Some remarks on the homological algebra of multiple complexes (Q1084140)

Let \(R\) be an associative ring with identity element and modules be unitary \(R\)-modules. Every element \(I=(i_ 1,i_ 2,\dots)\) of \(\mathbb Z^{\mathbb N}\) is called an index, the index \((0,0,\dots,1,0,\dots)\) with 1 in the \(i\)-th position is denoted \(E_ i\). A multiple complex (or complex) \({\mathcal M}\) consists of a collection of modules \({\mathcal M}_ I\) for each index \(I\) and a collection of boundary maps \(d^ i_ I: {\mathcal M}_ I\to {\mathcal M}_{I-E_ i}\) for each index \(I\) and each positive integer \(i\) which satisfy \(d^ i_{I-E_ i}d^ i_ I=0\) for all \(i\in \mathbb N\) and all indices \(I\) and \(d^ j_{I-E_ i}d^ i_ I=d^ i_{I-E_ j}d^ j_ I\) for all \(i\neq j\) and all indices \(I\). The author studies some properties of projective complexes, complexes over a field, projective resolutions, minimal resolutions over a local Noetherian ring and finite homological dimension of regular Noetherian local rings.