Comptes Rendus
no. 10
Algebra/Homological Algebra
A criterion for regularity of local rings
[Une critère pour la régularité des anneaux locaux]

Présenté par : Jean-Pierre Serre

Tom Bridgeland1, 2 ; Srikanth Iyengar1, 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, Volume 342 (2006) no. 10, pp. 723-726.

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 : Mi=0 pour i∉[0,dimA] ; l'homologie de M est de longueur finie ; H0(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: Mi=0 for i∉[0,dimA]; the homology of M has finite length; H0(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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.03.019
Tom Bridgeland 1, 2 ; Srikanth Iyengar 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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Tom Bridgeland; Srikanth Iyengar. A criterion for regularity of local rings. Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 723-726. doi : 10.1016/j.crma.2006.03.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.019/

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