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Tight closure, invariant theory, and the Briançon-Skoda theorem
Author(s):
Melvin
Hochster;
Craig
Huneke
Journal:
J. Amer. Math. Soc.
3
(1990),
31-116.
MSC:
Primary 13C05;
Secondary 13A15, 13A50, 13B99, 13D02
MathSciNet review:
1017784
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References:
-
- [A]
- M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Étude Sci. Publ. Math. 36 (1969), 23-56. MR 0268188 (42:3087)
- [AuBr]
- M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc., no. 94, Amer. Math. Soc., Providence, RI, 1969. MR 0269685 (42:4580)
- [B]
- J.-F. Boutot, Singularités rationelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65-68. MR 877006 (88a:14005)
- [BE]
- D. Buchsbaum and D. Eisenbud, What makes a complex exact, J. Algebra 25 (1973), 259-268. MR 0314819 (47:3369)
- [Bor]
- A. Borel, Linear algebraic groups, Benjamin, New York, 1969. MR 0251042 (40:4273)
- [BrS]
- J. Briançon and H. Skoda, Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de
, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949-951. MR 0340642 (49:5394) - [BrV]
- W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math., vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963 (89i:13001)
- [DEP]
- C. De Concini, D. Eisenbud, and C. Procesi, Hodge algebras, Astérisque 91 (1982), 1-87. MR 680936 (85d:13009)
- [Du1]
- S. P. Dutta, Frobenius and multiplicities, J. Algebra 85 (1983), 424-448. MR 725094 (85f:13022)
- [Du2]
- -, On the canonical element conjecture, Trans. Amer. Math. Soc. 299 (1987), 803-811. MR 869233 (87m:13028)
- [Du3]
- -, Ext and Frobenius, J. Algebra (to appear). MR 1029410 (91g:13018)
- [EHo]
- J. A. Eagon and M. Hochster,
-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61-71. MR 0337934 (49:2703) - [Ei]
- D. Eisenbud, Homological algebra on a complete intersection with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35-64. MR 570778 (82d:13013)
- [EvG1]
- E. G. Evans and P. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), 323-333. MR 632842 (83i:13006)
- [EvG2]
- -, Syzygies, London Math. Soc. Lecture Note Ser., no. 106, Cambridge Univ. Press, Cambridge, 1985.
- [EvG3]
- -, Order ideals, Commutative Algebra, Proc. Microprogram, June 15-July 12, 1987, Math. Sci. Res. Inst. Publ., no. 15, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989, pp. 213-225. MR 1015519 (90i:13011)
- [FeW]
- R. Fedder and K. Watanabe, A characterization of
-regularity in terms of -purity, Commutative Algebra, Proc. Microprogram, June 15-July 12, 1987, Math. Sci. Res. Inst. Publ., no. 15, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989, pp. 227-245. MR 1015520 (91k:13009) - [Fu]
- W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984. MR 732620 (85k:14004)
- [GraR]
- H. Grauert and O. Riemenschneider, Verschwindungsätze für analytische kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263-290. MR 0302938 (46:2081)
- [GrH]
- A. Grothendieck (notes by R. Hartshorne), Local cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag, Heidelberg, 1967. MR 0224620 (37:219)
- [He]
- J. Herzog, Ringe der Charakteristik
und Frobenius-funktoren, Math. Z. 140 (1974), 67-78. MR 0352081 (50:4569) - [HH1]
- M. Hochster and C. Huneke, Tightly closed ideals, Bull. Amer. Math. Soc. 18 (1988), 45-48. MR 919658 (89b:13003)
- [HH2]
- -, Tight closure, Commutative Algebra, Proc. Microprogram, June 15-July 12, 1987, Math. Sci. Res. Inst. Publ., no. 15, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989, pp. 305-324. MR 1015524 (91f:13022)
- [HH3]
- -, Tight closure and strong
-regularity, Mem. Soc. Math. France (N.S.), numéro consacré au colloque en l'honneur de P. Samuel (to appear). MR 1044348 (91i:13025) - [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 0304376 (46:3511)
- [Ho2]
- -, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25-43. MR 0349656 (50:2149)
- [Ho3]
- -, Topics in the homological theory of modules over commutative rings, C.B.M.S. Regional Conf. Ser. in Math., no. 24, Amer. Math. Soc., Providence, RI, 1975. MR 0371879 (51:8096)
- [Ho4]
- -, Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors, Proc. Queen's University Commutative Algebra Conference, Queen's Papers in Pure and Appl. Math. 42 (1975), 106-195. MR 0396544 (53:407)
- [Ho5]
- -, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488. MR 0463152 (57:3111)
- [Ho6]
- -, Some applications of the Frobenius in characteristic 0, Bull. Amer. Math. Soc. 84 (1978), 886-912. MR 0485848 (58:5650)
- [Ho7]
- -, Cohen-Macaulay rings and modules, Proc. Internat. Congr. Math., Helsinki, Finland, Vol. I, Academia Scientarium Fennica, 1980, pp. 291-298. MR 562618 (83c:13010)
- [Ho8]
- -, Associated graded rings derived from integrally closed ideals, Proc. Conf. Commutative Algebra (Rennes, France, May 1981), pp. 1-27.
- [Ho9]
- -, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), 503-553. MR 723406 (85j:13021)
- [HoE]
- M. Hochster and J. A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. MR 0302643 (46:1787)
- [HR1]
- M. Hochster and J. L. 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)
- [HR2]
- -, The purity of the Frobenius and local cohomology, Adv. in Math. 21 (1976), 117-172. MR 0417172 (54:5230)
- [Hu1]
- C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293-318. MR 894879 (89b:13037)
- [Hu2]
- -, An algebraist commuting in Berkeley, Math. Intelligencer 11 (1989), 40-52. MR 979023 (89m:13001)
- [It1]
- S. Itoh, Integral closures of ideals generated by regular sequences, J. Algebra 117 (1988), 390-401. MR 957448 (90g:13013)
- [It2]
- -, Integral closures of ideals of the principal class, Hiroshima Math. J. 17 (1987), 373-375. MR 909622 (88j:13009)
- [Kap]
- I. Kaplansky, Commutative algebra, Allyn and Bacon, 1970; revised edition, Univ. Chicago Press, Chicago, IL, 1974. MR 0345945 (49:10674)
- [Ke]
- G. Kempf, The Hochster-Roberts theorem of invariant theory, Michigan Math. J. 26 (1979), 19-32. MR 514958 (80g:14040)
- [Ku1]
- E. Kunz, Characterizations of regular local rings of characteristic
, Amer. J. Math. 91 (1969), 772-784. MR 0252389 (40:5609) - [Ku2]
- -, On Noetherian rings of characteristic
, Amer. J. Math. 98 (1976), 999-1013. MR 0432625 (55:5612) - [L]
- J. Lipman, Relative Lipschitz saturation, Amer. J. Math. 97 (1975), 791-813. MR 0417169 (54:5228a)
- [LS]
- J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), 199-222. MR 616270 (83m:13001)
- [LT]
- 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 600418 (82f:14004)
- [Ma]
- F. Ma, Splitting in integral extensions, Cohen-Macaulay modules and algebras, J. Algebra 116 (1988), 176-195. MR 944154 (89e:13010)
- [Mac]
- S. Mac Lane, Homology, Springer-Verlag, Berlin, Gottingen, and Heidelberg, 1963. MR 0156879 (28:122)
- [Mat]
- H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 0266911 (42:1813)
- [MeSr]
- V. B. Mehta and V. Srinivas, Normal
-pure surface singularities, Tata Institute, Bombay, preprint. MR 1128650 (92j:14044) - [Mon]
- P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), 43-49. MR 697329 (84k:13012)
- [N]
- M. Nagata, Local rings, Interscience, New York, 1972. MR 0155856 (27:5790)
- [NoR1]
- D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 0059889 (15:596a)
- [NoR2]
- -, A note on reductions of ideals with an application to the generalized Hilbert function, Proc. Cambridge Philos. Soc. 50 (1954), 353-359. MR 0062115 (15:929e)
- [PS1]
- C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 323-395. MR 0374130 (51:10330)
- [PS2]
- -, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1421-1424. MR 0349659 (50:2152)
- [Rat]
- L. J. Ratliff, Chain conjectures in ring theory, Springer-Verlag, Berlin and New York, 1978. MR 496884 (80c:13008)
- [Re1]
- D. Rees, A note on asymptotically equidimensional ideals, Math. Proc. Cambridge Philos. Soc. 98 (1985), 33-35. MR 789716 (86k:13015)
- [Re2]
- -, Reductions of modules, Math. Proc. Cambridge Philos. Soc. 101 (1987), 431-449. MR 878892 (88a:13001)
- [Ro1]
- P. Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. École Norm. Sup. (4) 9 (1976), 103-106. MR 0399075 (53:2926)
- [Ro2]
- -, Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture, J. Algebra 66 (1980), 225-230. MR 591254 (82e:14009)
- [Ro3]
- -, Homological invariants of modules over commutative rings, Sém. Math. Sup., Presses Univ. Montréal, 1980. MR 569936 (82j:13020)
- [Ro4]
- -, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. 13 (1985), 127-130. MR 799793 (87c:13030)
- [Ro5]
- -, Le théorème d'intersection, C. R. Acad. Sci. Paris Sér I Math. 304 (1987), 177-180. MR 880574 (89b:14008)
- [Ro6]
- -, Intersection theorems, Commutative Algebra, Proc. Microprogram, June 15-July 12, 1987, Math. Sci. Res. Inst. Publ., no. 15, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989, pp. 417-436. MR 1015532 (90j:13024)
- [S]
- J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Math., no. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965.
- [Se]
- G. Seibert, Complexes with homology of finite length and Frobenius functors, preprint. MR 1018945 (90j:13012)
- [Sk]
- H. Skoda, Applications des techniques
a la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids, Ann. Sci. Ecole Norm. Sup (4) 5 (1972), 545-579. MR 0333246 (48:11571) - [Sr]
- V. Srinivas, Normal surface singularities of
-pure type, Tata Institute, Bombay, preprint. MR 1127067 (93b:14060) - [Sz]
- L. Szpiro, Sur la théorie des complexes parfaits, Commutative Algebra, Durham, 1981, London Math. Soc. Lecture Notes Ser., no 72, Cambridge Univ. Press, 1982, pp. 83-90. MR 693628 (84m:13015)
- [Tay]
- D. Taylor, Thesis, Ideals generated by monomials in an
-sequence, Univ. of Chicago, 1966. - [Wal]
- C. T. C. Wall, Lectures on
stability and classification, Proc. Liverpool Singularities-Symposium I, Lecture Notes in Math., no. 192, Springer-Verlag, Heidelberg, 1971. MR 0285020 (44:2244) - [W]
- K. Watanabe, Study of
-purity in dimension two, Algebraic Geometry and Commutative Algebra in honor of Masayoshi Nagata, North-Holland, Amsterdam, New York, Oxford, 1987, pp. 791-800. MR 977783 (90b:14005) - [ZS]
- O. Zariski and P. Samuel, Commutative algebra, Vols. I and II, Van Nostrand, Princeton, 1958 and 1960.
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Additional Information:
DOI:
10.1090/S0894-0347-1990-1017784-6
PII:
S0894-0347-1990-1017784-6
Copyright of article:
Copyright
1990,
American Mathematical Society
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