Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Definition

For

$\cdots ↪{X}_{\left(n\right)}↪{X}_{\left(n+1\right)}↪\cdots ↪X$\cdots \hookrightarrow X_{(n)} \hookrightarrow X_{(n+1)} \hookrightarrow \cdots \hookrightarrow X

a filtered object in an abelian category $𝒞$, the associated graded object $\mathrm{Gr}\left(X\right)$ is the graded object which in degree $n$ is the cokernel of the $n$th inclusion, fitting into a short exact sequence

$0\to {X}_{\left(n-1\right)}\to {X}_{\left(n\right)}\to {\mathrm{Gr}}_{n}\left(X\right)\to 0\phantom{\rule{thinmathspace}{0ex}},$0 \to X_{(n-1)} \to X_{(n)} \to Gr_n(X) \to 0 \,,

hence the quotient of the $n$th layer of $X$ by the next lower one:

$\mathrm{Gr}\left(X{\right)}_{n}:={X}_{\left(n\right)}/{X}_{\left(n-1\right)}\phantom{\rule{thinmathspace}{0ex}},$Gr(X)_n := X_{(n)}/X_{(n-1)} \,,

Examples

• For $𝒜$ an abelian category and ${C}_{•,•}$ a double complex in $𝒜$, let $X=\mathrm{Tot}\left(C\right)$ be the corresponding total complex. This is naturally filtered by either row-degree or by column-degree. The corresponding associated graded complex is what the terms in the spectral sequence of a filtered complex compute.
Revised on October 18, 2012 15:22:52 by Zoran Škoda (161.53.130.104)