Chapter I — The Language of Schemes

§6. Finiteness Conditions

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

6.1. Noetherian and locally Noetherian preschemes

Definition (6.1.1). A prescheme $X$ is locally Noetherian if every $x\in X$ admits an affine open neighborhood $\mathrm{Spec}(A)$ with $A$ Noetherian. $X$ is Noetherian if it is locally Noetherian and quasi-compact.

Proposition (6.1.2). A prescheme $X$ is locally Noetherian iff $X$ admits a covering by affine opens $\mathrm{Spec}(A_i)$ with each $A_i$ Noetherian.

Proposition (6.1.3). For $X$ locally Noetherian: every open subset is locally Noetherian; the underlying topological space of $X$ is locally Noetherian.

Proposition (6.1.4). $X$ is Noetherian iff its underlying space is Noetherian and $X$ is covered by finitely many affine opens $\mathrm{Spec}(A_i)$ with each $A_i$ Noetherian.

Proposition (6.1.6). A closed subprescheme of a (locally) Noetherian prescheme is (locally) Noetherian.

Corollary (6.1.7). A locally closed subprescheme of a (locally) Noetherian prescheme is (locally) Noetherian.

Proposition (6.1.10). A locally Noetherian prescheme has only finitely many irreducible components in each quasi-compact open set.

Corollary (6.1.11). Every closed subset of a locally Noetherian prescheme has the topology of a Noetherian space locally.

Proposition (6.1.13). Every open subset of a Noetherian prescheme is quasi-compact.

6.2. Artinian preschemes

Definition (6.2.1). A prescheme $X$ is Artinian if $X=\mathrm{Spec}(A)$ for $A$ an Artinian ring.

Proposition (6.2.2). An Artinian prescheme is Noetherian and 0-dimensional; its underlying space is finite, discrete.

6.3. Morphisms of finite type

Definition (6.3.1). A morphism $f:X\to Y$ is locally of finite type if for every $x\in X$ there are affine opens $\mathrm{Spec}(A)\ni x$ and $\mathrm{Spec}(B)\ni f(x)$ with $f(\mathrm{Spec}(A))\subset \mathrm{Spec}(B)$ such that $A$ is a finitely generated $B$-algebra. $f$ is of finite type if additionally $f$ is quasi-compact (i.e., the inverse image of every quasi-compact open is quasi-compact).

Proposition (6.3.2). Locally finite type is preserved under composition.

Proposition (6.3.3). Locally finite type is preserved under base change.

Proposition (6.3.4). Locally finite type can be checked on an affine open cover.

Corollary (6.3.5). Finite-type morphisms are stable under composition, base change, and product.

Corollary (6.3.6). Closed immersions are of finite type.

Proposition (6.3.7). A morphism of finite type between Noetherian preschemes has Noetherian source and target compatibility.

Proposition (6.3.10). Finite-type morphisms with Noetherian target have Noetherian source.

6.4. Algebraic preschemes

Definition (6.4.1). A $K$-prescheme algebraic over a field $K$ (or algebraic $K$-prescheme) is a $K$-prescheme of finite type. $K$ is the base field.

Proposition (6.4.2). Every algebraic $K$-prescheme is Noetherian.

Corollary (6.4.3). Every closed subprescheme of an algebraic $K$-prescheme is algebraic.

Proposition (6.4.4) (Hilbert's Nullstellensatz). For an algebraic $K$-prescheme $X$ and any $K$-point $x\in X$ (closed point in particular), the residue field $\kappa (x)$ is a finite algebraic extension of $K$.

Corollary (6.4.6). A $K$-prescheme is finite over $K$ (a finite $K$-scheme) iff it is affine, of finite type over $K$, and its space is finite.

(6.4.5) A finite $K$-scheme is $\mathrm{Spec}(A)$ with $A$ a finite $K$-algebra; its rank is ${\dim }_{K}A$. The separable rank is ${\dim }_K(A_{sep})$, where $A_{sep}$ is the maximal separable subalgebra.

(6.4.8) The geometric number of points of an algebraic $K$-prescheme $X$ is the cardinality of the set of $\bar{K}$-rational points, where $\bar{K}$ is an algebraic closure.

Proposition (6.4.9). Algebraic $K$-preschemes form a category, closed under fiber products over $K$.

Proposition (6.4.11). Properties of algebraic $K$-preschemes pass to extension fields $K\to L$.

6.5. Local determination of a morphism

Proposition (6.5.1). A morphism of preschemes is determined by its restrictions to an open cover of the source (i.e., morphisms can be glued from compatible restrictions).

(6.5.6) A morphism $f:X\to Y$ is birational if X, Y have finitely many irreducible components and the induced map on generic points is bijective with isomorphisms of local rings; see (2.2.9).

6.6. Quasi-compact morphisms

Definition (6.6.1). A morphism $f:X\to Y$ is quasi-compact if $f^{-1}(U)$ is quasi-compact for every quasi-compact open $U\subset Y$. Equivalently, $f^{-1}(V)$ is quasi-compact for every affine open $V\subset Y$.

(6.6.2) Morphism locally of finite type: defined in (6.3.1); the combination of "locally of finite type" + "quasi-compact" yields "of finite type."

Composition and base change preserve quasi-compactness.