## Koszul duality of operads and homology of partition posets

**by Benoit Fresse**

**Published ***in* "Homotopy theory and its applications
(Evanston, 2002)", Contemp. Math. 346 (2004), 115-215.
**Abstract:**
We consider partitions of a set with $r$ elements ordered by
refinement.
We consider the simplicial complex $\bar{K}(r)$ formed by chains of
partitions
which starts at the smallest element and ends
at the largest element of the partition poset.
A classical theorem asserts that $\bar{K}(r)$ is equivalent to a wedge
of $r-1$-dimensional spheres.
In addition, the poset of partitions is equipped with a natural
action of the symmetric group in $r$ letters.
Consequently, the associated homology modules are representations of
the symmetric groups.
One observes that the $r-1$th homology modules of $\bar{K}(r)$,
where $r = 1,2,...$, are dual to the Lie representation of the
symmetric groups. In this article, we would like to point out that this
theorem occurs a by-product
of the theory of \emph{Koszul operads}.
For that purpose, we improve results of V. Ginzburg and M. Kapranov in
several directions.
More particularly, we extend the Koszul duality of operads to operads
defined over a field of positive characteristic (or over a ring).
In addition, we obtain more conceptual proofs of theorems of V.
Ginzburg and M. Kapranov.

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