nLab associated graded object

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:

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

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.

• associated graded vector space

Related concepts

filtered objects

• filtered topological space

  • spectral sequence of a tower of fibrations

• filtered vector space

• filtered chain complex

  • spectral sequence of a filtered complex

  • Hodge filtration

• filtered ring

• filtered object in an (∞,1)-category

• filtered stable homotopy type

  • spectral sequence of a filtered stable homotopy type

• filtered probability space

associated graded objects

• associated graded vector space

• associated graded ring

References

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