Some applications of the Frobenius in characteristic 0

Hochster, Melvin

doi:10.1090/S0002-9904-1978-14531-5

Some applications of the Frobenius in characteristic 0
HTML articles powered by AMS MathViewer

by Melvin Hochster PDF

Bull. Amer. Math. Soc. 84 (1978), 886-912

References

Shreeram Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $p\ne 0$, Ann. of Math. (2) 63 (1956), 491–526. MR 78017, DOI 10.2307/1970014
Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 232018, DOI 10.1007/BF01389777
M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188, DOI 10.1007/BF02684596
M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. MR 179211, DOI 10.1215/ijm/1255631585
Maurice Auslander, Modules oever unramified regular local rings, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 230–233. MR 0175930
Maurice Auslander and David A. Buchsbaum, Homological dimension in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 36–38. MR 75190, DOI 10.1073/pnas.42.1.36
Maurice Auslander and David A. Buchsbaum, Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957), 390–405. MR 86822, DOI 10.1090/S0002-9947-1957-0086822-7
Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI 10.2307/1970159
Maurice Auslander and D. A. Buchsbaum, Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 733–734. MR 103906, DOI 10.1073/pnas.45.5.733
M. Auslander and D. S. Rim, Ramification index and multiplicity, Illinois J. Math. 7 (1963), 566–581. MR 155853, DOI 10.1215/ijm/1255645095
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
Hyman Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155–196. MR 432781, DOI 10.1007/BF01390137
David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285, DOI 10.1515/9781400882069
Marie-José Bertin, Anneaux d’invariants d’anneaux de polynomes, en caractéristique $p$, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A653–A656 (French). MR 215826
Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
Jean-François Boutot, Schéma de Picard local, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A691–A694 (French). MR 345968
David A. Buchsbaum and David Eisenbud, Lifting modules and a theorem on finite free resolutions, Ring theory (Proc. Conf., Park City, Utah, 1971) Academic Press, New York, 1972, pp. 63–74. MR 0340343
David A. Buchsbaum and David Eisenbud, Remarks on ideals and resolutions, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 193–204. MR 0337946
David A. Buchsbaum and David Eisenbud, Some structure theorems for finite free resolutions, Advances in Math. 12 (1974), 84–139. MR 340240, DOI 10.1016/S0001-8708(74)80019-8
Wei Liang Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799–822. MR 171804, DOI 10.2307/2373158
I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. MR 16094, DOI 10.1090/S0002-9947-1946-0016094-3
C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), no. 3, 330–354. MR 422314, DOI 10.1016/S0001-8708(76)80003-5
Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697, DOI 10.24033/asens.1261
Jean A. Dieudonné and James B. Carrell, Invariant theory, old and new, Academic Press, New York-London, 1971. MR 0279102
John A. Eagon and M. Hochster, $R$-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61–71. MR 337934, DOI 10.1093/qmath/25.1.61
J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
David Eisenbud, Some directions of recent progress in commutative algebra, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 111–128. MR 0384767
David Eisenbud and E. Graham Evans Jr., A generalized principal ideal theorem, Nagoya Math. J. 62 (1976), 41–53. MR 409440, DOI 10.1017/S0027763000024740
Daniel Ferrand and Michel Raynaud, Fibres formelles d’un anneau local noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295–311 (French). MR 272779, DOI 10.24033/asens.1195
Robert Fossum, Hans-Bjørn Foxby, Phillip Griffith, and Idun Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 193–215. MR 396529, DOI 10.1007/BF02684302
Robert M. Fossum and Phillip A. Griffith, Complete local factorial rings which are not Cohen-Macaulay in characteristic $p$, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 189–199. MR 382257, DOI 10.24033/asens.1285
Hans-Bjørn Foxby and Anders Thorup, Minimal injective resolutions under flat base change, Proc. Amer. Math. Soc. 67 (1977), no. 1, 27–31. MR 453724, DOI 10.1090/S0002-9939-1977-0453724-1
Eberhard Freitag and Reinhardt Kiehl, Algebraische Eigenschaften der lokalen Ringe in den Spitzen der Hilbertschen Modulgruppen, Invent. Math. 24 (1974), 121–148 (German). MR 347823, DOI 10.1007/BF01404302
William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
Robert Gilmer, Dimension sequences of commutative rings, Ring theory (Proc. Conf., Univ. Oklahoma, Norman, Okla., 1973) Lecture Notes in Pure and Applied Mathematics, Vol. 7, Dekker, New York, 1974, pp. 31–46. MR 0332767
Phillip Griffith, A representation theorem for complete local rings, J. Pure Appl. Algebra 7 (1976), no. 3, 303–315. MR 412176, DOI 10.1016/0022-4049(76)90056-6
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, DOI 10.1007/BFb0073971
Tor Holtedahl Gulliksen and Odd Guttorm Negȧrd, Un complexe résolvant pour certains idéaux déterminantiels, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A16–A18 (French). MR 296063
W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102 (1975), no. 1, 67–83. MR 382294, DOI 10.2307/1970974
Robin Hartshorne and Arthur Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415–437. MR 347821, DOI 10.1080/00927877408548627
Robin Hartshorne and Robert Speiser, Local cohomological dimension in characteristic $p$, Ann. of Math. (2) 105 (1977), no. 1, 45–79. MR 441962, DOI 10.2307/1971025
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
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, DOI 10.2307/1970791
Melvin Hochster, Cohen-Macaulay modules, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 120–152. MR 0340251
M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25–43. MR 349656, DOI 10.1017/S0027763000015701
M. Hochster, Grassmannians and their Schubert subvarieties are arithmetically Cohen-Macaulay, J. Algebra 25 (1973), 40–57. MR 314833, DOI 10.1016/0021-8693(73)90074-4
Melvin Hochster, Associated graded rings derived from integrally closed ideals and the local homological conjectures, Algebra Colloquium (Rennes, 1980) Univ. Rennes, Rennes, 1980, pp. 1–27. MR 645209
M. Hochster, Grade-sensitive modules and perfect modules, Proc. London Math. Soc. (3) 29 (1974), 55–76. MR 374118, DOI 10.1112/plms/s3-29.1.55
Melvin Hochster, Constraints on systems of parameters, Ring theory (Proc. Conf., Univ. Oklahoma, Norman, Okla., 1973) Lecture Notes in Pure and Applied Mathematics, Vol. 7, Dekker, New York., 1974, pp. 121–161. MR 0330156
Melvin Hochster, The equicharacteristic case of some homological conjectures on local rings, Bull. Amer. Math. Soc. 80 (1974), 683–686. MR 342510, DOI 10.1090/S0002-9904-1974-13548-2
Melvin Hochster, Topics in the homological theory of modules over commutative rings, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24, 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. MR 0371879, DOI 10.1090/cbms/024
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 Papers on Pure and Applied Math., No. 42, Queen’s Univ., Kingston, Ont., 1975, pp. 106–195. MR 0396544
Melvin Hochster, An obstruction to lifting cyclic modules, Pacific J. Math. 61 (1975), no. 2, 457–463. MR 412169, DOI 10.2140/pjm.1975.61.457
Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977, pp. 171–223. MR 0441987
Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463–488. MR 463152, DOI 10.1090/S0002-9947-1977-0463152-5
M. Hochster, Properties of Noetherian rings stable under general grade reduction, Arch. Math. (Basel) 24 (1973), 393–396. MR 330147, DOI 10.1007/BF01228228
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, DOI 10.2307/2373744
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, DOI 10.1016/0001-8708(74)90067-X
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, DOI 10.1016/0001-8708(76)90073-6
Birger Iversen, Amplitude inequalities for complexes, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 547–558. MR 568903, DOI 10.24033/asens.1336
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Irving Kaplansky, Commutative rings, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 153–166. MR 0360550
George R. Kempf, Images of homogeneous vector bundles and varieties of complexes, Bull. Amer. Math. Soc. 81 (1975), no. 5, 900–901. MR 384817, DOI 10.1090/S0002-9904-1975-13878-X
George R. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), no. 3, 229–239. MR 424841, DOI 10.1007/BF01390321
George R. Kempf, Some quotient varieties have rational singularities, Michigan Math. J. 24 (1977), no. 3, 347–352. MR 491675
G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518, DOI 10.1007/BFb0070318
Jan O. Kleppe, Deformations of graded algebras, Math. Scand. 45 (1979), no. 2, 205–231. MR 580600, DOI 10.7146/math.scand.a-11837
Ronald E. Kutz, Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups, Trans. Amer. Math. Soc. 194 (1974), 115–129. MR 352082, DOI 10.1090/S0002-9947-1974-0352082-2
Dan Laksov, The arithmetic Cohen-Macaulay character of Schubert schemes, Acta Math. 129 (1972), no. 1-2, 1–9. MR 382297, DOI 10.1007/BF02392211
G. Levin and W. V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323. MR 230715, DOI 10.2140/pjm.1968.25.315
Stephen Lichtenbaum, On the vanishing of $\textrm {Tor}$ in regular local rings, Illinois J. Math. 10 (1966), 220–226. MR 188249
Joseph Lipman, Unique factorization in complete local rings, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 531–546. MR 0374125
Marie-Paule Malliavin-Brameret, Structure d’anneaux locaux réguliers et caractéristiques d’Euler-Poincaré, Bull. Sci. Math. (2) 97 (1973), 81–88 (French). MR 330153
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360, DOI 10.2140/pjm.1958.8.511
Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, London Mathematical Society Lecture Note Series, vol. 3, Cambridge University Press, London-New York, 1971. Prepared in collaboration with J. E. Reeve and A. A. Ball. MR 0301635
Shigefumi Mori, On affine cones associated with polarized varieties, Japan. J. Math. (N.S.) 1 (1975), no. 2, 301–309. MR 439859, DOI 10.4099/math1924.1.301
David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285, DOI 10.1515/9781400882069
David Mumford, Hilbert’s fourteenth problem–the finite generation of subrings such as rings of invariants, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc.., Providence, R.I., 1976, pp. 431–444. MR 0435076
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
C. Musili, Postulation formula for Schubert varieties, J. Indian Math. Soc. (N.S.) 36 (1972), 143–171. MR 330177
Masayoshi Nagata, Complete reducibility of rational representations of a matric group, J. Math. Kyoto Univ. 1 (1961/62), 87–99. MR 142667, DOI 10.1215/kjm/1250525107
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828
Hans-Joachim Nastold, Zur Serreschen Multiplizitätstheorie in der arithmetischen Geometrie, Math. Ann. 143 (1961), 333–343 (German). MR 125858, DOI 10.1007/BF01470614
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, DOI 10.1007/BF02685877
Christian Peskine and Lucien Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1421–1424 (French). MR 349659
Gerhard Pfister and Dorin Popescu, Die strenge Approximationseigenschaft lokaler Ringe, Invent. Math. 30 (1975), no. 2, 145–174 (German). MR 379490, DOI 10.1007/BF01425506
C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
D. Rees, The grade of an ideal or module, Proc. Cambridge Philos. Soc. 53 (1957), 28–42. MR 82967, DOI 10.1017/s0305004100031960
Gerald Allen Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), no. 1, 30–49. MR 407036, DOI 10.1016/0001-8708(76)90114-6
Paul Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 103–106. MR 399075, DOI 10.24033/asens.1305
P. Samuel, Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, No. 30, Tata Institute of Fundamental Research, Bombay, 1964. Notes by M. Pavman Murthy. MR 0214579
A. Seidenberg, A note on the dimension theory of rings, Pacific J. Math. 3 (1953), 505–512. MR 54571, DOI 10.2140/pjm.1953.3.505
A. Seidenberg, On the dimension theory of rings. II, Pacific J. Math. 4 (1954), 603–614. MR 65540, DOI 10.2140/pjm.1954.4.603
Jean-Pierre Serre, Sur la dimension homologique des anneaux et des modules noethériens, Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955, Science Council of Japan, Tokyo, 1956, pp. 175–189 (French). MR 0086071
Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 68874, DOI 10.2307/1969915
Jean-Pierre Serre, Exemples de variétés projectives en caractéristique $p$ non relevables en caractéristique zéro, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 108–109 (French). MR 132067, DOI 10.1073/pnas.47.1.108
Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468, DOI 10.1007/978-3-662-21576-0
R. Y. Sharp, Local cohomology theory in commutative algebra, Quart. J. Math. Oxford Ser. (2) 21 (1970), 425–434. MR 276217, DOI 10.1093/qmath/21.4.425
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
Richard P. Stanley, Cohen-Macaulay rings and constructible polytopes, Bull. Amer. Math. Soc. 81 (1975), 133–135. MR 364231, DOI 10.1090/S0002-9904-1975-13670-6
Richard P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), no. 2, 135–142. MR 458437, DOI 10.1002/sapm1975542135
Torgny Svanes, Coherent cohomology on Schubert subschemes of flag schemes and applications, Advances in Math. 14 (1974), 369–453. MR 419469, DOI 10.1016/0001-8708(74)90039-5
André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689. MR 159, DOI 10.2307/1968949
Oscar Zariski, Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc. 53 (1943), 490–542. MR 8468, DOI 10.1090/S0002-9947-1943-0008468-9
Oscar Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947), 1–52. MR 21694, DOI 10.1090/S0002-9947-1947-0021694-1