Documents

102 pages

Le téléchargement nécessite un accès à la bibliothèque YouScribe

__
Tout savoir sur nos offres
__

Description

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 ...

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 ...

Sujets

Informations

Publié par | Ernoi |

Nombre de visites sur la page | 105 |

Langue | English |

Signaler un problème