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Homology of generalized partition posets. (English) Zbl 1109.18002

For an operad (it is a monoid in a monoidal category \(({\mathcal S}\text{-Mod},\circ ,1)\) of \({\mathcal S}\)-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:

18D50 Operads (MSC2010)
05E25 Group actions on posets, etc. (MSC2000)
18G35 Chain complexes (category-theoretic aspects), dg categories
55U10 Simplicial sets and complexes in algebraic topology
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