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When does the F -signature exist?

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).

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 2, page 195-201
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Aberbach, Ian M., and Enescu, Florian. "When does the $F$-signature exist?." Annales de la faculté des sciences de Toulouse Mathématiques 15.2 (2006): 195-201. <http://eudml.org/doc/10043>.

@article{Aberbach2006,
abstract = {We show that the $F$-signature of an $F$-finite local ring $R$ of characteristic $p &gt;0$ exists when $R$ is either the localization of an $\mathbf\{N\}$-graded ring at its irrelevant ideal or $\mathbf\{Q\}$-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.},
affiliation = {Department of Mathematics, University of Missouri, Columbia, MO 65211.; Department of Mathematics and Statistics, Georgia State University, Atlanta, 30303 and The Institute of Mathematics of the Romanian Academy (Romania).},
author = {Aberbach, Ian M., Enescu, Florian},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {F-signature; F-finite ring},
language = {eng},
number = {2},
pages = {195-201},
publisher = {Université Paul Sabatier, Toulouse},
title = {When does the $F$-signature exist?},
url = {http://eudml.org/doc/10043},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Aberbach, Ian M.
AU - Enescu, Florian
TI - When does the $F$-signature exist?
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 2
SP - 195
EP - 201
AB - We show that the $F$-signature of an $F$-finite local ring $R$ of characteristic $p &gt;0$ exists when $R$ is either the localization of an $\mathbf{N}$-graded ring at its irrelevant ideal or $\mathbf{Q}$-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.
LA - eng
KW - F-signature; F-finite ring
UR - http://eudml.org/doc/10043
ER -

References

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  1. I. M. Aberbach, Some conditions for the equivalence of weak and strong F -regularity, Comm. Alg. 30 (2002), 1635-1651 Zbl1070.13005MR1894033
  2. I. M Aberbach, F. Enescu, The structure of F -pure rings, Math. Zeit. Zbl1102.13001MR2180375
  3. I. M. Aberbach, G. Leuschke, The F -signature and strong F -regularity, Math. Res. Lett. 10 (2003), 51-56 Zbl1070.13006MR1960123
  4. W. Bruns, J. Herzog, Cohen-Macaulay Rings, (1993), Cambridge University Press, Cambridge Zbl0788.13005MR1251956
  5. M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488 Zbl0369.13005MR463152
  6. M. Hochster, C. Huneke, Tight closure and strong F -regularity, Mémoires de la Soc. Math. France (1989), 119-133 Zbl0699.13003MR1044348
  7. C. Huneke, G. Leuschke, Two theorems about maximal Cohen-Macaulay modules, Math. Ann. 324 (2002), 391-404 Zbl1007.13005MR1933863
  8. G. Lyubeznik, K.E. Smith, Strong and weak F -regularity are equivalent for graded rings, Amer. J. Math. 121 (1999), 1279-1290 Zbl0970.13003MR1719806
  9. A.K. Singh, The F -signature of an affine semigroup ring, J. Pure Appl. Algebra 196 (2005), 313-321 Zbl1080.13001MR2110527
  10. K.E. Smith, M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic, Proc. London. Math. Soc. (3) 75 (1997), 32-62 Zbl0948.16019MR1444312
  11. Y. Yao, Modules with finite F -representation type, Jour. London Math. Soc. Zbl1108.13004MR2145728
  12. Y. Yao, Observations on the F -signature of local rings of characteristic p &gt; 0 , (2003) 

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