spectral sequence

谱序列

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

Translation evidence: Literature usage

Meaning in context

In an abelian category, a sequence of differential objects $(E_r,d_r)$ with $d_r^2=0$ and specified isomorphisms $H(E_r,d_r)\cong E_{r+1}$; the cohomological bigraded version has differential of bidegree $(r,1-r)$.

在 Abel 范畴中，由微分平方为零的对象 $(E_r,d_r)$ 以及指定同构 $H(E_r,d_r)\cong E_{r+1}$ 构成的序列；上同调双分次版本的微分次数为 $(r,1-r)$。

Example

For $C^0=ka$, $C^1=kb$, $d(a)=b$, place $a$ in filtration degree $0$ and $b$ in degree $2$; the filtered-complex spectral sequence has $d_2^{0,0}:ka\to kb$ an isomorphism and $E_3=0$.

Usage and distinctions

A target and convergence require additional hypotheses; even convergent $E_\infty$ terms usually determine only associated graded pieces, with extension problems remaining.

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Definition sources

1. Wen-Wei Li — Methods in Algebra II, author source谱序列的一般定义 (sec:spectral-sequence), def:SS-abstract and def:graded-spectral-sequence · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷二，作者源文件def:SS-abstract, index entry 谱序列 (spectral sequence) and definition; def:graded-spectral-sequence names the graded and bigraded variants · Checked 2026-10-15

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