When does the F-signature exist?
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 2, pp. 195-201.

We show that the F-signature of an F-finite local ring R of characteristic p>0 exists when R is either the localization of an N-graded ring at its irrelevant ideal or Q-Gorenstein on its punctured spectrum. This extends results by Huneke, Leuschke, Yao and Singh and proves the existence of the F-signature in the cases where weak F-regularity is known to be equivalent to strong F-regularity.

Nous prouvons dans cet article l’existence de la F-signature d’un anneau local F-fini R, de caractéristique positive p, quand R est la localisation à l’unique idéal homogène maximal d’un anneau N-gradué ou quand R est Q-Gorenstein sur son spectre épointé. Ceci généralise les résultats de Huneke, Leuschke, Yao et Singh et prouve l’existence de la F-signature dans les cas où faible et forte F-régularité sont équivalentes.

DOI: 10.5802/afst.1118
Aberbach, Ian M. 1; Enescu, Florian 2

1 Department of Mathematics, University of Missouri, Columbia, MO 65211.
2 Department of Mathematics and Statistics, Georgia State University, Atlanta, 30303 and The Institute of Mathematics of the Romanian Academy (Romania).
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Aberbach, Ian M.; Enescu, Florian. When does the $F$-signature exist?. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 2, pp. 195-201. doi : 10.5802/afst.1118. http://archive.numdam.org/articles/10.5802/afst.1118/

[1] Aberbach, I. M. Some conditions for the equivalence of weak and strong F-regularity, Comm. Alg., Volume 30 (2002), pp. 1635-1651 | MR | Zbl

[2] Aberbach, I. M; Enescu, F. The structure of F-pure rings, Math. Zeit. (to appear) | MR | Zbl

[3] Aberbach, I. M.; Leuschke, G. The F-signature and strong F-regularity, Math. Res. Lett., Volume 10 (2003), pp. 51-56 | MR | Zbl

[4] Bruns, W.; Herzog, J. Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[5] Hochster, M. Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc., Volume 231 (1977) no. 2, pp. 463-488 | MR | Zbl

[6] Hochster, M.; Huneke, C. Tight closure and strong F-regularity, Mémoires de la Soc. Math. France (1989) no. 38, pp. 119-133 | Numdam | MR | Zbl

[7] Huneke, C.; Leuschke, G. Two theorems about maximal Cohen-Macaulay modules, Math. Ann., Volume 324 (2002), pp. 391-404 | MR | Zbl

[8] Lyubeznik, G.; Smith, K.E. Strong and weak F-regularity are equivalent for graded rings, Amer. J. Math., Volume 121 (1999), pp. 1279-1290 | MR | Zbl

[9] Singh, A.K. The F-signature of an affine semigroup ring, J. Pure Appl. Algebra, Volume 196 (2005), pp. 313-321 | MR | Zbl

[10] Smith, K.E.; Van den Bergh, M. Simplicity of rings of differential operators in prime characteristic, Proc. London. Math. Soc., Volume (3) 75 (1997) no. 1, pp. 32-62 | MR | Zbl

[11] Yao, Y. Modules with finite F-representation type, Jour. London Math. Soc. (to appear) | MR | Zbl

[12] Yao, Y. Observations on the F-signature of local rings of characteristic p>0 (2003) (preprint) | Zbl

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