nLab connective chain complex

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

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

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

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

Examples

• The singular chain complex of a topological space is connective. This is directly related to the fact that the homotopy groups of topological spaces are in non-negative degree, as opposed to their stabilization by spectra, which may have homotopy groups also in negative degrees.

Properties

Relation to simplicial abelian groups

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

Related concepts

• connective object, connective spectrum

• category of chain complexes

  • chain complex

  • chain map, quasi-isomorphism

  • chain homotopy

  • model structure on chain complexes

  • elliptic chain complex

• cochain complex

• bounded chain complex

• filtered chain complex

• perfect chain complex