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Comptes Rendus Mathématique
Volume 353, n° 8
pages 689-693 (août 2015)
Doi : 10.1016/j.crma.2015.05.008
Received : 20 Mars 2015 ;  accepted : 19 May 2015
Quasi-hereditary property of double Burnside algebras
Propriété quasi-héréditaire des algèbres de Burnside doubles
 

Baptiste Rognerud
 EPFL/SB/MATHGEOM/CTG, Station 8, CH-1015 Lausanne, Switzerland 

Abstract

In this short note, we investigate some consequences of the vanishing of simple biset functors. As a corollary, if there is no non-trivial vanishing of simple biset functors (e.g., if the group G is commutative), then we show that   is a quasi-hereditary algebra in characteristic zero. In general, this is not true without the non-vanishing condition, as over a field of characteristic zero, the double Burnside algebra of the alternating group of degree 5 has infinite global dimension.

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Résumé

Dans cette note, on s'intéresse à quelques conséquences du phénomène dit de disparition des foncteurs à bi-ensembles simples. On démontre que, dans le cas où il n'y a pas de disparitions non triviales de foncteurs simples (par exemple, si le groupe est commutatif), alors l'algèbre de Burnside double en caractéristique zéro est quasi-héréditaire. Sans l'hypothèse de non-disparitions triviales, ce résultat est en général faux. En effet, l'algèbre de Burnside double du groupe alterné de degré 5 en caractéristique zéro est de dimension globale infinie.

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