# nLab associated graded object

Contents

### Context

#### Homological algebra

homological algebra

Introduction

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:

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

## References

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

Last revised on November 25, 2019 at 09:34:08. See the history of this page for a list of all contributions to it.