Fundamental groups of F-regular singularities via F-signature
[Groupes fondamentaux des singularités F-régulières via F-signature]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 993-1016.

Nous montrons que le groupe fondamental local étale d'une singularité F-régulière est fini. Ce théorème représente l'analogue en caractéristique p des résultats obtenus par Xu et Greb-Kebekus-Peternell pour les singularités KLT. Nous montrons que le cardinal du groupe fundamental est majoré par l'inverse de la F-signature de la singularité. En particulier, notre résultat principal est effectif. Pour cela, nous établissons des nouvelles formules de transformation de la F-signature par rapport aux extensions étale en codimension un. Nous obtenons également un nouveau critère de pureté du lieu de branchement sur les anneauux à singularités faibles. Ceci s'applique en particulier aux anneaux dont la F-signature est supérieure à 1/2.

We prove that the local étale fundamental group of a strongly F-regular singularity is finite. These results are analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic 0. Our result is effective, we show that the reciprocal of the F-signature of the singularity gives a bound on the size of this fundamental group. To prove these results we develop new transformation rules for the F-signature under finite étale-in-codimension-one extensions. We also obtain purity of the branch locus over rings with mild singularities (particularly if the F-signature is >1/2).

DOI : 10.24033/asens.2370
Classification : 14F18, 14F35, 13A35, 14B05
Keywords: Étale fundamental group, $F$-regular singularities, $F$-signature.
Mot clés : Groupe fondamental étale, singularités $F$-régulières, $F$-signature. 
@article{ASENS_2018__51_4_993_0,
     author = {Carvajal-Rojas, Javier and Schwede, Karl and Tucker, Kevin},
     title = {Fundamental groups of~$F$-regular singularities via $F$-signature},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {993--1016},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {4},
     year = {2018},
     doi = {10.24033/asens.2370},
     mrnumber = {3861567},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2370/}
}
TY  - JOUR
AU  - Carvajal-Rojas, Javier
AU  - Schwede, Karl
AU  - Tucker, Kevin
TI  - Fundamental groups of $F$-regular singularities via $F$-signature
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2018
SP  - 993
EP  - 1016
VL  - 51
IS  - 4
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2370/
DO  - 10.24033/asens.2370
LA  - en
ID  - ASENS_2018__51_4_993_0
ER  - 
%0 Journal Article
%A Carvajal-Rojas, Javier
%A Schwede, Karl
%A Tucker, Kevin
%T Fundamental groups of $F$-regular singularities via $F$-signature
%J Annales scientifiques de l'École Normale Supérieure
%D 2018
%P 993-1016
%V 51
%N 4
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2370/
%R 10.24033/asens.2370
%G en
%F ASENS_2018__51_4_993_0
Carvajal-Rojas, Javier; Schwede, Karl; Tucker, Kevin. Fundamental groups of $F$-regular singularities via $F$-signature. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 993-1016. doi : 10.24033/asens.2370. http://www.numdam.org/articles/10.24033/asens.2370/

Altman, A.; Kleiman, S., Lecture Notes in Math., 146, Springer, Berlin-New York, 1970, 185 pages | MR | Zbl

Aberbach, I. M.; Leuschke, G. J. The F-signature and strong F-regularity, Math. Res. Lett., Volume 10 (2003), pp. 51-56 (ISSN: 1073-2780) | DOI | MR | Zbl

Artin, M. Coverings of the rational double points in characteristic p , Complex analysis and algebraic geometry (1977), pp. 11-22 | DOI | MR | Zbl

Blickle, M. Test ideals via algebras of p-e-linear maps, J. Algebraic Geom., Volume 22 (2013), pp. 49-83 (ISSN: 1056-3911) | DOI | MR | Zbl

Blickle, M.; Schwede, K.; Tucker, K. F-signature of pairs and the asymptotic behavior of Frobenius splittings, Adv. Math., Volume 231 (2012), pp. 3232-3258 (ISSN: 0001-8708) | DOI | MR | Zbl

Chambert-Loir, A. Points rationnels et groupes fondamentaux: applications de la cohomologie p-adique (d'après P. Berthelot, T. Ekedahl, H. Esnault, etc.), Astérisque, Volume 294 (2004), pp. 125-146 (ISSN: 0303-1179) | Numdam | MR | Zbl

Cutkosky, S. D.; Srinivasan, H. Local fundamental groups of surface singularities in characteristic p , Comment. Math. Helv., Volume 68 (1993), pp. 319-332 (ISSN: 0010-2571) | DOI | MR | Zbl

Cutkosky, S. D. Purity of the branch locus and Lefschetz theorems, Compos. math., Volume 96 (1995), pp. 173-195 (ISSN: 0010-437X) | Numdam | MR | Zbl

Flenner, H. Reine lokale Ringe der Dimension zwei, Math. Ann., Volume 216 (1975), pp. 253-263 (ISSN: 0025-5831) | DOI | MR | Zbl

Greb, D.; Kebekus, S.; Peternell, T. Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J., Volume 165 (2016), pp. 1965-2004 (ISSN: 0012-7094) | DOI | MR

Gongyo, Y.; Li, Z.; Patakfalvi, Z.; Schwede, K.; Tanaka, H.; Zong, R. On rational connectedness of globally F-regular threefolds, Adv. Math., Volume 280 (2015), pp. 47-78 (ISSN: 0001-8708) | DOI | MR

Grothendieck, A.; Murre, J. P., Lecture Notes in Math., 208, Springer, Berlin-New York, 1971, 133 pages | MR | Zbl

Grothendieck, A., Documents Mathématiques (Paris), 4, Soc. Math. France, Paris, 2005, 208 pages (ISBN: 2-85629-169-4) | MR

Grothendieck, A., Séminaire de Géométrie Algébrique, 1960/61, Institut des Hautes Études Scientifiques, Paris, 1963 | MR

Hara, N. A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math., Volume 120 (1998), pp. 981-996 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5hara.pdf (ISSN: 0002-9327) | DOI | MR | Zbl

Huneke, C.; Leuschke, G. J. Two theorems about maximal Cohen-Macaulay modules, Math. Ann., Volume 324 (2002), pp. 391-404 (ISSN: 0025-5831) | DOI | MR | Zbl

Hara, N.; Watanabe, K.-I. F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom., Volume 11 (2002), pp. 363-392 (ISSN: 1056-3911) | DOI | MR | Zbl

Hochster, M.; Yao, Y. The F-rational signature and drops in the Hilbert-Kunz multiplicity (2009) (preprint http://www.math.lsa.umich.edu/~hochster/r-sig.pdf ) | MR

Kollár, J. New examples of terminal and log canonical singularities (preprint arXiv:1107.2864 )

Kollár, J., M. B. Porter Lectures, Princeton Univ. Press, Princeton, NJ, 1995, 201 pages (ISBN: 0-691-04381-7) | DOI | MR | Zbl

Kerz, M.; Schmidt, A. On different notions of tameness in arithmetic geometry, Math. Ann., Volume 346 (2010), pp. 641-668 (ISSN: 0025-5831) | DOI | MR | Zbl

Kunz, E. Remark on the purity of the branch locus, Proc. Amer. Math. Soc., Volume 20 (1969), pp. 378-380 (ISSN: 0002-9939) | DOI | MR | Zbl

Milne, J. S., Princeton Mathematical Series, 33, Princeton Univ. Press, Princeton, N.J., 1980, 323 pages (ISBN: 0-691-08238-3) | MR | Zbl

Monsky, P. The Hilbert-Kunz function, Math. Ann., Volume 263 (1983), pp. 43-49 (ISSN: 0025-5831) | DOI | MR | Zbl

Mumford, D. The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. IHÉS, Volume 9 (1961), pp. 5-22 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Murre, J.-P., Tata Institute of Fundamental Research Lectures on Mathematics, 40, 1967 | MR | Zbl

Nagata, M. Remarks on a paper of Zariski on the purity of branch loci, Proc. Nat. Acad. Sci. U.S.A., Volume 44 (1958), pp. 796-799 (ISSN: 0027-8424) | DOI | MR | Zbl

Stacks Project Authors The Stacks Project (2016)

Schwede, K. F-adjunction, Algebra Number Theory, Volume 3 (2009), pp. 907-950 (ISSN: 1937-0652) | DOI | MR | Zbl

Schwede, K. Test ideals in non-ℚ-Gorenstein rings, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 5925-5941 (ISSN: 0002-9947) | DOI | MR | Zbl

Speyer, D. E. Frobenius split subvarieties pull back in almost all characteristics (preprint arXiv:1611.06290 ) | MR

Schwede, K.; Smith, K. E. Globally F-regular and log Fano varieties, Adv. Math., Volume 224 (2010), pp. 863-894 (ISSN: 0001-8708) | DOI | MR | Zbl

Schwede, K.; Tucker, K. On the behavior of test ideals under finite morphisms, J. Algebraic Geom., Volume 23 (2014), pp. 399-443 (ISSN: 1056-3911) | DOI | MR | Zbl

Smith, K. E.; Van den Bergh, M. Simplicity of rings of differential operators in prime characteristic, Proc. London Math. Soc., Volume 75 (1997), pp. 32-62 (ISSN: 0024-6115) | DOI | MR | Zbl

Tucker, K. F-signature exists, Invent. math., Volume 190 (2012), pp. 743-765 (ISSN: 0020-9910) | DOI | MR | Zbl

Tomari, M.; Watanabe, K. Normal Zr-graded rings and normal cyclic covers, Manuscripta Math., Volume 76 (1992), pp. 325-340 (ISSN: 0025-2611) | DOI | MR | Zbl

Tian, Z.; Xu, C. Finiteness of fundamental groups, Compos. Math., Volume 153 (2017), pp. 257-273 (ISSN: 0010-437X) | DOI | MR

Vraciu, A. A new version of 𝔞-tight closure, Nagoya Math. J., Volume 192 (2008), pp. 1-25 (ISSN: 0027-7630) | DOI | MR | Zbl

Watanabe, K. F-regular and F-pure normal graded rings, J. Pure Appl. Algebra, Volume 71 (1991), pp. 341-350 (ISSN: 0022-4049) | DOI | MR | Zbl

Watanabe, K.-i.; Yoshida, K.-i. Hilbert-Kunz multiplicity of three-dimensional local rings, Nagoya Math. J., Volume 177 (2005), pp. 47-75 (ISSN: 0027-7630) | DOI | MR | Zbl

Xu, C. Finiteness of algebraic fundamental groups, Compos. Math., Volume 150 (2014), pp. 409-414 (ISSN: 0010-437X) | DOI | MR | Zbl

Yao, Y. Observations on the F-signature of local rings of characteristic p , J. Algebra, Volume 299 (2006), pp. 198-218 (ISSN: 0021-8693) | DOI | MR | Zbl

Zariski, O. On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A., Volume 44 (1958), pp. 791-796 (ISSN: 0027-8424) | DOI | MR | Zbl

Cité par Sources :