ is surjective, $\alpha {\times }_S\beta$ is a (closed) immersion. If $\mathfrak{b}$ (resp. $\mathfrak{c}$) is the kernel of $\rho$ (resp. $\sigma$), the kernel of $\tau$ is $u(\mathfrak{b})+v(\mathfrak{c})$ ($u:b\mapsto b\otimes 1$, $v:c\mapsto 1\otimes c$); this corresponds in $\mathrm{Spec}(B{\otimes }_{A}C)$ to the closed set $p^{-1}(X^{\prime })\cap q^{-1}(Y^{\prime })$, completing the proof.

Corollary (4.3.2). If $f:X\to Y$ is an immersion (resp. open, resp. closed immersion) and an $S$-morphism, then $f_{(S^{\prime })}$ is an immersion (resp. open, resp. closed immersion) for every extension $S^{\prime }\to S$ of the base prescheme.

4.4. Inverse image of a subprescheme

Proposition (4.4.1). Let $f:X\to Y$ be a morphism, $Y^{\prime }$ a subprescheme (resp. closed subprescheme, resp. prescheme induced on an open) of $Y$, and $j:Y^{\prime }\to Y$ the injection morphism. Then the projection $p:X{\times }_Y{Y}^{\prime }\to X$ is an immersion (resp. closed, resp. open); the associated subprescheme of $X$ has underlying space $f^{-1}(Y^{\prime })$, and if $j^{\prime }$ is its injection morphism, a morphism $h:Z\to X$ is such that $f\circ h$ is dominated by $j$ if and only if $h$ is dominated by $j^{\prime }$.

Proof. Since $p=1_X{\times }_{Y}j$ (3.3.4), the first assertion follows from (4.3.1); the second is a special case of (3.5.10) (exchanging the roles of $X$ and $Y^{\prime }$). Finally, if $f\circ h=j\circ h^{\prime }$ with $h^{\prime }:Z\to Y^{\prime }$, the definition of the product gives $h=p\circ u$ for a morphism $u:Z\to X{\times }_Y{Y}^{\prime }$, whence the last assertion.

We say the subprescheme of $X$ so defined is the inverse image of the subprescheme $Y^{\prime }$ of $Y$ by $f$, consistent with the terminology of (3.3.6). When $f^{-1}(Y^{\prime })=X$, $f$ factors as $X\to Y^{\prime }\to Y$. When $y$ is a closed point of $Y$ and $Y^{\prime }=\mathrm{Spec}(\kappa (y))$ the smallest closed subprescheme with $\{y\}$ as underlying space (4.1.9), the closed subprescheme $f^{-1}(Y^{\prime })$ is canonically isomorphic to the fiber $f^{-1}(y)$ of (3.6.2), with which it is identified.

Corollary (4.4.2). Let $f:X\to Y$, $g:Y\to Z$ be two morphisms, $h=g\circ f$. For every subprescheme $Z^{\prime }$ of $Z$, the subpreschemes $f^{-1}(g^{-1}(Z^{\prime }))$ and $h^{-1}(Z^{\prime })$ of $X$ are identical. This follows from the canonical isomorphism $X{\times }_Y(Y{\times }_Z{Z}^{\prime })\cong X{\times }_Z{Z}^{\prime }$ (3.3.9.1).

Corollary (4.4.3). Let $X^{\prime }$, $X^{\prime \prime }$ be two subpreschemes of $X$ with injections $j^{\prime }:X^{\prime }\to X$, $j^{\prime \prime }:X^{\prime \prime }\to X$. Then $j^{\prime -1}(X^{\prime \prime })$ and $j^{\prime \prime -1}(X^{\prime })$ both equal the infimum $\inf (X^{\prime },X^{\prime \prime })$ for the order relation between subpreschemes, and are canonically isomorphic to $X^{\prime }{\times }_X{X}^{\prime \prime }$. This follows from (4.4.1) and (4.1.10).

Corollary (4.4.4). Let $f:X\to Y$ be a morphism and $Y^{\prime }$, $Y^{\prime \prime }$ two subpreschemes of $Y$; then $f^{-1}(\inf (Y^{\prime },Y^{\prime \prime }))=\inf (f^{-1}(Y^{\prime }),f^{-1}(Y^{\prime \prime }))$. This follows from the canonical isomorphism between $(X{\times }_Y{Y}^{\prime }){\times }_X(X{\times }_Y{Y}^{\prime \prime })$ and $X{\times }_Y(Y^{\prime }{\times }_Y{Y}^{\prime \prime })$ (3.3.9.1).

Proposition (4.4.5). Let $f:X\to Y$ be a morphism and $Y^{\prime }$ a closed subprescheme of $Y$ defined by a quasi-coherent sheaf $\mathcal{J}$ of ideals of ${\mathcal{O}}_Y$ (4.1.3). Then the closed subprescheme $f^{-1}(Y^{\prime })$ of $X$ is defined by the quasi-coherent sheaf of ideals $f^{\ast }(\mathcal{J}){\mathcal{O}}_X$ of ${\mathcal{O}}_X$.

Proof. The question is local on $X$ and $Y$; it suffices to remark that if $B$ is an $A$-algebra and $\mathfrak{J}$ an ideal of $A$, then $B{\otimes }_A(A/\mathfrak{J})=B/\mathfrak{J}B$, and to apply (1.6.9).

Corollary (4.4.6). Let $X^{\prime }$ be a closed subprescheme of $X$ defined by a quasi-coherent sheaf of ideals $\mathcal{J}$ of ${\mathcal{O}}_X$, and $i:X^{\prime }\to X$ the injection. For the restriction $f\circ i$ of $f$ to $X^{\prime }$ to be dominated by the injection $j:Y^{\prime }\to Y$ (i.e. to factor as $j\circ g$ with $g:X^{\prime }\to Y^{\prime }$), it is necessary and sufficient that $f^{\ast }({\mathcal{J}}^{\prime }){\mathcal{O}}_X\subset \mathcal{J}$, where ${\mathcal{J}}^{\prime }$ is the ideal sheaf of $Y^{\prime }$. It suffices to apply (4.4.1) to $i$, taking (4.4.5) into account.

4.5. Local immersions and local isomorphisms

Definition (4.5.1). Let $f:X\to Y$ be a morphism of preschemes. We say $f$ is a local immersion at a point $x\in X$ if there exist an open neighborhood $U$ of $x$ in $X$ and an open neighborhood $V$ of $f(x)$ in $Y$ such that the restriction of $f$ to the induced prescheme $U$ is a closed immersion of $U$ into the induced prescheme $V$. We say $f$ is a local immersion if it is so at every point of $X$.

Definition (4.5.2). We say a morphism $f:X\to Y$ is a local isomorphism at a point $x\in X$ if there exists an open neighborhood $U$ of $x$ in $X$ such that the restriction of $f$ to the induced prescheme $U$ is an open immersion of $U$ into $Y$. We say $f$ is a local isomorphism if it is so at every point of $X$.

(4.5.3) An immersion (resp. a closed immersion) $f:X\to Y$ may thus be characterized as a local immersion such that $f$ is a homeomorphism of the underlying space of $X$ onto a subset (resp. a closed subset) of $Y$. An open immersion $f$ may be characterized as an injective local isomorphism.

Proposition (4.5.4). Let $X$ be an irreducible prescheme and $f:X\to Y$ an injective dominant morphism. If $f$ is a local immersion, then $f$ is an immersion and $f(X)$ is open in $Y$.

Proof. Let $x\in X$, and let $U$, $V$ be open neighborhoods of $x$ and $f(x)$ such that $f|U$ is a closed immersion into $V$. Since $U$ is dense in $X$, $f(U)$ is dense in $Y$ by hypothesis, so $f(U)=V$ and $f$ is a homeomorphism of $U$ onto $V$; injectivity of $f$ gives $f^{-1}(V)=U$, whence the proposition.

Proposition (4.5.5). (i) The composite of two local immersions (resp. two local isomorphisms) is a local immersion (resp. a local isomorphism). (ii) Let $f:X\to X^{\prime }$, $g:Y\to Y^{\prime }$ be two $S$-morphisms. If $f$ and $g$ are local immersions (resp. local isomorphisms), so is $f{\times }_{S}g$. (iii) If an $S$-morphism $f$ is a local immersion (resp. a local isomorphism), so is $f_{(S^{\prime })}$ for every extension $S^{\prime }\to S$ of the base prescheme.

Proof. By (3.5.1) it suffices to prove (i) and (ii). (i) follows at once from the transitivity of closed (resp. open) immersions (4.2.4) and the fact that if $f$ is a homeomorphism of $X$ onto a closed subset of $Y$, then for every open $U\subset X$, $f(U)$ is open in $f(X)$, hence $f(U)=V\cap f(X)$ for some open $V$ of $Y$, so $f(U)$ is closed in $V$. For (ii), let $p$, $q$ be the projections of $X{\times }_{S}Y$ and $p^{\prime }$, $q^{\prime }$ those of $X^{\prime }{\times }_S{Y}^{\prime }$. There are open neighborhoods $U$, $U^{\prime }$, $V$, $V^{\prime }$ of $x=p(\xi )$, $x^{\prime }=p^{\prime }({\xi }^{\prime })$, $y=q(\xi )$, $y^{\prime }=q^{\prime }({\xi }^{\prime })$ such that the restrictions of $f$ and $g$ to $U$ and $V$ are closed (resp. open) immersions into $U^{\prime }$ and $V^{\prime }$. Since the underlying spaces of $U{\times }_{S}V$ and $U^{\prime }{\times }_S{V}^{\prime }$ are identified with the open neighborhoods $p^{-1}(U)\cap q^{-1}(V)$ and $p^{\prime -1}(U^{\prime })\cap q^{\prime -1}(V^{\prime })$ (3.2.7), the proposition follows from (4.3.1).