nLab
connective chain complex

Contents

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

An unbounded chain complex X Ch X_\bullet \in Ch_\bullet

X 2X 1X 0X 1X 2 \cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} X_{-1} \stackrel{\partial}{\to} X_{-2} \stackrel{\partial}{\to} \cdots

is called connective if it has no nontrivial homology groups in negative degree. Often one means more strictly that it is connective if it is concentrated in non-negative degree, hence if X n0X_{-n} \simeq 0 for all n1n \geq 1, hence if it is of the form

X 2X 1X 000. \cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} \cdots \,.

This is connective as in connective spectrum. Hence a connective chain complex is in particular a bounded chain complex, bounded from below.

Examples

Properties

Relation to simplicial abelian groups

The Dold-Kan correspondence asserts that connective chain complexes of abelian groups are equivalent to abelian simplicial groups.

Last revised on March 13, 2015 at 07:42:30. See the history of this page for a list of all contributions to it.