Chapter I — The Language of Schemes

§3. Product of Preschemes

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3.1. Sums of preschemes

(3.1.1) Let $(X_{\lambda }{)}_{\lambda \in L}$ be a family of preschemes. The topological sum $X={⨆}_{\lambda }{X}_{\lambda }$ carries a sheaf of rings ${\mathcal{O}}_X$ whose restriction to each $X_{\lambda }$ is ${\mathcal{O}}_{X_{\lambda }}$. The ringed space $(X,{\mathcal{O}}_X)$ is a prescheme, called the sum of the $(X_{\lambda })$. The canonical injections $X_{\lambda }\to X$ are open immersions. For an $S$-prescheme structure, the sum is the coproduct in the category of $S$-preschemes.

3.2. Products of preschemes

Definition (3.2.1). Let $X$, $Y$ be $S$-preschemes with structure morphisms $f:X\to S$ and $g:Y\to S$. A product of $X$ and $Y$ over $S$ is an $S$-prescheme $Z$ together with $S$-morphisms $p:Z\to X$ and $q:Z\to Y$ (the canonical projections) such that, for every $S$-prescheme $T$, the map

Hom_S(T, Z) → Hom_S(T, X) × Hom_S(T, Y),   u ↦ (p ∘ u, q ∘ u)

is a bijection. By the universal property, when it exists, $Z$ is unique up to unique $S$-isomorphism. Write $Z=X{\times }_{S}Y$.

Proposition (3.2.2). The product $X{\times }_{S}Y$ exists when $X$, $Y$, and $S$ are affine. Specifically, if $X=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(B)$, $S=\mathrm{Spec}(C)$, then $X{\times }_{S}Y=\mathrm{Spec}(A{\otimes }_{C}B)$, with projections corresponding to $a\mapsto a\otimes 1$ and $b\mapsto 1\otimes b$.

Corollary (3.2.3). For affine $S$, $X{\times }_{S}Y$ exists if $X$ is affine over $S$, etc.

Proposition (3.2.4). Existence of products is local: if it holds when $X$, $Y$, $S$ are replaced by affine opens forming covers, then it holds in general.

Theorem (3.2.6). For any $S$-preschemes $X$, $Y$, the product $X{\times }_{S}Y$ exists.

The proof glues affine local products using a cocycle argument and (3.2.4); see Lemmas (3.2.6.1)–(3.2.6.4).

Corollary (3.2.7). Products are functorial: for $S$-morphisms $u:X\to X^{\prime }$, $v:Y\to Y^{\prime }$, there is a unique $S$-morphism $u\times v:X{\times }_{S}Y\to X^{\prime }{\times }_S{Y}^{\prime }$ compatible with projections.

3.3. Formal properties of the product; change of base prescheme

(3.3.1) Commutativity: there is a canonical isomorphism $X{\times }_{S}Y\cong Y{\times }_{S}X$.

(3.3.2) Associativity: for three $S$-preschemes, $(X{\times }_{S}Y){\times }_{S}Z\cong X{\times }_S(Y{\times }_{S}Z)\cong X{\times }_{S}Y{\times }_{S}Z$.

Proposition (3.3.3). Identity: $X{\times }_{S}S\cong X$ canonically.

Corollary (3.3.4). For $S^{\prime }\to S$ a morphism, the base change $X_{(S^{\prime })}=X{\times }_S{S}^{\prime }$ is an $S^{\prime }$-prescheme, with structure morphism the second projection. This $X\mapsto X_{(S^{\prime })}$ is a functor (S\text{-preschemes}) → (S′\text{-preschemes}).

Proposition (3.3.9). Transitivity of base change: for $S^{\prime \prime }\to S^{\prime }\to S$, $(X{\times }_S{S}^{\prime }){\times }_{S^{\prime }}{S}^{\prime \prime }\cong X{\times }_S{S}^{\prime \prime }$.

Corollary (3.3.10). Base change commutes with finite products: $(X{\times }_{S}Y{)}_{(S^{\prime })}=X_{(S^{\prime })}{\times }_{S^{\prime }}{Y}_{(S^{\prime })}$.

Corollary (3.3.11). Open immersions are preserved by base change: if $j:U\to X$ is an open immersion, so is $j{\times }_S{1}_{S^{\prime }}:U_{(S^{\prime })}\to X_{(S^{\prime })}$.

3.4. Points of a prescheme with values in a prescheme; geometric points

Definition (3.4.1). Let $X$ be an $S$-prescheme, $T$ an $S$-prescheme. A $T$-valued point of $X$ (or $T$-point) is an $S$-morphism $T\to X$; the set of such is $X(T)={\mathrm{Hom}}_S(T,X)$.

(3.4.2) A fiber product of sets: for sets $A\to C\leftarrow B$, $A{\times }_{C}B=(a,b)\in A\times B:f(a)=g(b)$.

Proposition (3.4.3). $(X{\times }_{S}Y)(T)=X(T){\times }_{S(T)}Y(T)$ for every $S$-prescheme $T$.

(3.4.4) Points with values in a ring $A$: if $T=\mathrm{Spec}(A)$, write $X(A)=X(T)$. For $S=\mathrm{Spec}(C)$, X(A) = Hom_{C\text{-alg}}(\text{some algebra}, A) when $X$ is affine.

(3.4.5) Geometric points. A geometric point of $X$ is a morphism $\mathrm{Spec}(K)\to X$ for $K$ a field. The value field of the geometric point is $K$. A geometric point above $s\in S$ is a geometric point $\mathrm{Spec}(K)\to X$ whose composition with $X\to S$ factors through $\mathrm{Spec}(K)\to \mathrm{Spec}(\kappa (s))\to S$. A geometric point localized at $x\in X$ is one whose underlying map sends the unique point to $x$.

Lemma (3.4.6). Geometric points of $X$ above $s\in S$ correspond to pairs $(x,K_x)$ with $x\in X$ over $s$ and $K_x$ an extension of $\kappa (x)$.

Proposition (3.4.7). For $f:X\to Y$ an $S$-morphism, every geometric point of $X$ maps via $f$ to a geometric point of $Y$ with the same value field.

3.5. Surjections and injections

Proposition (3.5.2). A morphism $f:X\to Y$ is surjective iff every geometric point of $Y$ lifts to a geometric point of $X$.

Proposition (3.5.3). Surjectivity is preserved by base change: if $f:X\to Y$ is surjective and $g:Y^{\prime }\to Y$ is a morphism, then $f_{(Y^{\prime })}:X{\times }_Y{Y}^{\prime }\to Y^{\prime }$ is surjective.

Definition (3.5.4). A morphism $f:X\to Y$ is radicial (or universally injective) if for every base change $Y^{\prime }\to Y$, $f_{(Y^{\prime })}$ is injective; equivalently, for every field $K$, $X(K)\to Y(K)$ is injective. A radicial morphism is injective.

Proposition (3.5.6). $f$ is radicial iff it is injective and induces purely inseparable residue-field extensions at every point.

Proposition (3.5.7). Composition and base change preserve radicial morphisms.

Proposition (3.5.8). Geometric injection: $f:X\to Y$ is radicial iff the diagonal $X\to X{\times }_{Y}X$ is surjective.

3.6. Fibers

Proposition (3.6.1). For an $S$-prescheme $X$ with structure morphism $f$, and a geometric point $s:\mathrm{Spec}(K)\to S$, the fiber $X_s=X{\times }_S\mathrm{Spec}(K)$ is a $K$-prescheme. Set-theoretically, the underlying space of $X_s$ (where $K=\kappa (s)$) is $f^{-1}(s)$.

Proposition (3.6.4). Fibers commute with base change: $(X{\times }_{S}Y{)}_s=X_s{\times }_K{Y}_s$ ($K=\kappa (s)$).

Proposition (3.6.5). Surjectivity and radiciality of a morphism are detected fiber-by-fiber.

3.7. Reduction mod 𝔍

For an ideal $\mathfrak{J}\subset A$ and an $A$-prescheme $X$, the reduction $Xmod\mathfrak{J}$ is $X{\times }_{\mathrm{Spec}(A)}\mathrm{Spec}(A/\mathfrak{J})$. This is a functor from $A$-preschemes to $(A/\mathfrak{J})$-preschemes.