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Homology of generalized partition posets. (English) Zbl 1109.18002
For an operad (it is a monoid in a monoidal category $({\cal S}\text{-Mod},\circ ,1)$ of ${\cal S}$-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 05E25 Group actions on posets, etc. (MSC2000) 18G35 Chain complexes (homological algebra) 55U10 Semisimplicial complexes
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##### References:
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