Homology of multiple complexes and Mayer-Vietoris spectral sequences

Similarities are noted in two Mayer-Vietoris spectral sequences that generalize to any number of ideals the Mayer-Vietoris exact sequence in local cohomology for two ideals. One has as first terms Čech cohomology with respect to sums of the given ideals and converge to cohomology with respect to the product of the ideals, the other has as first terms Čech cohomology with respect to products of the given ideals and converge to cohomology with respect to the sum of the ideals. The first one was obtained by Lyubeznik [Adv. Math. 213 (2007), pp. 621–643], while the second is constructed by Holanda [ Some aspects of local cohomology theory , PhD Thesis, Universidade Federal da Para’iba, 2022, Chapter 2] and could also be deduced from results by Godement [ Topologie alg’ebrique et th’eorie des faisceaux (French), Publications de l’Institut de Math’ematique de l’Universit’e de Strasbourg, XIII, Hermann, Paris, 1973]. We present results on the cohomology of multiple complexes that enables to deduce both from two related constructions on multiple complexes. A key ingredient is a fact that seems not to have been noticed before: cohomology with respect to a product of ideals is the one of a subcomplex of the Čech complex computing cohomology with respect to the sum of the given ideals; this provides a much shorter complex to compute cohomology with respect to product of ideals.