nLab connective chain complex

Contents

Context

Homological algebra

homological algebra

Introduction

diagram chasing

Contents

Definition

An unbounded chain complex $X_\bullet \in Ch_\bullet$

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

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 $X_{-n} \simeq 0$ for all $n \geq 1$, hence if it is of the form

$\cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} \cdots \,.$

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

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.