nLab associated graded object

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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.