associated graded object

(also nonabelian homological algebra)

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:

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

- 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.

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)