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

@article{CRMATH_2006__342_10_723_0,
     author = {Tom Bridgeland and Srikanth Iyengar},
     title = {A criterion for regularity of local rings},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {723--726},
     publisher = {Elsevier},
     volume = {342},
     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.03.019},
     language = {en},
}

TY  - JOUR
AU  - Tom Bridgeland
AU  - Srikanth Iyengar
TI  - A criterion for regularity of local rings
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 723
EP  - 726
VL  - 342
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2006.03.019
LA  - en
ID  - CRMATH_2006__342_10_723_0
ER  -

%0 Journal Article
%A Tom Bridgeland
%A Srikanth Iyengar
%T A criterion for regularity of local rings
%J Comptes Rendus. Mathématique
%D 2006
%P 723-726
%V 342
%N 10
%I Elsevier
%R 10.1016/j.crma.2006.03.019
%G en
%F CRMATH_2006__342_10_723_0

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

Nicolas Addington; Daniel Bragg; Alexander Petrov Hodge numbers are not derived invariants in positive characteristic, Mathematische Annalen, Volume 387 (2023) no. 1-2, pp. 847-878 | DOI:10.1007/s00208-022-02474-w | Zbl:1532.14036
Christopher D. Hacon; Daniel Huybrechts; Richard P. W. Thomas; Chenyang Xu Algebraic geometry: moduli spaces, birational geometry and derived aspects. Abstracts from the workshop held July 10–16, 2022, Oberwolfach Rep. 19, No. 3, 1805-1864, 2022 | DOI:10.4171/owr/2022/32 | Zbl:1520.00027
Mohsen Gheibi; David A. Jorgensen; Ryo Takahashi Quasi-projective dimension, Pacific Journal of Mathematics, Volume 312 (2021) no. 1, pp. 113-147 | DOI:10.2140/pjm.2021.312.113 | Zbl:1469.13033
Yves André The direct factor conjecture, Publications Mathématiques, Volume 127 (2018), pp. 71-93 | DOI:10.1007/s10240-017-0097-9 | Zbl:1419.13029
Michael Groechenig Hilbert Schemes as Moduli of Higgs Bundles and Local Systems, International Mathematics Research Notices, Volume 2014 (2014) no. 23, p. 6523 | DOI:10.1093/imrn/rnt167
Jinjia Li Characterizations of regular local rings in positive characteristics, Proceedings of the American Mathematical Society, Volume 136 (2008) no. 5, pp. 1553-1558 | DOI:10.1090/s0002-9939-07-09158-7 | Zbl:1135.13002

Cité par 6 documents. Sources : Crossref, zbMATH

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: