# 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

## Access Full Article

top## Abstract

top## How to cite

topAberbach, 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 $p >0$ 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 $p >0$ 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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.