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

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.

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.

DOI : 10.5802/afst.1118
Ian M. Aberbach 1 ; Florian Enescu 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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     title = {When does the $F$-signature exist?},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Ian M. Aberbach; Florian Enescu. When does the $F$-signature exist?. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 2, pp. 195-201. doi : 10.5802/afst.1118. https://afst.centre-mersenne.org/articles/10.5802/afst.1118/

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

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

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

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

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

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

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

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

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

[10] K.E. Smith; M. Van den Bergh 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] Y. Yao Modules with finite F-representation type, Jour. London Math. Soc. (to appear) | MR | Zbl

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

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