scheme

概形

Algebra & Algebraic Geometry · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A locally ringed space in which every point has an open neighborhood isomorphic, as a locally ringed space, to the prime spectrum of a commutative unital ring equipped with its structure sheaf.

每个点都有开邻域，连同限制的结构层同构于某个交换含幺环的素谱及其结构层的局部赋环空间。

Example

$\operatorname{Spec}(k[\epsilon]/(\epsilon^2))$ for a field $k$ is a one-point scheme with a nonzero nilpotent global section.

Usage and distinctions

The underlying prime-spectrum topology does not determine the structure sheaf; being affine, reduced, Noetherian or separated is not required in the general scheme definition.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. The Stacks Project — Schemes; affine schemesDefinition 26.9.1, with affine schemes as in Definition 26.5.5 (tag 01HR) · Checked 2026-10-15
2. The Stacks Project — Affine schemesDefinition 26.5.3, Lemma 26.5.4 and Definition 26.5.5 · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，作者公开稿Section 10.2 几何模形式, opening discussion of R-概形 and X(R)=Hom_{R-概形}(Spec R,X), printed p.296 (PDF p.302); evidence of usage, not a normative translation ruling · Checked 2026-10-15

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