$F$-regularity, test elements, and smooth base change
Authors:
Melvin Hochster and Craig Huneke
Journal:
Trans. Amer. Math. Soc. 346 (1994), 1-62
MSC:
Primary 13A35; Secondary 13B99, 13F40
DOI:
https://doi.org/10.1090/S0002-9947-1994-1273534-X
MathSciNet review:
1273534
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Abstract: This paper deals with tight closure theory in positive characteristic. After a good deal of preliminary work in the first five sections, including a treatment of $F$-rationality and a treatment of $F$-regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in $\S 6$ and is then applied in $\S 7$ to prove that both tight closure and $F$-regularity commute with smooth base change under many circumstances (where "smooth" is used to mean flat with geometrically regular fibers). For example, it is shown in $\S 6$ that for a reduced ring $R$ essentially of finite type over an excellent local ring of characteristic $p$, if $c$ is not in any minimal prime of $R$ and ${R_c}$ is regular, then $c$ has a power that is a test element. It is shown in $\S 7$ that if $S$ is a flat $R$-algebra with regular fibers and $R$ is $F$-regular then $S$ is $F$-regular. The general problem of showing that tight closure commutes with smooth base change remains open, but is reduced here to showing that tight closure commutes with localization.
- I. M. Aberbach, Finite phantom projective dimension, Amer. J. Math. 116 (1994), no. 2, 447â477. MR 1269611, DOI https://doi.org/10.2307/2374936
- Ian M. Aberbach, Test elements in excellent rings with an application to the uniform Artin-Rees property, Proc. Amer. Math. Soc. 118 (1993), no. 2, 355â363. MR 1129869, DOI https://doi.org/10.1090/S0002-9939-1993-1129869-9 ---, Tight closure in $F$-rational rings, Nagoya Math. J. (to appear).
- Ian M. Aberbach, Melvin Hochster, and Craig Huneke, Localization of tight closure and modules of finite phantom projective dimension, J. Reine Angew. Math. 434 (1993), 67â114. MR 1195691, DOI https://doi.org/10.1515/crll.1993.434.67
- Jean-François Boutot, SingularitĂ©s rationnelles et quotients par les groupes rĂ©ductifs, Invent. Math. 88 (1987), no. 1, 65â68 (French). MR 877006, DOI https://doi.org/10.1007/BF01405091
- Henri Skoda and JoĂ«l Briançon, Sur la clĂŽture intĂ©grale dâun idĂ©al de germes de fonctions holomorphes en un point de ${\bf C}^{n}$, C. R. Acad. Sci. Paris SĂ©r. A 278 (1974), 949â951 (French). MR 340642
- Sankar P. Dutta, On the canonical element conjecture, Trans. Amer. Math. Soc. 299 (1987), no. 2, 803â811. MR 869233, DOI https://doi.org/10.1090/S0002-9947-1987-0869233-2 A. Grothendieck (rĂ©digĂ©s avec la collaboration de J. DieudonnĂ©), ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique IV: Ătude local des schĂ©mas et des morphismes de schĂ©mas (Seconde partie), Inst. Hautes Etudes Sci. Publ. Math. 24 (1965), 1-231.
- E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323â333. MR 632842, DOI https://doi.org/10.2307/1971296 ---, Syzygies, London Math. Soc. Lecture Note Ser., vol. 106, Cambridge Univ. Press, London and New York, 1985.
- E. Graham Evans Jr. and Phillip A. Griffith, Order ideals, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 213â225. MR 1015519, DOI https://doi.org/10.1007/978-1-4612-3660-3_10
- Richard Fedder and Keiichi Watanabe, A characterization of $F$-regularity in terms of $F$-purity, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 227â245. MR 1015520, DOI https://doi.org/10.1007/978-1-4612-3660-3_11 D. J. Glassbrenner, Invariant rings of group actions, determinantal rings, and tight closure, Thesis, Univ. of Michigan, 1992.
- Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
- M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25â43. MR 349656
- Melvin Hochster, Topics in the homological theory of modules over commutative rings, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24â28, 1974; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24. MR 0371879
- Melvin Hochster, Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors, Conference on Commutative Algebraâ1975 (Queenâs Univ., Kingston, Ont., 1975), Queenâs Univ., Kingston, Ont., 1975, pp. 106â195. Queenâs Papers on Pure and Applied Math., No. 42. MR 0396544
- Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463â488. MR 463152, DOI https://doi.org/10.1090/S0002-9947-1977-0463152-5
- Melvin Hochster, Some applications of the Frobenius in characteristic $0$, Bull. Amer. Math. Soc. 84 (1978), no. 5, 886â912. MR 485848, DOI https://doi.org/10.1090/S0002-9904-1978-14531-5
- Melvin Hochster, Cohen-Macaulay rings and modules, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 291â298. MR 562618
- Melvin Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), no. 2, 503â553. MR 723406, DOI https://doi.org/10.1016/0021-8693%2883%2990092-3
- Melvin Hochster, Solid closure, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 103â172. MR 1266182, DOI https://doi.org/10.1090/conm/159/01508
- Melvin Hochster, Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 173â196. MR 1266183, DOI https://doi.org/10.1090/conm/159/01507
- M. Hochster and John A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020â1058. MR 302643, DOI https://doi.org/10.2307/2373744
- Melvin Hochster and Craig Huneke, Tightly closed ideals, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 45â48. MR 919658, DOI https://doi.org/10.1090/S0273-0979-1988-15592-9
- Melvin Hochster and Craig Huneke, Tight closure, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 305â324. MR 1015524, DOI https://doi.org/10.1007/978-1-4612-3660-3_15 ---, Tight closure and strong $F$-regularity, MĂ©m. Soc. Math. France 38 (1989), 119-133.
- Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31â116. MR 1017784, DOI https://doi.org/10.1090/S0894-0347-1990-1017784-6
- Melvin Hochster and Craig Huneke, Absolute integral closures are big Cohen-Macaulay algebras in characteristic $P$, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 1, 137â143. MR 1056558, DOI https://doi.org/10.1090/S0273-0979-1991-15970-7
- Melvin Hochster and Craig Huneke, Tight closure and elements of small order in integral extensions, J. Pure Appl. Algebra 71 (1991), no. 2-3, 233â247. MR 1117636, DOI https://doi.org/10.1016/0022-4049%2891%2990149-V
- Melvin Hochster and Craig Huneke, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. (2) 135 (1992), no. 1, 53â89. MR 1147957, DOI https://doi.org/10.2307/2946563
- Melvin Hochster and Craig Huneke, Phantom homology, Mem. Amer. Math. Soc. 103 (1993), no. 490, vi+91. MR 1144758, DOI https://doi.org/10.1090/memo/0490
- Melvin Hochster and Craig Huneke, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599â670. MR 1297848 ---, Tight closure in equal characteristic zero (in preparation).
- Melvin Hochster and Craig Huneke, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math. 113 (1995), no. 1, 45â117. MR 1332808, DOI https://doi.org/10.1006/aima.1995.1035
- Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115â175. MR 347810, DOI https://doi.org/10.1016/0001-8708%2874%2990067-X
- Melvin Hochster and Joel L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2, 117â172. MR 417172, DOI https://doi.org/10.1016/0001-8708%2876%2990073-6
- Craig Huneke, An algebraist commuting in Berkeley, Math. Intelligencer 11 (1989), no. 1, 40â52. MR 979023, DOI https://doi.org/10.1007/BF03023775
- Craig Huneke, Absolute integral closure and big Cohen-Macaulay algebras, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 339â349. MR 1159222
- Craig Huneke, Uniform bounds in Noetherian rings, Invent. Math. 107 (1992), no. 1, 203â223. MR 1135470, DOI https://doi.org/10.1007/BF01231887
- George Kempf, The Hochster-Roberts theorem of invariant theory, Michigan Math. J. 26 (1979), no. 1, 19â32. MR 514958
- Ernst Kunz, Characterizations of regular local rings of characteristic $p$, Amer. J. Math. 91 (1969), 772â784. MR 252389, DOI https://doi.org/10.2307/2373351
- Ernst Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), no. 4, 999â1013. MR 432625, DOI https://doi.org/10.2307/2374038
- Joseph Lipman and Avinash Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), no. 2, 199â222. MR 616270
- Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97â116. MR 600418
- Frank Ma, Splitting in integral extensions, Cohen-Macaulay modules and algebras, J. Algebra 116 (1988), no. 1, 176â195. MR 944154, DOI https://doi.org/10.1016/0021-8693%2888%2990200-1
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
- C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications Ă la dĂ©monstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Ătudes Sci. Publ. Math. 42 (1973), 47â119 (French). MR 374130
- Christian Peskine and Lucien Szpiro, Syzygies et multiplicitĂ©s, C. R. Acad. Sci. Paris SĂ©r. A 278 (1974), 1421â1424 (French). MR 349659
- Paul Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. Ăcole Norm. Sup. (4) 9 (1976), no. 1, 103â106. MR 399075
- Paul Roberts, Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture, J. Algebra 66 (1980), no. 1, 220â225. MR 591254, DOI https://doi.org/10.1016/0021-8693%2880%2990121-0
- Paul Roberts, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 127â130. MR 799793, DOI https://doi.org/10.1090/S0273-0979-1985-15394-7
- Paul Roberts, Le thĂ©orĂšme dâintersection, C. R. Acad. Sci. Paris SĂ©r. I Math. 304 (1987), no. 7, 177â180 (French, with English summary). MR 880574
- Paul Roberts, Intersection theorems, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 417â436. MR 1015532, DOI https://doi.org/10.1007/978-1-4612-3660-3_23 J.-P. Serre, AlgĂšbre locale: MultiplicitĂ©s, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965. P. Deligne avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie, et J. L. Verdier, Cohomologie Ă©tale (SĂ©m. GĂ©omĂ©trie AlgĂ©brique du Bois-Marie $4\tfrac {1} {2}$), Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, Heidelberg, and New York, 1977.
- Henri Skoda, Application des techniques $L^{2}$ Ă la thĂ©orie des idĂ©aux dâune algĂšbre de fonctions holomorphes avec poids, Ann. Sci. Ăcole Norm. Sup. (4) 5 (1972), 545â579 (French). MR 333246
- K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41â60. MR 1248078, DOI https://doi.org/10.1007/BF01231753
- K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41â60. MR 1248078, DOI https://doi.org/10.1007/BF01231753 ---, $F$-rational rings have rational singularities, preprint.
- Irena Swanson, Joint reductions, tight closure, and the Briançon-Skoda theorem, J. Algebra 147 (1992), no. 1, 128â136. MR 1154678, DOI https://doi.org/10.1016/0021-8693%2892%2990256-L ---, Tight closure, joint reductions, and mixed multiplicities, Thesis, Purdue Univ., 1992. J. Velez, Openness of the $F$-rational locus, smooth base change, and Kohâs conjecture, Thesis, Univ. of Michigan, 1993. ---, Openness of the $F$-rational locus and smooth base change, J. Algebra (to appear). K.-I. Watanabe, Study of $F$-purity in dimension two, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Vol. II, Kinokuniya, Tokyo, 1988, pp. 791-800.
- Keiichi Watanabe, $F$-regular and $F$-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), no. 2-3, 341â350. MR 1117644, DOI https://doi.org/10.1016/0022-4049%2891%2990157-W
- Kei-ichi Watanabe, Takeshi Ishikawa, Sadao Tachibana, and Kayo Otsuka, On tensor products of Gorenstein rings, J. Math. Kyoto Univ. 9 (1969), 413â423. MR 257062, DOI https://doi.org/10.1215/kjm/1250523903 L. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, Thesis, Univ. of Michigan, 1992.
- Lori J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), no. 3, 721â743. MR 1324179, DOI https://doi.org/10.1006/jabr.1995.1067
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Keywords:
Tight closure,
<IMG WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$F$">-regular ring,
characteristic <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$p$">,
test element,
smooth base change
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American Mathematical Society