V2EX snake lemma

Snake Lemma

定义 Definition

Snake lemma(蛇形引理)是同调代数/代数拓扑中的一个基本引理:在一个“交换图”(通常是两行短正合列上下对应、竖直映射相容)里,它保证可以把若干个(kernel)和余核(cokernel)连接起来,得到一条长正合序列,并且给出关键的“连接同态”(connecting homomorphism)。它常用于证明同态性质、推出长正合列、比较两个序列的结构。

发音 Pronunciation (IPA)

/snek lm/

例句 Examples

The snake lemma helps us derive a long exact sequence from a commutative diagram.
蛇形引理帮助我们从一个交换图中推出一条长正合序列。

Using the snake lemma, one can construct the connecting homomorphism between the cokernel of one map and the kernel of another, which is essential in many diagram-chasing arguments.
利用蛇形引理,人们可以构造从某个映射的余核到另一个映射的核之间的连接同态,这在许多“追图”论证中至关重要。

词源 Etymology

lemma”在数学里指“引理”,意为证明更大定理时用到的辅助命题,词源可追溯到希腊语 lēmma(“前提/所取之物)。而“snake(蛇)”来自该引理常见证明方式在交换图里沿箭头“追逐元素”(diagram chase)时,路径弯弯曲曲像蛇一样穿行于核与余核之间,因此得名“蛇形引理”。

相关词 Related Words

文学与著作 Notable Works

  • Homological Algebra Henri Cartan & Samuel Eilenberg
  • An Introduction to Homological Algebra Charles A. Weibel
  • Homology Saunders Mac Lane
  • An Introduction to Algebraic Topology Joseph J. Rotman
  • Categories for the Working Mathematician(相关背景:图与函子语言中常提及) Saunders Mac Lane
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