Chapter I — The Language of Schemes

§5. Reduced Preschemes; Separation Condition

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

5.1. Reduced preschemes

Proposition (5.1.1). Let $X$ be a prescheme. The set $\mathcal{N}\subset {\mathcal{O}}_X$ of nilpotent germs is a quasi-coherent sheaf of ideals of ${\mathcal{O}}_X$, called the nilradical of ${\mathcal{O}}_X$. Locally on an affine open $\mathrm{Spec}(A)$, $\mathcal{N}$ corresponds to the nilradical of $A$.

Corollary (5.1.2). $\mathcal{N}$ defines a closed subprescheme $X_{red}$ of $X$ whose underlying space coincides with that of $X$.

Definition (5.1.3). A prescheme $X$ is reduced at $x\in X$ if ${\mathcal{O}}_x$ is a reduced ring (no nonzero nilpotents). $X$ is reduced if it is reduced at every point — equivalently, $\mathcal{N}=0$. $X_{red}$ is called the reduced prescheme associated with $X$.

Proposition (5.1.4). $X_{red}$ is reduced. The canonical immersion $X_{red}\to X$ is a homeomorphism, and every morphism $Y\to X$ with $Y$ reduced factors uniquely through $X_{red}$.

Proposition (5.1.6). Integral preschemes (2.1.8) are exactly the reduced and irreducible ones.

Proposition (5.1.7). For an affine scheme $\mathrm{Spec}(A)$: $\mathrm{Spec}(A)$ is integral iff $A$ is an integral domain.

Corollary (5.1.8). A prescheme $X$ is integral iff ${\mathcal{O}}_x$ is an integral domain for some/every $x$ (equivalently, the residue field $\kappa (\eta )$ at the generic point $\eta$ is the field of fractions of ${\mathcal{O}}_{X,\eta }$).

Proposition (5.1.9). For a prescheme $X$ of finite type over a Noetherian base, $X_{red}$ is again of finite type.

Corollary (5.1.10). Reducedness is preserved by open immersions, fiber products over reduced base, and base change to reduced bases.

5.2. Subprescheme with a given underlying space

Proposition (5.2.1). For every locally closed subset $Y\subset X$, there is a unique reduced subprescheme of $X$ whose underlying space is $Y$. Call it the reduced subprescheme structure on $Y$.

Proposition (5.2.2). For closed $Y\subset X$, the reduced structure on $Y$ corresponds to the largest radical ideal sheaf $\mathcal{J}\subset {\mathcal{O}}_X$ with $Supp({\mathcal{O}}_X/\mathcal{J})=Y$.

Corollary (5.2.3). Every immersion $Y\hookrightarrow X$ factors uniquely as a closed immersion into the reduced subprescheme structure on its image.

Corollary (5.2.4). Two subpreschemes with the same underlying space and the same ideal radical agree.

5.3. Diagonal; graph of a morphism

(5.3.1) For an $S$-prescheme $X$, the diagonal morphism is ${\Delta }_{X/S}:X\to X{\times }_{S}X$, the unique $S$-morphism with both projections equal to 1_X.

Proposition (5.3.2). ${\Delta }_{X/S}$ is an immersion. It is a closed immersion iff $X\to S$ is separated (see (5.4.1)).

Corollary (5.3.4). The diagonal is functorial in $X$.

Proposition (5.3.5). For a morphism $f:X\to Y$ of $S$-preschemes, the graph ${\Gamma }_f:X\to X{\times }_{S}Y$ defined by $(1_X,f)$ is an immersion.

Corollary (5.3.6). If $Y\to S$ is separated, every graph ${\Gamma }_f$ is a closed immersion.

Proposition (5.3.8). Diagonal and graph commute with base change.

Corollary (5.3.13). For composable morphisms $X\to Y\to S$, the diagonal of $X\to S$ factors through the diagonal of $X\to Y$ followed by the inclusion $X{\times }_{Y}X\to X{\times }_{S}X$.

5.4. Separated morphisms and separated preschemes

Definition (5.4.1). A morphism $f:X\to S$ is separated (or $X$ is separated over $S$) if ${\Delta }_{X/S}:X\to X{\times }_{S}X$ is a closed immersion. An $S$-scheme is a separated $S$-prescheme. A scheme (without base) is a $\mathbb{Z}$-scheme — i.e., a separated prescheme.

Bracketed gloss: scheme [in EGA: separated prescheme].

Proposition (5.4.2). Affine schemes are separated. Closed immersions are separated.

Corollary (5.4.3). Composition of separated morphisms is separated.

Corollary (5.4.4). Separatedness is preserved under base change.

Corollary (5.4.5). A subprescheme of a separated prescheme is separated.

Corollary (5.4.6). If $g\circ f$ is separated, so is $f$.

Corollary (5.4.7). A morphism is separated iff its base change to every affine open of the target is separated.

5.5. Separation criteria

Proposition (5.5.1) (valuative criterion). $f:X\to Y$ is separated iff for every valuation ring $V$ with field of fractions $K$, every commutative square

Spec(K) ──→ X
   │        │
   ↓        ↓
Spec(V) ──→ Y

admits at most one lift $\mathrm{Spec}(V)\to X$.

Corollary (5.5.2). Equivalently: any two $S$-sections of $X\to S$ that agree on a dense open subset agree everywhere.

Corollary (5.5.3). For $X$, $Y$ separated $S$-schemes and $f,g:X\to Y$ two $S$-morphisms agreeing on a dense open: $f=g$.

Proposition (5.5.4). Sums of separated $S$-preschemes are separated.

Proposition (5.5.5). A subprescheme of $\mathrm{Spec}(A)$ (any $A$) is separated.

Proposition (5.5.6). A monomorphism is separated.

Corollaries (5.5.7)–(5.5.9). Immersions are separated; locally closed subpreschemes of separated preschemes are separated; if $f:X\to Y$ is separated and $g\circ f$ is separated, then $g$ is separated.