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

Schanuel's lemma

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 \,.

Such connective chain complexes are the connective objects in the \infty -category of chain complexes. Rearding under the stable Dold-Kan correspondence as HRH R-module spectra they are the connective spectra.

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 April 20, 2023 at 07:25:43. See the history of this page for a list of all contributions to it.