nLab
total complex

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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

Definition

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

Write Ch (𝒜)Ch_\bullet(\mathcal{A}) for the category of chain complexes in 𝒜\mathcal{A} and C ,Ch (Ch (𝒜))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 ,Ch (Ch (𝒜))C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A})) a double complex, it total complex Tot(C) Ch (𝒜)Tot(C)_\bullet \in Ch_\bullet(\mathcal{A}) is the chain complex whose components are the direct sums

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

and whose differentials are give by the linear combination

Tot vert C+(1) verticaldegree hor C. \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) Tot(C)_\bullet is sometimes called the total homology of the double complex C ,C_{\bullet, \bullet}.

Remark

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

Exactness

Proposition

If C ,C_{\bullet,\bullet} is bounded and has exact rows or coloumns then also Tot(C) 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 November 24, 2013 23:28:35 by Tim Porter (2.26.15.29)