A criterion for regularity of local rings

Tom Bridgeland; Srikanth Iyengar

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

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

Comptes Rendus. Mathématique, Volume 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 :

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/

Bibliographie
Cité par

[1] T. Bridgeland Flops and derived categories, Invent. Math., Volume 147 (2002), pp. 613-632

[2] T. Bridgeland; A. King; M. Reid The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc., Volume 14 (2001), pp. 535-554

[3] T. Bridgeland; A. Maciocia Fourier–Mukai transforms for K3 and elliptic fibrations, J. Alg. Geom., Volume 11 (2002), pp. 629-657

[4] W. Bruns; J. Herzog Cohen–Macaulay Rings, Cambridge Stud. Adv. Math., vol. 39, Cambridge Univ. Press, 1998

[5] E.G. Evans; P. Griffith Syzygies, London Math. Soc. Lecture Note Ser., vol. 106, Cambridge Univ. Press, 1985

[6] M. Hochster Topics in the Homological Theory of Modules Over Commutative Rings, CBMS Reg. Conf. Ser. Math., vol. 24, Amer. Math. Soc., 1975

[7] M. Hochster Big Cohen–Macaulay algebras in dimension three via Heitmann's theorem, J. Algebra, Volume 254 (2002), pp. 395-408

[8] S. Iyengar Depth for complexes, and intersection theorems, Math. Z., Volume 230 (1999), pp. 545-567

[9] H. Matsumura Commutative Ring Theory, Cambridge Stud. Adv. Math., vol. 8, Cambridge Univ. Press, Cambridge, 1986

[10] P. Roberts Le théorème d'intersection, C. R. Acad. Sci. Paris, Ser. I, Volume 304 (1987), pp. 177-180

[11] J.-P. Serre Local Algebra, Springer, 2000

[12] H. Schoutens On the vanishing of Tor of the absolute integral closure, J. Algebra, Volume 275 (2004), pp. 567-574

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