V2EX affine scheme

Affine Scheme

定义 Definition

Affine scheme(仿射概形)是代数几何中的基本对象:它是某个交换环 \(A\) 的素谱 \(\mathrm{Spec}(A)\) 连同其结构层(sheaf of rings)所组成的概形。直观上,它把“由多项式方程定义的几何空间”与“环的代数信息”紧密对应起来,是一般概形的局部模型(概形在局部看起来像仿射概形)。

发音 Pronunciation (IPA)

/fan skim/

例句 Examples

An affine scheme is determined by a commutative ring.
仿射概形由一个交换环所决定。

Every scheme can be covered by open subsets that are affine schemes, which makes local computations possible.
任何概形都可以被一些开子集覆盖,而这些开子集都是仿射概形,这使得局部计算成为可能。

词源 Etymology

Affine源自拉丁语 affinis(“相邻的、相关的”),在数学里逐渐固定为“仿射的”(与线性结构相关,但允许平移)。Scheme在数学语境中指一种“组织/结构化的构造”,由20世纪格罗滕迪克(Grothendieck)在代数几何中系统化并推广为“概形”理论;affine scheme则特指与环 \(A\) 直接对应的那类概形 \(\mathrm{Spec}(A)\)。

相关词 Related Words

文学与典籍 Literary Works

  • Grothendieck & Dieudonné, *léments de géométrie algébrique (EGA)*(系统奠定概形语言,仿射概形为核心起点)
  • Robin Hartshorne, Algebraic Geometry(早期章节大量使用“affine scheme/affine schemes”建立基础)
  • David Mumford, The Red Book of Varieties and Schemes(以仿射概形与其粘合构造为重要线索)
  • Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry(频繁以仿射概形作为局部模型与练习对象)
  • Ulrich Grtz & Torsten Wedhorn, Algebraic Geometry I: Schemes(以仿射概形概形的递进方式讲授)
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