Chapter I — The Language of Schemes

§8. Chevalley Schemes

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

8.1. Allied local rings

Lemma (8.1.1). Let $(A_{\lambda })$ be a family of local rings inside a fixed field $K$, with $A_{\lambda }\subset A_{\mu }$ if and only if the corresponding maximal ideals satisfy ${\mathfrak{m}}_{\lambda }\supset {\mathfrak{m}}_{\mu }\cap A_{\lambda }$. Then there is a unique smallest local ring containing all the $A_{\lambda }$ inside $K$.

(8.1.1) Two local subrings $A_1,A_2\subset K$ are allied (or related, apparentés in EGA) if there is a chain $A_1=B_0\subset B_1\supset B_2\subset B_3\supset \cdots \supset B_n=A_2$ of local subrings with each inclusion either an extension of dominating local rings or its inverse.

Lemma (8.1.3). Allied is an equivalence relation. Allied local rings have the same field of fractions.

Lemma (8.1.4). For local rings $A_1,A_2\subset K$ with the same fraction field, alliance is captured by intersection: A_1 and A_2 are allied iff there is a finite sequence of local rings between them with successive dominance/anti-dominance.

Definition (8.1.4). A local subring $A_2\subset K$ dominates A_1 if $A_1\subset A_2$ and ${\mathfrak{m}}_1\subset {\mathfrak{m}}_2$ (where ${\mathfrak{m}}_i$ is the maximal ideal of $A_i$). A local ring of integral type over $K$ is one that dominates a finitely generated subalgebra of $K$.

Proposition (8.1.5). Allied local rings have the same dimension and the same residue field up to isomorphism.

8.2. Local rings of an integral scheme

(8.2.1) Let $X$ be an integral prescheme with function field $K=R(X)$. For every $x\in X$, ${\mathcal{O}}_x$ is a local subring of $K$. The collection ${\mathcal{O}}_x:x\in X$ determines $X$ up to canonical isomorphism in a precise sense:

Proposition (8.2.2). Let $K$ be a field. A scheme $X$ over $K$ in Chevalley's sense is a family $(A_{\alpha }{)}_{\alpha \in I}$ of local subrings of $K$, with shared field of fractions $K$, satisfying:

(a) Every two $A_{\alpha },A_{\beta }$ are allied; (b) Every prime ideal of every $A_{\alpha }$ is the contraction of some $A_{\beta }$'s prime; (c) The corresponding "spectrum" is irreducible.

The set $X$ of these local subrings is then in canonical bijection with the points of an integral prescheme over $\mathrm{Spec}(K)$, with ${\mathcal{O}}_x=A_{\alpha }$ for the corresponding $\alpha$.

Corollary (8.2.3). An integral $K$-scheme is determined by its function field and the family of its local rings.

Corollary (8.2.4). Morphisms of integral schemes correspond to compatible families of local-ring inclusions.

Corollary (8.2.5). Integral subschemes of an integral scheme $X$ correspond to "irreducible closed" subfamilies of the local rings.

Proposition (8.2.7). Chevalley's classical schemes (in his Séminaire) are equivalent to integral algebraic schemes over a field.

Proposition (8.2.8). Algebraic varieties in the sense of Serre's (FAC) are precisely integral algebraic $K$-schemes.

8.3. Chevalley schemes

A Chevalley scheme over a field $K$ is an integral algebraic $K$-scheme (in the EGA sense) — i.e., a finite-type, separated, integral $K$-scheme. The terminology is preserved historically; see (8.2.7).