Koszul complexes and relative homological algebra of functors over posets

In relative homological algebra of vector space valued functors indexed by a poset, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in relative minimal resolutions. In this article we show that, under certain conditions, grading the chosen family of functors by an upper semilattice guarantees the existence of relative minimal resolutions and the uniqueness of direct sum decompositions in these resolutions. These conditions are necessary for defining relative Betti diagrams and computing these diagrams using Koszul complexes.

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