Ext group of modules

模的 Ext 群

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Translation evidence: Literature usage

Meaning in context

For left RR-modules M,NM,N, Ext⁡Rn(M,N)\operatorname{Ext}^n_R(M,N) is degree-nn cohomology of Hom⁡R(P∙,N)\operatorname{Hom}_R(P_\bullet,N) for a projective resolution P∙→MP_\bullet\to M.

将第一变量的投射解消施以到第二变量的 Hom 后取第 nn 次上同调所得的群。

Example

Ext⁡Z1(Z/2,Z)≅Z/2\operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/2,\mathbb Z)\cong\mathbb Z/2.

Usage and distinctions

Ext⁡0\operatorname{Ext}^0 is Hom, while Ext⁡1\operatorname{Ext}^1 classifies extensions with fixed outer terms.

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Definition sources

  1. Wen-Wei Li — Methods in Algebra II, author source实例: Ext 和 Tor (sec:Ext-Tor), definition-proposition def:Ext-classical (Ext 函子), preceding projective-resolution definition and following R-Mod specialization · Checked 2026-10-15

Translation sources

  1. 李文威 — 代数学方法 卷二，作者源文件实例: Ext 和 Tor (sec:Ext-Tor), def:Ext-classical (Ext 函子), value in Ab and following notation Ext_R^n for module categories · Checked 2026-10-15
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