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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.

MSC:
18D50Operads
05E25Group actions on posets, etc. (MSC2000)
18G35Chain complexes (homological algebra)
55U10Semisimplicial complexes