Tight closure, invariant theory, and the Briançon-Skoda theorem

Hochster, Melvin; Huneke, Craig

doi:10.1090/S0894-0347-1990-1017784-6

Tight closure, invariant theory, and the Briançon-Skoda theorem

Authors: Melvin Hochster and Craig Huneke
Journal: J. Amer. Math. Soc. 3 (1990), 31-116
MSC: Primary 13C05; Secondary 13A15, 13A50, 13B99, 13D02
DOI: https://doi.org/10.1090/S0894-0347-1990-1017784-6
MathSciNet review: 1017784
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[A] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188
[AuBr] Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
[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
[BE] David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259–268. MR 314819, https://doi.org/10.1016/0021-8693(73)90044-6
[Bor] Armand Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0251042
[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
[BrV] Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963
[DEP] Corrado De Concini, David Eisenbud, and Claudio Procesi, Hodge algebras, Astérisque, vol. 91, Société Mathématique de France, Paris, 1982. With a French summary. MR 680936
[Du1] Sankar P. Dutta, Frobenius and multiplicities, J. Algebra 85 (1983), no. 2, 424–448. MR 725094, https://doi.org/10.1016/0021-8693(83)90106-0
[Du2] 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
[Du3] S. P. Dutta, Ext and Frobenius, J. Algebra 127 (1989), no. 1, 163–177. MR 1029410, https://doi.org/10.1016/0021-8693(89)90281-0
[EHo] John A. Eagon and M. Hochster, 𝑅-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61–71. MR 337934, https://doi.org/10.1093/qmath/25.1.61
[Ei] David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. MR 570778, https://doi.org/10.1090/S0002-9947-1980-0570778-7
[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., no. 106, Cambridge Univ. Press, Cambridge, 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
[Fu] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
[GraR] Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 (German). MR 302938, https://doi.org/10.1007/BF01403182
[GrH] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. MR 0224620
[He] Jürgen Herzog, Ringe der Charakteristik 𝑝 und Frobeniusfunktoren, Math. Z. 140 (1974), 67–78 (German). MR 352081, https://doi.org/10.1007/BF01218647
[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] Melvin Hochster and Craig Huneke, Tight closure and strong 𝐹-regularity, Mém. Soc. Math. France (N.S.) 38 (1989), 119–133. Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044348
[HH4] -, Phantom homology, preprint, 1989.
[HH5] -, Tight closure, -regularity, test elements and smooth base change (in preparation).
[HH6] -, Tight closures of parameter ideals and splitting in module-finite extensions (in preparation).
[HH7] -, Tight closure in characteristic zero (in preparation).
[HH8] -, Tight closure and elments of small order in integral extensions, preprint, 1989.
[HH9] -, Infinite integral extensions and big Cohen-Macaulay algebras, preprint, 1989.
[Ho1] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. MR 304376, https://doi.org/10.2307/1970791
[Ho2] M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25–43. MR 349656
[Ho3] 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
[Ho4] 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
[Ho5] 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
[Ho6] 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
[Ho7] 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
[Ho8] -, Associated graded rings derived from integrally closed ideals, Proc. Conf. Commutative Algebra (Rennes, France, May 1981), pp. 1-27.
[Ho9] 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
[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
[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, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), no. 2, 293–318. MR 894879, https://doi.org/10.1307/mmj/1029003560
[Hu2] Craig Huneke, An algebraist commuting in Berkeley, Math. Intelligencer 11 (1989), no. 1, 40–52. MR 979023, https://doi.org/10.1007/BF03023775
[It1] Shiroh Itoh, Integral closures of ideals generated by regular sequences, J. Algebra 117 (1988), no. 2, 390–401. MR 957448, https://doi.org/10.1016/0021-8693(88)90114-7
[It2] Shiroh Itoh, Integral closures of ideals of the principal class, Hiroshima Math. J. 17 (1987), no. 2, 373–375. MR 909622
[Kap] Irving Kaplansky, Commutative rings, Revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
[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
[L] Joseph Lipman, Relative Lipschitz-saturation, Amer. J. Math. 97 (1975), no. 3, 791–813. MR 417169, https://doi.org/10.2307/2373777
[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
[Mac] Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
[Mat] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
[MeSr] V. B. Mehta and V. Srinivas, Normal 𝐹-pure surface singularities, J. Algebra 143 (1991), no. 1, 130–143. MR 1128650, https://doi.org/10.1016/0021-8693(91)90255-7
[Mon] P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), no. 1, 43–49. MR 697329, https://doi.org/10.1007/BF01457082
[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
[NoR1] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, https://doi.org/10.1017/s0305004100029194
[NoR2] D. G. Northcott and D. Rees, A note on reductions of ideals with an application to the generalized Hilbert function, Proc. Cambridge Philos. Soc. 50 (1954), 353–359. MR 62115, https://doi.org/10.1017/s0305004100029455
[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
[Rat] Louis J. Ratliff Jr., Chain conjectures in ring theory, Lecture Notes in Mathematics, vol. 647, Springer, Berlin, 1978. An exposition of conjectures on catenary chains. MR 496884
[Re1] D. Rees, A note on asymptotically unmixed ideals, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 1, 33–35. MR 789716, https://doi.org/10.1017/S0305004100063210
[Re2] D. Rees, Reduction of modules, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 431–449. MR 878892, https://doi.org/10.1017/S0305004100066810
[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, Homological invariants of modules over commutative rings, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 72, Presses de l’Université de Montréal, Montreal, Que., 1980. MR 569936
[Ro4] 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
[Ro5] 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
[Ro6] 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., no. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965.
[Se] Gerhard Seibert, Complexes with homology of finite length and Frobenius functors, J. Algebra 125 (1989), no. 2, 278–287. MR 1018945, https://doi.org/10.1016/0021-8693(89)90164-6
[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
[Sr] V. Srinivas, Normal surface singularities of 𝐹-pure type, J. Algebra 142 (1991), no. 2, 348–359. MR 1127067, https://doi.org/10.1016/0021-8693(91)90311-U
[Sz] L. Szpiro, Sur la théorie des complexes parfaits, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 83–90. MR 693628
[Tay] D. Taylor, Thesis, Ideals generated by monomials in an -sequence, Univ. of Chicago, 1966.
[Wal] C. T. C. Wall, Lectures on 𝐶^{∞}-stability and classification, Proceedings of Liverpool Singularities–Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 178–206. MR 0285020
[W] Keiichi Watanabe, Study of 𝐹-purity in dimension two, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 791–800. MR 977783
[ZS] O. Zariski and P. Samuel, Commutative algebra, Vols. I and II, Van Nostrand, Princeton, 1958 and 1960.