A criterion for regularity of local rings

Bridgeland, Tom; Iyengar, Srikanth

doi:10.1016/j.crma.2006.03.019

Algebra/Homological Algebra

A criterion for regularity of local rings
[Une critère pour la régularité des anneaux locaux]

Présenté par : Jean-Pierre Serre

Bridgeland, Tom ^1,² ; Iyengar, Srikanth ^1,²

¹ Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK
² Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA

Comptes Rendus. Mathématique, Tome 342 (2006) no. 10, pp. 723-726.

Résumé
Abstract

On démontre qu'un anneau local noethérien commutatif A contenant un corps est régulier s'il existe un complexe M de A-modules libres avec les propriétés suivantes : $M_{i} = 0$ pour $i \notin [0, \dim A]$ ; l'homologie de M est de longueur finie ; $H_{0} (M)$ contient le corps résiduel de A en tant que facteur direct. Ce résultat est une composante essentielle dans les démonstrations de la correspondance de McKay en dimension 3 et du fait que les flops de dimension trois induisent des équivalences de catégories dérivées.

It is proved that a noetherian commutative local ring A containing a field is regular if there is a complex M of free A-modules with the following properties: $M_{i} = 0$ for $i \notin [0, \dim A]$ ; the homology of M has finite length; $H_{0} (M)$ contains the residue field of A as a direct summand. This result is an essential component in the proofs of the McKay correspondence in dimension 3 and of the statement that threefold flops induce equivalences of derived categories.

Reçu le : 2005-12-06
Accepté le : 2006-03-14
Publié le : 2006-04-18

DOI : 10.1016/j.crma.2006.03.019

Affiliations des auteurs :

Bridgeland, Tom ^{1, 2} ; Iyengar, Srikanth ^{1, 2}

¹ Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK
² Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA

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Bridgeland, Tom; Iyengar, Srikanth. A criterion for regularity of local rings. Comptes Rendus. Mathématique, Tome 342 (2006) no. 10, pp. 723-726. doi : 10.1016/j.crma.2006.03.019. http://www.numdam.org/articles/10.1016/j.crma.2006.03.019/

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