nLab total complex

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Idea

For $C_{\bullet, \bullet}$ a double complex (in some abelian category $\mathcal{A}$), its total complex $Tot(C)_\bullet$ is an ordinary complex which in degree $k$ is the direct sum of all components of total degree $k$.

Definition

Let $\mathcal{A}$ be an abelian category with arbitrary direct sums.

Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes in $\mathcal{A}$ and $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ for the category of double complexes. (Hence we use the convention that in a double complex the vertical and horizontal differential commute with each other.)

Definition

For $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ a double complex, its associated total complex $Tot(C)_\bullet \in Ch_\bullet(\mathcal{A})$ is the chain complex whose components are the direct sums

$Tot(C)_n = \bigoplus_{k+l = n} C_{k,l}$

and whose differentials are given by the linear combination

$\partial^{Tot} \coloneqq \partial^C_{vert} + (-1)^{vertical\;degree} \partial^C_{hor} \,.$

Properties

Total homology and spectral sequences

Remark

The chain homology of the total complex $Tot(C)_\bullet$ is sometimes called the total homology of the double complex $C_{\bullet, \bullet}$.

Remark

A tool for computing the homology of a total complex, hence for computing the total homology of a double complex, is the spectral sequence of a double complex. See there for more details.

Exactness

Proposition

If $C_{\bullet,\bullet}$ is bounded and has exact rows or coloumns then also $Tot(C)_\bullet$ is exact.

Proof

Use the acyclic assembly lemma.

Relation to total simplicial sets and homotopy colimits

The total chain complex is, under the Dold-Kan correspondence, equivalent to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the total simplicial set of a bisimplicial set.

References

For instance secton 1.2 of

Revised on March 11, 2015 11:17:27 by marcel? (141.39.226.234)