nLab
associated graded object

Definition

For

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

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

$$0\to X_{(n-1)}\to X_{(n)}\to {\mathrm{Gr}}_n(X)\to 0,$$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:

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

Examples

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

Related concepts

• spectral sequence of a filtered complex
• associated graded ring (or algebra)