Transactions of the American Mathematical Society

Author(s): Hans Schoutens
Journal: Trans. Amer. Math. Soc. 360 (2008), 609-627.
MSC (2000): Primary 14B05, 13H10, 03C20
Posted: September 21, 2007
Retrieve article in: PDF DVI PostScript

Abstract: We prove a generalization conjectured by Aschenbrenner and Schoutens (2003) of the Hochster-Roberts-Boutot-Kawamata Theorem: let $R\to S$ be a pure homomorphism of equicharacteristic zero Noetherian local rings. If

is regular, then

is pseudo-rational, and if

is moreover $\mathbb{Q}$ -Gorenstein, then it is pseudo-log-terminal.

1.: M. Aschenbrenner and H. Schoutens, Lefschetz extensions, tight closure and big Cohen-Macaulay algebras, Israel Journal of Mathematics (2007), to appear.
2.: J.-F. Boutot, Singularités rationelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65-68. MR 0877006 (88a:14005)
3.: W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993.MR 1251956 (95h:13020)
4.: J. Eagon and M. Hochster, -sequences and indeterminates, Quart. J. Math. Oxford Ser. 25 (1974), 61-71.MR 0337934 (49:2703)
5.: P. Eklof, Ultraproducts for algebraists, Handbook of Mathematical Logic, North-Holland Publishing, 1977, pp. 105-137.MR 0491125 (58:10395)
6.: N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996.MR 1646049 (99h:13005)
7.: R. Hartshorne, Local cohomology, Lect. Notes in Math., vol. 41, Springer-Verlag, 1967.MR 0224620 (37:219)
8.: -, Algebraic geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
9.: H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964), 109-326. MR 0199184 (33:7333)
10.: M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488. MR 0463152 (57:3111)
11.: -, The tight integral closure of a set of ideals, J. Algebra 230 (2000), no. 1, 184-203.MR 1774763 (2002f:13009)
12.: M. Hochster and C. Huneke, Applications of the existence of big Cohen-Macaulay algebras, Adv. in Math. 113 (1995), 45-117. MR 1332808 (96d:13014)
13.: M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. in Math. 13 (1974), 115-175.MR 0347810 (50:311)
14.: W. Hodges, Model theory, Cambridge University Press, Cambridge, 1993. MR 1221741 (94e:03002)
15.: C. Huneke, Tight closure and its applications, CBMS Regional Conf. Ser. in Math, vol. 88, Amer. Math. Soc., 1996.MR 1377268 (96m:13001)
16.: Y. Kawamata, The cone of curves of algebraic varieties, Ann. of Math. 119 (1984), 603-633.MR 0744865 (86c:14013b)
17.: J. Kollár and S. Mori, Birational geometry and algebraic varieties, Cambridge University Press, Cambridge, 1998.MR 1658959 (2000b:14018)
18.: J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97-116.MR 0600418 (82f:14004)
19.: H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.MR 0879273 (88h:13001)
20.: J. Milne, Etale cohomology, 33, Princeton Math., 1980. MR 0559531 (81j:14002)
21.: H. Schoutens, Existentially closed models of the theory of Artinian local rings, J. Symbolic Logic 64 (1999), 825-845. MR 1777790 (2001g:03068)
22.: -, Lefschetz principle applied to symbolic powers, J. Algebra Appl. 2 (2003), 177-187.MR 1980407 (2004c:13040)
23.: -, Non-standard tight closure for affine $\mathbb{C}$ -algebras, Manuscripta Math. 111 (2003), 379-412.MR 1993501 (2004m:13019)
24.: -, A non-standard proof of the Briançon-Skoda theorem, Proc. Amer. Math. Soc. 131 (2003), 103-112.MR 1929029 (2003i:13004)
25.: -, Projective dimension and the singular locus, Comm. Algebra 31 (2003), 217-239.MR 1969220 (2005e:13020)
26.: -, Canonical big Cohen-Macaulay algebras and rational singularities, Illinois J. Math. 48 (2004), 131-150. MR 2048219 (2005e:13012)
27.: -, Log-terminal singularities and vanishing theorems via non-standard tight closure, J. Alg. Geom. 14 (2005), 357-390. MR 2123234 (2006e:13005)
28.: K. Smith, F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159-180.MR 1428062 (97k:13004)
29.: -, Vanishing, singularities and effective bounds via prime characteristic local algebra, Algebraic geometry--Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 289-325.MR 1492526 (99a:14026)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14B05, 13H10, 03C20

Hans Schoutens
Affiliation: Department of Mathematics, City University of New York, 365 Fifth Avenue, New York, New York 10016
Email: hschoutens@citytech.cuny.edu

DOI: 10.1090/S0002-9947-07-04134-7
PII: S 0002-9947(07)04134-7
Keywords: Tight closure, non-standard Frobenius, rational singularities, Boutot's Theorem, log-terminal singularities
Received by editor(s): July 22, 2005
Posted: September 21, 2007
Additional Notes: The author was partially supported by a grant from the National Science Foundation and a PSC-CUNY grant.
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS