Chapter I — The Language of Schemes

§2. Preschemes and Morphisms of Preschemes

2.1. Definition of preschemes

(2.1.1) Given a ringed space $(X,{\mathcal{O}}_X)$, an open subset $V\subset X$ is called an affine open if the ringed space $(V,{\mathcal{O}}_X|V)$ is an affine scheme (1.7.1).

Definition (2.1.2). A prescheme [modern: scheme] is a ringed space $(X,{\mathcal{O}}_X)$ such that every point of $X$ admits an affine open neighborhood.

Proposition (2.1.3). If $(X,{\mathcal{O}}_X)$ is a prescheme, the affine open subsets form a basis of the topology of $X$.

Proof. Let $V$ be an open neighborhood of $x$, and $W$ an affine open neighborhood with ring $A$. Then $V\cap W$ is open in $W$, so some $D(f)\subset V\cap W$ ($f\in A$) is an affine open neighborhood of $x$, by (1.1.10) and (1.3.6).

Proposition (2.1.4). The underlying space of a prescheme is a Kolmogorov space.

Proposition (2.1.5). In a prescheme $(X,{\mathcal{O}}_X)$, every closed irreducible subset of $X$ admits a unique generic point; thus $x\mapsto \bar{x}$ is a bijection of $X$ onto the set of closed irreducible subsets.

Proof. If $Y$ is closed irreducible and $y\in Y$, let $U$ be an affine open neighborhood of $y$. Then $U\cap Y$ is dense in $Y$, irreducible, and closed in $U$; by (1.1.14), $U\cap Y=\bar{x}\cap U$ for some $x\in U$, so $Y=\bar{x}$. Uniqueness follows from (2.1.4) and (0.2.1.3).

(2.1.6) If $Y\subset X$ is closed irreducible with generic point $y$, the local ring ${\mathcal{O}}_y$ (also written ${\mathcal{O}}_{X/Y}$) is called the local ring of $X$ along $Y$ (or the local ring of $Y$ in $X$).

If $X$ is irreducible with generic point $x$, ${\mathcal{O}}_x$ is the ring of rational functions on $X$ (cf. §7).

Proposition (2.1.7). For every open $U\subset X$, the ringed space $(U,{\mathcal{O}}_X|U)$ is a prescheme — the induced prescheme (or restriction) on $U$.

(2.1.8) A prescheme $(X,{\mathcal{O}}_X)$ is irreducible (resp. connected) if $X$ is. It is integral if it is irreducible and reduced (cf. (5.1.4)). It is locally integral if every $x\in X$ admits an open neighborhood $U$ such that the induced prescheme on $U$ is integral.

2.2. Morphisms of preschemes

Definition (2.2.1). Given two preschemes $(X,{\mathcal{O}}_X)$ and $(Y,{\mathcal{O}}_Y)$, a morphism of preschemes $(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ is a morphism $(\psi ,\theta )$ of ringed spaces such that for every $x\in X$, ${\theta }_x^{♯}:{\mathcal{O}}_{\psi (x)}\to {\mathcal{O}}_x$ is a local homomorphism.

Passing to quotients, the stalk map gives a monomorphism ${\theta }^x:\kappa (\psi (x))\to \kappa (x)$, making $\kappa (x)$ an extension of $\kappa (\psi (x))$.

(2.2.2) Composition of two morphisms of preschemes is a morphism of preschemes (by ${\theta }^{\prime \prime ♯}={\theta }^{♯}\circ \psi \ast ({\theta }^{\prime ♯})$, (0.3.5.5)). Thus preschemes form a category; we write $\mathrm{Hom}(X,Y)$ for the set of morphisms.

Example (2.2.3). For an open $U\subset X$, the canonical injection $(U,{\mathcal{O}}_X|U)\to (X,{\mathcal{O}}_X)$ is a monomorphism of preschemes.

Proposition (2.2.4). Let $(X,{\mathcal{O}}_X)$ be a prescheme and $(S,{\mathcal{O}}_S)$ an affine scheme of ring $A$. There is a canonical bijection between morphisms of preschemes $(X,{\mathcal{O}}_X)\to (S,{\mathcal{O}}_S)$ and ring homomorphisms $A\to \Gamma (X,{\mathcal{O}}_X)$.

Proof sketch. Any morphism $(\psi ,\theta )$ gives $\Gamma (\theta ):A\to \Gamma (X,{\mathcal{O}}_X)$. Conversely, given $\varphi :A\to \Gamma (X,{\mathcal{O}}_X)$, cover $X$ by affine opens $(V_{\alpha })$; composing with restriction $\Gamma (X,{\mathcal{O}}_X)\to \Gamma (V_{\alpha },{\mathcal{O}}_X|V_{\alpha })$ gives ${\varphi }_{\alpha }$, which corresponds by (1.7.3) to a morphism $({\psi }_{\alpha },{\theta }_{\alpha }):V_{\alpha }\to S$. These agree on overlaps (by (2.1.3) and stalkwise compatibility), so they glue to a morphism $(\psi ,\theta ):X\to S$ of preschemes with $\Gamma (\theta )=\varphi$.

For $f\in A$, ${\psi }^{-1}(D(f))=X_{\varphi (f)}$ (the open set where the section $\varphi (f)$ does not vanish; see (0.5.5.2)). (2.2.4.1)

Proposition (2.2.5). Under the hypotheses of (2.2.4), let $\varphi :A\to \Gamma (X,{\mathcal{O}}_X)$, $f:X\to S$ the corresponding morphism, $\mathcal{G}$ an ${\mathcal{O}}_X$-module, $\mathcal{F}$ an ${\mathcal{O}}_S$-module, and $M=\Gamma (S,\mathcal{F})$. There is a canonical bijection between $f$-morphisms $\mathcal{F}\to \mathcal{G}$ (0.4.4.1) and $A$-homomorphisms $M\to (\Gamma (X,\mathcal{G}){)}_{[\varphi ]}$.

(2.2.6) A morphism $(\psi ,\theta ):X\to Y$ of preschemes is open (resp. closed) if for every open $U\subset X$ (resp. closed $F\subset X$), $\psi (U)$ is open (resp. $\psi (F)$ is closed) in $Y$. It is dominant if $\psi (X)$ is dense in $Y$, and surjective if $\psi$ is. These conditions depend only on $\psi$.

Proposition (2.2.7). Let $f:X\to Y$ and $g:Y\to Z$ be morphisms of preschemes.

(i) If $f$, $g$ are both open (resp. closed, dominant, surjective), so is $g\circ f$. (ii) If $f$ is surjective and $g\circ f$ closed, then $g$ is closed. (iii) If $g\circ f$ is surjective, so is $g$.

Proposition (2.2.8). Let $f:X\to Y$ and $(U_{\alpha })$ an open cover of $Y$. $f$ is open (resp. closed, surjective, dominant) iff each restriction ${\psi }^{-1}(U_{\alpha })\to U_{\alpha }$ is.

(2.2.9) Let X, Y have finitely many irreducible components $X_i,Y_i$ ($1\le i\le n$) with generic points ${\xi }_i,{\eta }_i$ (2.1.5). A morphism $f=(\psi ,\theta ):X\to Y$ is birational if for every $i$, ${\psi }^{-1}({\eta }_i)={\xi }_i$ and ${\theta }_{{\xi }_i}^{♯}:{\mathcal{O}}_{{\eta }_i}\to {\mathcal{O}}_{{\xi }_i}$ is an isomorphism. A birational morphism is dominant (0.2.1.8), and surjective if it is closed.

Notation (2.2.10). When no confusion threatens, we suppress the structure sheaf (resp. the morphism of structure sheaves) from notation. For an open $U\subset X$ of a prescheme, "the prescheme $U$" means the induced prescheme on $U$.

2.3. Gluing of preschemes

(2.3.1) By (2.1.2), every ringed space obtained by gluing preschemes (0.4.1.7) is a prescheme. In particular, every prescheme can be obtained by gluing affine schemes.

Example (2.3.2). Let $K$ be a field, $B=K[s]$, $C=K[t]$ polynomial rings; set $X_1=\mathrm{Spec}(B)$, $X_2=\mathrm{Spec}(C)$. In X_1 (resp. X_2), let $U_{12}=D(s)$ (resp. $U_{21}=D(t)$), with ring $B_s$ (resp. $C_t$). Let $u_{12}:U_{21}\to U_{12}$ be the isomorphism corresponding to $B_s\to C_t$ sending $f(s)/s^m$ to $f(1/t)/(1/t^m)$. Glue $X_1,X_2$ along $U_{12},U_{21}$ via $u_{12}$ (no cocycle condition). The resulting prescheme $X$ is not affine: $\Gamma (X,{\mathcal{O}}_X)=K$, since a global section is a polynomial $f(s)=g(t)=f(1/t)$ on the overlap, forcing $f=g\in K$. (This is the projective line ${\mathbb{P}}_K^1$; see (II.2.4.3).)

2.4. Local schemes

(2.4.1) An affine scheme $X=\mathrm{Spec}(A)$ is a local scheme if $A$ is a local ring; $X$ then has a unique closed point $a$, and $a\in \bar{b}$ for every $b\in X$ (1.1.7).

For a prescheme $Y$ and $y\in Y$, the local scheme $\mathrm{Spec}({\mathcal{O}}_y)$ is the local scheme of $Y$ at $y$. For an affine open $V\subset Y$ with ring $B$ containing $y$, ${\mathcal{O}}_y\cong B_y$, and the canonical $B\to B_y$ gives a morphism $\mathrm{Spec}({\mathcal{O}}_y)\to V\to Y$, independent of $V$ — the canonical morphism $\mathrm{Spec}({\mathcal{O}}_y)\to Y$.

Proposition (2.4.2). Let $Y$ be a prescheme. For $y\in Y$, let $(\psi ,\theta ):\mathrm{Spec}({\mathcal{O}}_y)\to Y$ be the canonical morphism. Then $\psi$ is a homeomorphism of $\mathrm{Spec}({\mathcal{O}}_y)$ onto the subspace $S_y=z\in Y:y\in \bar{z}$ (the generizations of $y$); for $z=\psi (\mathfrak{p})$, ${\theta }_z^{♯}:{\mathcal{O}}_z\to ({\mathcal{O}}_y{)}_{\mathfrak{p}}$ is an isomorphism. So $(\psi ,\theta )$ is a monomorphism of ringed spaces.

There is thus a bijection between $\mathrm{Spec}({\mathcal{O}}_y)$ and the set of closed irreducible subsets of $Y$ containing $y$.

Corollary (2.4.3). $y\in Y$ is the generic point of an irreducible component of $Y$ iff ${\mathcal{O}}_y$ has only its maximal ideal as prime — i.e., ${\mathcal{O}}_y$ has dimension zero.

Proposition (2.4.4). Let $X=\mathrm{Spec}(A)$ be a local scheme with closed point $a$, and $Y$ a prescheme. Every morphism $u=(\psi ,\theta ):X\to Y$ factors uniquely as $X\to \mathrm{Spec}({\mathcal{O}}_{\psi (a)})\to Y$, where the second arrow is the canonical morphism and the first corresponds to a local homomorphism ${\mathcal{O}}_{\psi (a)}\to A$. This gives a canonical bijection between $\mathrm{Hom}(X,Y)$ and the set of local homomorphisms ${\mathcal{O}}_y\to A$ (for $y\in Y$).

(2.4.5) The spectrum of a field $K$ is a single point. For $A$ a local ring with maximal ideal $\mathfrak{m}$, a local homomorphism $A\to K$ has kernel $\mathfrak{m}$, factoring as $A\to A/\mathfrak{m}\to K$ with the second map a field monomorphism. Thus morphisms $\mathrm{Spec}(K)\to \mathrm{Spec}(A)$ correspond bijectively to field monomorphisms $A/\mathfrak{m}\to K$.

For $y\in Y$ and an ideal ${\mathfrak{a}}_y\subset {\mathcal{O}}_y$, the canonical ${\mathcal{O}}_y\to {\mathcal{O}}_y/{\mathfrak{a}}_y$ gives a morphism $\mathrm{Spec}({\mathcal{O}}_y/{\mathfrak{a}}_y)\to \mathrm{Spec}({\mathcal{O}}_y)\to Y$, the canonical morphism. When ${\mathfrak{a}}_y={\mathfrak{m}}_y$, this gives the morphism $\mathrm{Spec}(\kappa (y))\to Y$.

Corollary (2.4.6). Let $X=\mathrm{Spec}(K)$ for $K$ a field with unique point $\xi$, and $Y$ a prescheme. Every morphism $u:X\to Y$ factors uniquely as $X\to \mathrm{Spec}(\kappa (\psi (\xi )))\to Y$, with the first arrow given by a field monomorphism $\kappa (\psi (\xi ))\to K$. Hence Hom(X, Y) ↔ ⨆_{y ∈ Y} Hom_{field}(κ(y), K).

Corollary (2.4.7). For every $y\in Y$, the canonical morphism $\mathrm{Spec}({\mathcal{O}}_y/{\mathfrak{a}}_y)\to Y$ is a monomorphism of ringed spaces.

Remark (2.4.8). On a local scheme, every invertible ${\mathcal{O}}_X$-module is trivial (isomorphic to ${\mathcal{O}}_X$), since every affine open containing the closed point equals $X$. This fails for general affine schemes; for $A$ normal, invertibility implies triviality iff $A$ is a UFD.

2.5. Preschemes over a prescheme

Definition (2.5.1). Given a prescheme $S$, a prescheme over $S$ (or $S$-prescheme) is a prescheme $X$ together with a morphism $\varphi :X\to S$, the structure morphism. $S$ is the base prescheme. When $S=\mathrm{Spec}(A)$, $X$ is an $A$-prescheme; this is equivalent to making ${\mathcal{O}}_X$ a sheaf of $A$-algebras.

Every prescheme is canonically a $\mathbb{Z}$-prescheme.

For $\varphi :X\to S$ and $x\in X$, $s\in S$, $x$ is over $s$ if $\varphi (x)=s$. $X$ dominates $S$ if $\varphi$ is dominant.

(2.5.2) For $S$-preschemes X, Y, a morphism $u:X\to Y$ is an $S$-morphism (or morphism over $S$) if ${\varphi }_Y\circ u={\varphi }_X$. $S$-preschemes form a category, with morphism set ${\mathrm{Hom}}_S(X,Y)$; the identity is 1_X. When $S=\mathrm{Spec}(A)$, we say $A$-morphism.

(2.5.3) A morphism $v:X^{\prime }\to X$ makes any $S$-prescheme $X$ into an $S$-prescheme $X^{\prime }$; restrictions of $S$-morphisms to open subsets are $S$-morphisms; $S$-morphisms glue from compatible restrictions on an open cover.

(2.5.4) A morphism $S^{\prime }\to S$ makes every $S^{\prime }$-prescheme into an $S$-prescheme. If $S^{\prime }\subset S$ is open and an $S$-prescheme $X$ has $\varphi (X)\subset S^{\prime }$, then $X$ is naturally an $S^{\prime }$-prescheme.

(2.5.5) An $S$-section of an $S$-prescheme $X$ is an $S$-morphism $S\to X$ — i.e., $\psi :S\to X$ with $\varphi \circ \psi =1_S$. Write $\Gamma (X/S)$ for the set of $S$-sections.