Un théorème d'annulation en cohomologie de MacLane

Gaudens, Gerald; Schwartz, Lionel

doi:10.1016/j.crma.2005.06.008

Topologie/Algèbre

Un théorème d'annulation en cohomologie de MacLane
[A vanishing theorem in MacLane cohomology]

Presented by: Christophe Soulé

Gaudens, Gerald ¹; Schwartz, Lionel ²

¹ Mathematisches Institut, Universität Bonn, Beringstraße 1, 53115 Bonn, Allemagne
² Laboratoire analyse, géométrie et applications, université Paris 13, 99, avenue J.-B. Clément, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 119-122.

Abstract
Résumé

The aim of this Note is to give a vanishing theorem in MacLane cohomology that generalizes a theorem of the second author in the appendix of Powell [G. Powell, The Artinian conjecture for $I \otimes I$ , J. Pure Appl. Algebra 128 (1998) 291–310]. Let $F$ be the category of functors from the category of finite dimensional vector spaces over $F_{2}$ to the one of all $F_{2}$ -vector spaces. Let F be a polynomial functor such that $F {0} = {0}$ , and let K and L be functors taking values in Boolean algebras, F, K, L taking finite dimensional values. Then

{Ext}_{F}^{*} (K, L \otimes F) = {0} .

Le but de cette Note est de donner un résultat d'annulation en cohomologie de MacLane généralisant celui donné dans l'appendice de Powell [G. Powell, The Artinian conjecture for $I \otimes I$ , J. Pure Appl. Algebra 128 (1998) 291–310] par le second auteur. Notons $F$ la catégorie des foncteurs depuis la catégorie des espaces vectoriels de dimension finie sur $F_{2}$ vers celle de tous les $F_{2}$ -espaces vectoriels. Soit F un foncteur polynomial tel que $F {0} = {0}$ , et soient K et L deux foncteurs en algèbres de Boole, F, K, L prenant des valeurs de dimension finie. Alors on a

{Ext}_{F}^{*} (K, L \otimes F) = {0} .

Received: 2005-02-16
Accepted: 2005-05-31
Published online: 2005-07-11

DOI: 10.1016/j.crma.2005.06.008

Author's affiliations:

Gaudens, Gerald ¹; Schwartz, Lionel ²

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     title = {Un th\'eor\`eme d'annulation en cohomologie de {MacLane}},
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Gaudens, Gerald; Schwartz, Lionel. Un théorème d'annulation en cohomologie de MacLane. Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 119-122. doi : 10.1016/j.crma.2005.06.008. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.008/

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