nLab
associated graded object

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

For

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

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

0X (n1)X (n)Gr n(X)0, 0 \to X_{(n-1)} \to X_{(n)} \to Gr_n(X) \to 0 \,,

hence the quotient of the nnth layer of XX by the next lower one:

Gr(X) n:=X (n)/X (n1), Gr(X)_n := X_{(n)}/X_{(n-1)} \,,

Examples

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)