Koszul duality of operads and homology of partition posets

Koszul duality of operads
and
homology of partition posets
Benoit Fresse
Abstract. Weconsiderpartitionsofasetwith r elementsorderedbyrefine-
¯ment. Weconsiderthesimplicialcomplex K(r)formedbychainsofpartitions
which starts at the smallest element and ends at the largest element of the
¯partitionposet. Aclassicaltheoremassertsthat K(r)isequivalenttoawedge
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−1th homology modules of K(r), where r = 1,2,...,
are dualto the Lie representationofthe symmetricgroups. In thisarticle, we
would like to point out that this theorem occurs a by-product of the theory
of 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.
Contents
Prologue and introduction
0. Conventions
1. Composition products and operad structures
2. Chain complexes of modules over an operad
3. The reduced bar construction
4. Bar constructions with coefficients
5. Koszul duality for operads
6. Epilogue: partition posets
References
Glossary and notation index
Prologue
We consider the set of partitions ...