Chain complexes {#chain-complexes .unnumbered}
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For our computations, we will work with nonnegatively graded chain complexes $C_\ast$ of abelian groups on which $G$ acts by chain maps; denoting by ${{{\mathbb Z}G}}$ the integral group ring of $G$, one might equivalently say that $C_*$ is a chain complex of ${{{\mathbb Z}G}}$-modules. We will adopt this terminology from now on. We will also assume that these chain complexes are *cellular*, i.e. equipped with a distinguished ${{{\mathbb Z}G}}$-basis; this means that for each ${{n}}\geq 0$ there is a collection of distinguished elements of $C_{{n}}$ such that the elements of the form $ay$, with $a\in G$ and $y$ distinguished, are all distinct and form a ${{\mathbb Z}}$-basis of $C_{{n}}$.

In the locally effective version, we assume that the elements of the chain complex have a finite encoding, and there is an algorithm expressing arbitrary elements as (unique) ${{{\mathbb Z}G}}$-linear combinations of the elements of the distinguished bases. We require that the operations of zero, addition, inverse, multiplication by elements of ${{{\mathbb Z}G}}$, and differentials are computable.[^10]

A basic example on which these assumptions are modelled is that of the normalized chain complex $C_*X$ of a simplicial set $X$. For each ${{n}}\geq 0$, a ${{\mathbb 