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Homology of generalized partition posets. (English) Zbl 1109.18002
For an operad (it is a monoid in a monoidal category (𝒮-Mod,,1) of 𝒮-modules) finite partition posets are defined. A finite poset with exactly one minimal and exactly one maximal element such that maximal chains have the same length and its homology is concentrated in top dimension is called Cohen-Macauley. The main result says that an operad is Kozsul if and only if its posets are Cohen-Macauley. This characterization allows us to compute homology of Cohen-Macauley posets (the homology groups are isomorphic to the Kozsul dual cooperad) and provides combinatorial methods for recognization of Kozsul operads. Many examples illustrating these results are presented.

05E25Group actions on posets, etc. (MSC2000)
18G35Chain complexes (homological algebra)
55U10Semisimplicial complexes