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Similarities are noted in two Mayer-Vietoris spectral sequences thatgeneralize to any number of ideals in the Mayer-Vietoris exact sequence inlocal cohomology for two ideals. One has as first terms \v{C}ech cohomologywith respect to sums of the given ideals and converge to cohomology withrespect to the product of the ideals, the other has as first terms \v{C}echcohomology with respect to products of the given ideals and converge tocohomology with respect to the sum of the ideals. The first one was obtained byLyubeznik in \cite{Lyu}, while the second is constructed in \cite[Chapter2]{Hol} and could also be deduced from results in \cite{God}. We presentresults on the cohomology of multiple complexes that enables us to deduce bothfrom two related constructions on multiple complexes. A key ingredient is afact that seems not to have been noticed before: cohomology with respect to aproduct of ideals is the one of a subcomplex of the \v{C}ech complex computingcohomology with respect to the sum of the given ideals; this provides a muchshorter complex to compute cohomology with respect to the product of ideals.

Homology of multiple complexes and Mayer-Vietoris spectral sequences