nLab associated graded object

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

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

Examples

filtered objects

associated graded objects

References

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

Last revised on November 25, 2019 at 14:34:08. See the history of this page for a list of all contributions to it.