𝐹-regularity, test elements, and smooth base change

Hochster, Melvin; Huneke, Craig

doi:10.1090/S0002-9947-1994-1273534-X

-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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 -rationality and a treatment of -regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in $\S6$ and is then applied in $\S7$ to prove that both tight closure and -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 $\S6$ that for a reduced ring essentially of finite type over an excellent local ring of characteristic , if is not in any minimal prime of and ${R_c}$ is regular, then has a power that is a test element. It is shown in $\S7$ that if is a flat -algebra with regular fibers and is -regular then is -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.

[Ab1] I. M. Aberbach, Finite phantom projective dimension, Amer. J. Math. 116 (1994), no. 2, 447–477. MR 1269611, https://doi.org/10.2307/2374936
[Ab2] 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, 355–363. MR 1129869, https://doi.org/10.1090/S0002-9939-1993-1129869-9
[Ab3] -, Tight closure in -rational rings, Nagoya Math. J. (to appear).
[AHH] 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), 67–114. MR 1195691, https://doi.org/10.1515/crll.1993.434.67
[B] Jean-François Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), no. 1, 65–68 (French). MR 877006, https://doi.org/10.1007/BF01405091
[BrS] Henri Skoda and Joël Briançon, Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de 𝐶ⁿ, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949–951 (French). MR 0340642
[Du] Sankar P. Dutta, On the canonical element conjecture, Trans. Amer. Math. Soc. 299 (1987), no. 2, 803–811. MR 869233, https://doi.org/10.1090/S0002-9947-1987-0869233-2
[EGA] A. Grothendieck (rédigés avec la collaboration de J. Dieudonné), Éléments de géométrie algébrique IV: Étude local des schémas et des morphismes de schémas (Seconde partie), Inst. Hautes Etudes Sci. Publ. Math. 24 (1965), 1-231.
[EvG1] E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, https://doi.org/10.2307/1971296
[EvG2] -, Syzygies, London Math. Soc. Lecture Note Ser., vol. 106, Cambridge Univ. Press, London and New York, 1985.
[EvG3] 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. 213–225. MR 1015519, https://doi.org/10.1007/978-1-4612-3660-3_10
[FeW] 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. 227–245. MR 1015520, https://doi.org/10.1007/978-1-4612-3660-3_11
[Gla] D. J. Glassbrenner, Invariant rings of group actions, determinantal rings, and tight closure, Thesis, Univ. of Michigan, 1992.
[GrHa] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. MR 0224620
[Ho1] M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25–43. MR 349656
[Ho2] 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 24–28, 1974; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24. MR 0371879
[Ho3] Melvin Hochster, Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors, Conference on Commutative Algebra–1975 (Queen’s Univ., Kingston, Ont., 1975), Queen’s Univ., Kingston, Ont., 1975, pp. 106–195. Queen’s Papers on Pure and Applied Math., No. 42. MR 0396544
[Ho4] Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463–488. MR 463152, https://doi.org/10.1090/S0002-9947-1977-0463152-5
[Ho5] Melvin Hochster, Some applications of the Frobenius in characteristic 0, Bull. Amer. Math. Soc. 84 (1978), no. 5, 886–912. MR 485848, https://doi.org/10.1090/S0002-9904-1978-14531-5
[Ho6] Melvin Hochster, Cohen-Macaulay rings and modules, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 291–298. MR 562618
[Ho7] Melvin Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), no. 2, 503–553. MR 723406, https://doi.org/10.1016/0021-8693(83)90092-3
[Ho8] 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. 103–172. MR 1266182, https://doi.org/10.1090/conm/159/01508
[Ho9] 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. 173–196. MR 1266183, https://doi.org/10.1090/conm/159/01507
[HoE] M. Hochster and John A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058. MR 302643, https://doi.org/10.2307/2373744
[HH1] Melvin Hochster and Craig Huneke, Tightly closed ideals, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 45–48. MR 919658, https://doi.org/10.1090/S0273-0979-1988-15592-9
[HH2] Melvin Hochster and Craig Huneke, Tight closure, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 305–324. MR 1015524, https://doi.org/10.1007/978-1-4612-3660-3_15
[HH3] -, Tight closure and strong -regularity, Mém. Soc. Math. France 38 (1989), 119-133.
[HH4] Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, https://doi.org/10.1090/S0894-0347-1990-1017784-6
[HH5] 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, 137–143. MR 1056558, https://doi.org/10.1090/S0273-0979-1991-15970-7
[HH6] Melvin Hochster and Craig Huneke, Tight closure and elements of small order in integral extensions, J. Pure Appl. Algebra 71 (1991), no. 2-3, 233–247. MR 1117636, https://doi.org/10.1016/0022-4049(91)90149-V
[HH7] Melvin Hochster and Craig Huneke, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. (2) 135 (1992), no. 1, 53–89. MR 1147957, https://doi.org/10.2307/2946563
[HH8] Melvin Hochster and Craig Huneke, Phantom homology, Mem. Amer. Math. Soc. 103 (1993), no. 490, vi+91. MR 1144758, https://doi.org/10.1090/memo/0490
[HH9] Melvin Hochster and Craig Huneke, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599–670. MR 1297848
[HH10] -, Tight closure in equal characteristic zero (in preparation).
[HH11] Melvin Hochster and Craig Huneke, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math. 113 (1995), no. 1, 45–117. MR 1332808, https://doi.org/10.1006/aima.1995.1035
[HR1] Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, https://doi.org/10.1016/0001-8708(74)90067-X
[HR2] Melvin Hochster and Joel L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2, 117–172. MR 417172, https://doi.org/10.1016/0001-8708(76)90073-6
[Hu1] Craig Huneke, An algebraist commuting in Berkeley, Math. Intelligencer 11 (1989), no. 1, 40–52. MR 979023, https://doi.org/10.1007/BF03023775
[Hu2] 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. 339–349. MR 1159222
[Hu3] Craig Huneke, Uniform bounds in Noetherian rings, Invent. Math. 107 (1992), no. 1, 203–223. MR 1135470, https://doi.org/10.1007/BF01231887
[Ke] George Kempf, The Hochster-Roberts theorem of invariant theory, Michigan Math. J. 26 (1979), no. 1, 19–32. MR 514958
[Ku1] Ernst Kunz, Characterizations of regular local rings of characteristic 𝑝, Amer. J. Math. 91 (1969), 772–784. MR 252389, https://doi.org/10.2307/2373351
[Ku2] Ernst Kunz, On Noetherian rings of characteristic 𝑝, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625, https://doi.org/10.2307/2374038
[LS] Joseph Lipman and Avinash Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), no. 2, 199–222. MR 616270
[LT] 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, 97–116. MR 600418
[Ma] Frank Ma, Splitting in integral extensions, Cohen-Macaulay modules and algebras, J. Algebra 116 (1988), no. 1, 176–195. MR 944154, https://doi.org/10.1016/0021-8693(88)90200-1
[Mat] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
[N] 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
[PS1] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130
[PS2] Christian Peskine and Lucien Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1421–1424 (French). MR 0349659
[Ro1] Paul Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 103–106. MR 399075
[Ro2] Paul Roberts, Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture, J. Algebra 66 (1980), no. 1, 220–225. MR 591254, https://doi.org/10.1016/0021-8693(80)90121-0
[Ro3] Paul Roberts, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 127–130. MR 799793, https://doi.org/10.1090/S0273-0979-1985-15394-7
[Ro4] Paul Roberts, Le théorème d’intersection, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 7, 177–180 (French, with English summary). MR 880574
[Ro5] Paul Roberts, Intersection theorems, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 417–436. MR 1015532, https://doi.org/10.1007/978-1-4612-3660-3_23
[S] J.-P. Serre, Algèbre locale: Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965.
[SGA $4\tfrac{1} {2}$ ] P. Deligne avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie, et J. L. Verdier, Cohomologie étale (Sém. Géométrie Algébrique du Bois-Marie $4\tfrac{1} {2}$ ), Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, Heidelberg, and New York, 1977.
[Sk] Henri Skoda, Application des techniques 𝐿² à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids, Ann. Sci. École Norm. Sup. (4) 5 (1972), 545–579 (French). MR 333246
[Sm1] K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41–60. MR 1248078, https://doi.org/10.1007/BF01231753
[Sm2] K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41–60. MR 1248078, https://doi.org/10.1007/BF01231753
[Sm3] -, -rational rings have rational singularities, preprint.
[Sw1] Irena Swanson, Joint reductions, tight closure, and the Briançon-Skoda theorem, J. Algebra 147 (1992), no. 1, 128–136. MR 1154678, https://doi.org/10.1016/0021-8693(92)90256-L
[Sw2] -, Tight closure, joint reductions, and mixed multiplicities, Thesis, Purdue Univ., 1992.
[Ve11] J. Velez, Openness of the -rational locus, smooth base change, and Koh's conjecture, Thesis, Univ. of Michigan, 1993.
[Vel2] -, Openness of the -rational locus and smooth base change, J. Algebra (to appear).
[W1] K.-I. Watanabe, Study of -purity in dimension two, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Vol. II, Kinokuniya, Tokyo, 1988, pp. 791-800.
[W2] Keiichi Watanabe, 𝐹-regular and 𝐹-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), no. 2-3, 341–350. MR 1117644, https://doi.org/10.1016/0022-4049(91)90157-W
[WITO] Kei-ichi Watanabe, Takeshi Ishikawa, Sadao Tachibana, and Kayo Otsuka, On tensor products of Gorenstein rings, J. Math. Kyoto Univ. 9 (1969), 413–423. MR 257062, https://doi.org/10.1215/kjm/1250523903
[Wil1] L. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, Thesis, Univ. of Michigan, 1992.
[Wil2] Lori J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), no. 3, 721–743. MR 1324179, https://doi.org/10.1006/jabr.1995.1067