homology of a module complex

模复形的同调

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Translation evidence: Literature usage

Meaning in context

For a chain complex $C$, the module $H_n(C)=\ker d_n/\operatorname{im}d_{n+1}$.

链复形中某次微分的核对前一次微分的像取商所得的模。

Example

For $\mathbb Z\xrightarrow{2}\mathbb Z$, $H_0\cong\mathbb Z/2$ and $H_1=0$.

Usage and distinctions

The relation $d^2=0$ is needed for the denominator to be a submodule of the numerator.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Wen-Wei Li — Methods in Algebra I, author source正合列入门 (sec:exactseq-module), opening complex definition's 同调群 quotient and immediately following indexed formula H_i=Ker d_i / Im d_{i+1} · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷一，作者源文件正合列入门 (sec:exactseq-module), opening complex definition's 同调群 quotient and immediately following indexed formula H_i=Ker d_i / Im d_{i+1} · Checked 2026-10-15

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