### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

For

$\cdots \hookrightarrow X_{(n)} \hookrightarrow X_{(n+1)} \hookrightarrow \cdots \hookrightarrow X$

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

$0 \to X_{(n-1)} \to X_{(n)} \to Gr_n(X) \to 0 \,,$

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

$Gr(X)_n := X_{(n)}/X_{(n-1)} \,,$

## Examples

• For $\mathcal{A}$ an abelian category and $C_{\bullet, \bullet}$ a double complex in $\mathcal{A}$, let $X = Tot(C)$ 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.

## References

Discussion of the universal property of the associated graded construction on mathoverflow

Revised on July 28, 2016 01:21:43 by Bartek (130.216.30.132)