nLab total complex

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, direct products or a mixture of those

The sum total complex

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

the product total complex

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

the product-sum total complex

$$Tot^{\pi\oplus}(C)_n = \prod_{k+l = n, k \lt 0} C_{k,l}\oplus\bigoplus_{k+l = n, k\geq 0} C_{k,l}$$

and the sum-product total complex

$$Tot^{\oplus\pi}(C)_n = \bigoplus_{k+l = n, k \lt 0} C_{k,l}\oplus\prod_{k+l = n, k\geq 0} C_{k,l}$$

and whose differentials are given by the linear combination

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

Using the four different types of total complexes gives the flexibility to make more nuanced exactness statements for the total complex under various assumptions on the double complex.

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

First let $C$ be a double complex in any abelian category

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

Now let $C$ be a double complex of abelian groups.

• If $C_{\bullet,\bullet}$ has exact rows then the product-sum total complex is exact.
• If $C_{\bullet,\bullet}$ has exact rows and kernels (or equivalently) images between row complexes are exact, then the sum and product total complexes are exact
• If $C_{\bullet,\bullet}$ has exact rows and for each $i$ taking $H_i$ of the columns gives an exact complex, then the sum-product total complex 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:

• Charles Weibel, section 1.2 of: An Introduction to Homological Algebra