Homology of multiple complexes and Mayer-Vietoris spectral sequences
Abstract
Similarities are noted in two Mayer-Vietoris spectral sequences that generalize to any number of ideals in the Mayer-Vietoris exact sequence in local cohomology for two ideals. One has as first terms Cech 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 Cech 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 in Lyu, while the second is constructed in [Chapter 2]Hol and could also be deduced from results in God. We present results on the cohomology of multiple complexes that enables us 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 Cech complex computing cohomology with respect to the sum of the given ideals; this provides a much shorter complex to compute cohomology with respect to the product of ideals.
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