Some applications of the Frobenius in characteristic 0
Melvin Hochster
Source: Bull. Amer. Math. Soc. Volume 84, Number 5
(1978), 886-912.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.bams/1183541144
Mathematical Reviews number (MathSciNet): MR0485848
Zentralblatt MATH identifier: 0421.14001
References
[Ab] S. S. Abhyankar, Local uniformization on algebraic surfaces over a ground field of characteristic p≠0, Ann. of Math. 63 (1956), 491-526.
Zentralblatt MATH: 0108.16803
Mathematical Reviews (MathSciNet): MR78017
Digital Object Identifier: doi:10.2307/1970014
[AK] A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math., vol. 146, Springer-Verlag, Berlin and New York, 1970.
Zentralblatt MATH: 0215.37201
Mathematical Reviews (MathSciNet): MR274461
[Ar1] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291.
Zentralblatt MATH: 0172.05301
Mathematical Reviews (MathSciNet): MR232018
Digital Object Identifier: doi:10.1007/BF01389777
[Ar2] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Étudies Sci. Publ. Math. 36 (1969), 23-56.
Zentralblatt MATH: 0181.48802
Mathematical Reviews (MathSciNet): MR268188
Digital Object Identifier: doi:10.1007/BF02684596
[Au1] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631-645.
Zentralblatt MATH: 0104.26202
Mathematical Reviews (MathSciNet): MR179211
Project Euclid: euclid.ijm/1255631585
[Au2] M. Auslander, Modules over unramified regular local rings, Proc. Internat. Congress of Math., 1962, pp. 230-233.
Zentralblatt MATH: 0123.03702
Mathematical Reviews (MathSciNet): MR175930
[AB1] M. Auslander and D. A. Buchsbaum, Homological dimension in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 36-38.
Zentralblatt MATH: 0070.03503
Mathematical Reviews (MathSciNet): MR75190
Digital Object Identifier: doi:10.1073/pnas.42.1.36
[AB2] M. Auslander and D. A. Buchsbaum, Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957), 390-405.
Zentralblatt MATH: 0078.02802
Mathematical Reviews (MathSciNet): MR86822
Digital Object Identifier: doi:10.1090/S0002-9947-1957-0086822-7
[AB3] M. Auslander and D. A. Buchsbaum, Codimension and multiplicity, Ann. of Math. 68 (1958), 625-657; corrections, Ann. of Math. 70 (1959), 395-397.
Zentralblatt MATH: 0092.03902
Mathematical Reviews (MathSciNet): MR99978
Digital Object Identifier: doi:10.2307/1970159
[AB4] M. Auslander and D. A. Buchsbaum, Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 733-734.
Zentralblatt MATH: 0084.26504
Mathematical Reviews (MathSciNet): MR103906
Digital Object Identifier: doi:10.1073/pnas.45.5.733
[AR] M. Auslander and D. S. Rim, Ramification index and multiplicity, Illinois J. Math. 7 (1963), 566-581.
Zentralblatt MATH: 0123.03703
Mathematical Reviews (MathSciNet): MR155853
Project Euclid: euclid.ijm/1255645095
[Bs1] H. Bass, On the ubiquity of Gorensten rings, Math. Z. 82 (1963), 8-28.
Zentralblatt MATH: 0112.26604
Mathematical Reviews (MathSciNet): MR153708
Digital Object Identifier: doi:10.1007/BF01112819
[Bs2] H. Bass, Euler characteristics and characters of discrete groups, preprint.
Zentralblatt MATH: 0365.20008
Mathematical Reviews (MathSciNet): MR432781
Digital Object Identifier: doi:10.1007/BF01390137
[Bg] G. M. Bergman, Lecture 26 in Lectures on curves on an algebraic surface by D. Mumford, Ann. of Math. Studies No. 59, Princeton Univ. Press, Princeton, N. J., 1966.
Zentralblatt MATH: 0187.42701
Mathematical Reviews (MathSciNet): MR209285
[Bt] M.-J. Bertin, Anneaux d'invariants de polynômes, en caractéristique p, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), 653-656.
Zentralblatt MATH: 0147.29503
Mathematical Reviews (MathSciNet): MR215826
[Bor] A. Borel, Linear algebraic groups, Benjamin, New York, 1969.
Zentralblatt MATH: 0186.33201
Mathematical Reviews (MathSciNet): MR251042
[Bou1] J.-F. Boutot, Frobenius et cohomologie locale, Séminaire Bourbaki, Exp. 453, 1974.
Zentralblatt MATH: 0339.14009
[Bou2] J.-F. Boutot, Schéma de Picard local, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), 691-694.
Zentralblatt MATH: 0285.14010
Mathematical Reviews (MathSciNet): MR345968
[BE1] D. A. Buchsbaum and D. Eisenbud, Lifting modules and a theorem on finite free resolutions, Ring Theory, Academic Press, New York, 1972, pp. 63-74.
Zentralblatt MATH: 0248.13011
Mathematical Reviews (MathSciNet): MR340343
[BE2] D. A. Buchsbaum and D. Eisenbud, Remarks on ideals and resolutions, Symposia Math. 11 (1973), 193-204.
Zentralblatt MATH: 0294.13009
Mathematical Reviews (MathSciNet): MR337946
[BE3] D. A. Buchsbaum and D. Eisenbud, Some structure theorems for finite free resolutions, Advances in Math. 12 (1974), 84-139.
Zentralblatt MATH: 0297.13014
Mathematical Reviews (MathSciNet): MR340240
Digital Object Identifier: doi:10.1016/S0001-8708(74)80019-8
[Ch] W. L. Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799-822.
Zentralblatt MATH: 0146.17203
Mathematical Reviews (MathSciNet): MR171804
Digital Object Identifier: doi:10.2307/2373158
[C] I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106.
Zentralblatt MATH: 0060.07001
Mathematical Reviews (MathSciNet): MR16094
Digital Object Identifier: doi:10.1090/S0002-9947-1946-0016094-3
[DPr] C. DeConcini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), 330-354.
Zentralblatt MATH: 0347.20025
Mathematical Reviews (MathSciNet): MR422314
Digital Object Identifier: doi:10.1016/S0001-8708(76)80003-5
[D] M. Demazure, Désingularisations de variétés de Schubert généralisés, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88.
Zentralblatt MATH: 0312.14009
Mathematical Reviews (MathSciNet): MR354697
[DC] J. A. Dieudonné and J. B. Carrell, Invariant theory, old and new, Academic Press, New York, 1971.
Zentralblatt MATH: 0196.05802
Mathematical Reviews (MathSciNet): MR279102
[EH] J. A. Eagon and M. Hochster, R-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61-71.
Zentralblatt MATH: 0278.13008
Mathematical Reviews (MathSciNet): MR337934
Digital Object Identifier: doi:10.1093/qmath/25.1.61
[EN] 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.
Zentralblatt MATH: 0106.25603
Mathematical Reviews (MathSciNet): MR142592
Digital Object Identifier: doi:10.1098/rspa.1962.0170
[Ei] D. Eisenbud, Some directions of recent progress in commutative algebra, Proc. Sympos. Pure Math., vol 29, Amer. Math. Soc., Providence, R.I., 1975, pp. 111-128.
Zentralblatt MATH: 0308.13002
Mathematical Reviews (MathSciNet): MR384767
[EE] D. Eisenbud and E. G. Evans, A generalized principal ideal theorem, Nagoya Math. J. 62 (1976), 41-53.
Zentralblatt MATH: 0313.13018
Mathematical Reviews (MathSciNet): MR409440
Project Euclid: euclid.nmj/1118795845
[FR] D. Ferrand and M. Raynaud, Fibres formelles d'un anneau local noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295-311.
Zentralblatt MATH: 0204.36601
Mathematical Reviews (MathSciNet): MR272779
[FFGR] R. Fossum, H.-B. Foxby, P. Griffith, and I. Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 193-215.
Zentralblatt MATH: 0321.13013
Mathematical Reviews (MathSciNet): MR396529
Digital Object Identifier: doi:10.1007/BF02684302
[FG] R. Fossum and P. Griffith, Complete local factorial rings which are not Cohen-Macaulay in characteristic p, Ann. Sci. École Norm. Sup. (4) 8 (1975), 189-200.
Zentralblatt MATH: 0303.13015
Mathematical Reviews (MathSciNet): MR382257
[F1] H.-B. Foxby, On the µi in a minimal injective resolution, Math. Scand. 29 (1971), 175-186.
Zentralblatt MATH: 0235.13006
Mathematical Reviews (MathSciNet): MR309919
[F2] H.-B. Foxby, Applications of isomorphisms between complexes, Copenhagen University, preprint.
[F3] H.-B. Foxby, On the µi in a minimal injective resolution. II, Copenhagen Math. Inst. Preprint Series, No. 20. 1976.
Zentralblatt MATH: 0373.13013
Mathematical Reviews (MathSciNet): MR476801
[FT] H.-B. Foxby and A. Thorup, Minimal injective resolutions under flat base change, Copenhagen Math. Inst. Preprint Series, No. 22, 1976.
Zentralblatt MATH: 0381.13006
Mathematical Reviews (MathSciNet): MR453724
Digital Object Identifier: doi:10.1090/S0002-9939-1977-0453724-1
[FK] E. Freitag and R. Kiel, Algebraische Eigenschaften der lokalen Ringe in den Spitzen der Hilbertschen Modulgruppen, Invent. Math. 24 (1974), 121-148.
Zentralblatt MATH: 0304.32018
Mathematical Reviews (MathSciNet): MR347823
Digital Object Identifier: doi:10.1007/BF01404302
[Fu] W. Fulton, Algebraic curves, Benjamin, New York, 1969.
Zentralblatt MATH: 0181.23901
Mathematical Reviews (MathSciNet): MR313252
[Gil] R. Gilmer, Dimension sequences of commutative rings, Ring Theory, Proc. of the Oklahoma Conference, Marcel Dekker, New York, 1974.
Zentralblatt MATH: 0275.13014
Mathematical Reviews (MathSciNet): MR332767
[Gr] P. Griffith, A representation theorem for complete local rings, J. Pure Appl. Algebra 7 (1976), 303-315.
Zentralblatt MATH: 0338.13023
Mathematical Reviews (MathSciNet): MR412176
Digital Object Identifier: doi:10.1016/0022-4049(76)90056-6
[G] A. Grothendieck (with J. Dieudonné), Éléments de géomtrie algébrique. I-IV, Inst. Hautes Études Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32, (1960)-(1967).
[GH] A. Grothendieck (notes by R. Hartshorne), Local cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag, Berlin and New York, 1967.
Zentralblatt MATH: 0185.49202
Mathematical Reviews (MathSciNet): MR224620
[GN] T. Gulliksen and O. Negard, Un complexe résolvent pour certains idéaux déterminantiels, C. R. Acad. Sci. Paris Ser. A-B 274 (1972), 16-19.
Zentralblatt MATH: 0238.13015
Mathematical Reviews (MathSciNet): MR296063
[Ha] W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. 102 (1975), 67-83.
Zentralblatt MATH: 0316.14016
Mathematical Reviews (MathSciNet): MR382294
Digital Object Identifier: doi:10.2307/1970974
[HO] R. Hartshorne and A. Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415-437.
Zentralblatt MATH: 0286.13013
Mathematical Reviews (MathSciNet): MR347821
Digital Object Identifier: doi:10.1080/00927877408548627
[HS] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic p, preprint.
Zentralblatt MATH: 0362.14002
Mathematical Reviews (MathSciNet): MR441962
Digital Object Identifier: doi:10.2307/1971025
[Hi] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic 0, Ann. of Math. 79 (1964), 205-326.
Zentralblatt MATH: 0122.38603
Mathematical Reviews (MathSciNet): MR199184
Digital Object Identifier: doi:10.2307/1970547
[Ho1] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1976), 318-337.
Zentralblatt MATH: 0237.14019
Mathematical Reviews (MathSciNet): MR304376
Digital Object Identifier: doi:10.2307/1970791
[Ho2] M. Hochster, Cohen-Macaulay modules, Proc. Kansas Commutative Algebra Conference, Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973, pp. 120-152.
Zentralblatt MATH: 0254.13030
Mathematical Reviews (MathSciNet): MR340251
[Ho3] M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25-43.
Zentralblatt MATH: 0245.13012
Mathematical Reviews (MathSciNet): MR349656
Project Euclid: euclid.nmj/1118794784
[Ho4] M. Hochster, Grassmannians and their Schubert subvarieties are arithmetically Cohen-Macaulay, J. Algebra 25 (1973), 40-57.
Zentralblatt MATH: 0256.14024
Mathematical Reviews (MathSciNet): MR314833
Digital Object Identifier: doi:10.1016/0021-8693(73)90074-4
[Ho6] M. Hochster, Deep local rings, preliminary preprint (no longer available), Aarhus University preprint series, 1973.
Mathematical Reviews (MathSciNet): MR645209
Zentralblatt MATH: 0479.13005
[Ho6] M. Hochster, Grade-sensitive modules and perfect modules, Proc. London Math. Soc. (3) 29 (1974), 55-76.
Zentralblatt MATH: 0289.13006
Mathematical Reviews (MathSciNet): MR374118
Digital Object Identifier: doi:10.1112/plms/s3-29.1.55
[Ho7] M. Hochster, Constraints on systems of parameters, Ring Theory, Proc. of the Oklahoma Conference, Marcel Dekker, New York, 1974, pp. 121-161.
Zentralblatt MATH: 0278.13007
Mathematical Reviews (MathSciNet): MR330156
[Ho8] M. Hochster, The equicharacteristic case of some homological conjectures on local rings, Bull. Amer. Math. Soc. 80 (1974), 683-686.
Zentralblatt MATH: 0289.13007
Mathematical Reviews (MathSciNet): MR342510
Digital Object Identifier: doi:10.1090/S0002-9904-1974-13548-2
Project Euclid: euclid.bams/1183535698
[Ho9] M. Hochster, 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, R. I., 1975.
Zentralblatt MATH: 0302.13003
Mathematical Reviews (MathSciNet): MR371879
[Ho10] M. Hochster, Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors, Proc. of the Queen's University Commutative Algebra Conference (Kingston, Ontario, Canada, 1975), Queen's Papers in Pure and Appl. Math. 42, 106-195.
Zentralblatt MATH: 0342.13009
Mathematical Reviews (MathSciNet): MR396544
[Ho11] M. Hochster, An obstruction to lifting cyclic modules, Pacific J. Math. 61 (1975), 457-463.
Zentralblatt MATH: 0335.13011
Mathematical Reviews (MathSciNet): MR412169
Project Euclid: euclid.pjm/1102868039
[Ho12] M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Proc. of the Second Oklahoma Ring Theory Conference (March, 1976), Marcel Dekker, New York, 1977.
Zentralblatt MATH: 0351.13009
Mathematical Reviews (MathSciNet): MR441987
[Ho13] M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488.
Zentralblatt MATH: 0369.13005
Mathematical Reviews (MathSciNet): MR463152
Digital Object Identifier: doi:10.1090/S0002-9947-1977-0463152-5
[Ho14] M. Hochster, Properties of Noetherian rings stable under general grade reduction, Arch. Math. 24 (1973), 53-65.
Zentralblatt MATH: 0268.13013
Mathematical Reviews (MathSciNet): MR330147
Digital Object Identifier: doi:10.1007/BF01228228
[HE] 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.
Zentralblatt MATH: 0244.13012
Mathematical Reviews (MathSciNet): MR302643
Digital Object Identifier: doi:10.2307/2373744
[HR1] M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115-175.
Zentralblatt MATH: 0289.14010
Mathematical Reviews (MathSciNet): MR347810
Digital Object Identifier: doi:10.1016/0001-8708(74)90067-X
[HR2] M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 20 (1976), 117-172.
Zentralblatt MATH: 0348.13007
Mathematical Reviews (MathSciNet): MR417172
Digital Object Identifier: doi:10.1016/0001-8708(76)90073-6
[HP] W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. I, II, Cambridge Univ. Press, 1947 and 1952.
Zentralblatt MATH: 0055.38705
[Iv] B. Iversen, Amplitude inequalities for complexes, Aarhus University Preprint Series No. 36 (1976/77).
Zentralblatt MATH: 0356.13005
Mathematical Reviews (MathSciNet): MR568903
[K1] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1970. Revised ed., 1974.
Zentralblatt MATH: 0203.34601
Mathematical Reviews (MathSciNet): MR254021
[K2] I. Kaplansky, Commutative rings, First Jeffery-Williams Lecture, University of Manitoba, 1968, Canadian Math. Congress; reproduced in Proc. Kansas Commutative Algebra Conference, Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973.
Zentralblatt MATH: 0254.13003
Mathematical Reviews (MathSciNet): MR360550
[Ke1] G. Kempf, Images of homogeneous vector bundles and varieties of complexes, Bull. Amer. Math. Soc. 81 (1975), 900-901.
Zentralblatt MATH: 0322.14020
Mathematical Reviews (MathSciNet): MR384817
Digital Object Identifier: doi:10.1090/S0002-9904-1975-13878-X
Project Euclid: euclid.bams/1183537242
[Ke2] G. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229-239.
Zentralblatt MATH: 0338.14015
Mathematical Reviews (MathSciNet): MR424841
Digital Object Identifier: doi:10.1007/BF01390321
[Ke3] G. Kempf, Some quotient varieties have rational singularities, preprint.
Zentralblatt MATH: 0385.14016
Mathematical Reviews (MathSciNet): MR491675
Digital Object Identifier: doi:10.1307/mmj/1029001952
Project Euclid: euclid.mmj/1029001952
[KKMS] G. Kempf, F. Knudsen, D. Mumford and B. St. Donat, Toroidal embeddings. I, Lecture Notes in Math., vol. 339, Springer-Verlag, Berlin and New York, 1973.
Zentralblatt MATH: 0271.14017
Mathematical Reviews (MathSciNet): MR335518
[Kl] J. Kleppe, Liftings (deformations) of graded algebras, Oslo University Math. Preprint Series, No. 101 (1976).
Zentralblatt MATH: 0436.14004
Mathematical Reviews (MathSciNet): MR580600
[Kr] W. Krull, Primidealketten in allgemeinen Ringbereichen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. 7 (1928).
[Ku] R. E. Kutz, Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups, Thesis, University of Minnesota, 1977; and Trans. Amer. Math. Soc. 194 (1974), 115-129.
Zentralblatt MATH: 0288.13004
Mathematical Reviews (MathSciNet): MR352082
Digital Object Identifier: doi:10.1090/S0002-9947-1974-0352082-2
[Lk] D. Laksov, The arithmetic Cohen-Macaulay character of the Schubert schemes, Acta Math. 129 (1972), 1-9.
Zentralblatt MATH: 0233.14012
Mathematical Reviews (MathSciNet): MR382297
Digital Object Identifier: doi:10.1007/BF02392211
[Las] A. Lascoux, Polynômes symétriques, foncteurs de Schur, et grassmanniennes, Thesis, Université Paris VII, 1977.
[Lau] O. A. Laudal, Sections of functors and the problem of lifting (deforming) algebraic structures. I, II, III, Oslo University Math. Preprint Series, No. 18 (1975), No. 24 (1975), No. 6 (1976).
[LV] G. Levin and W. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315-328.
Zentralblatt MATH: 0161.03903
Mathematical Reviews (MathSciNet): MR230715
Project Euclid: euclid.pjm/1102986272
[Lic] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220-226.
Zentralblatt MATH: 0139.26601
Mathematical Reviews (MathSciNet): MR188249
Project Euclid: euclid.ijm/1256055103
[Lip] J. Lipman, Unique factorization in complete local rings, Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, R. I., 1975.
Zentralblatt MATH: 0306.13005
Mathematical Reviews (MathSciNet): MR374125
[Mac1] F. S. Macaulay, Algebraic theory of modular systems, Cambridge Tracts No. 19, Cambridge, 1916.
[Mac2] F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.
[Mal1] M.-P. Malliavin-Brameret, Une remarque sur les anneaux locaux reguliers, Sem. Dubreil-Pisot, 24e annee, 1970/71, no. 13.
Zentralblatt MATH: 0244.13017
[Mal2] M.-P. Malliavin-Brameret, Structure d'anneaux locaux reguliers et caracteristiques d'Euler-Poincaré, Bull. Soc. Math. France (2) 97 (1973), 81-88.
Zentralblatt MATH: 0304.13019
Mathematical Reviews (MathSciNet): MR330153
[Mat] E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511-528.
Zentralblatt MATH: 0084.26601
Mathematical Reviews (MathSciNet): MR99360
Project Euclid: euclid.pjm/1103039896
[M] H. Matsumura, Commutative algebra, Benjamin, New York, 1970.
Zentralblatt MATH: 0211.06501
Mathematical Reviews (MathSciNet): MR266911
[MS] P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, London Math. Soc. Lecture Note Series 3, Cambridge Univ. Press, 1971.
Zentralblatt MATH: 0217.46702
Mathematical Reviews (MathSciNet): MR301635
[Mol] T. Molien, Über die Invarianten der linearen Substitutionsgruppen, Sitzungsber. König. Preuss. Akad. Wiss. (1897), 1152-1156.
[Mo] S. Mori, On affine cones associated with polarized varieties, preprint.
Zentralblatt MATH: 0324.13019
Mathematical Reviews (MathSciNet): MR439859
[Mu1] D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin and New York, 1965.
Zentralblatt MATH: 0147.39304
Mathematical Reviews (MathSciNet): MR214602
[Mu2] D. Mumford, Introduction to algebraic geometry, Lecture Notes, Harvard University, 1966.
Mathematical Reviews (MathSciNet): MR209285
[Mu3] D. Mumford, Hilbert's fourteenth problem–the finite generation of subrings such as rings of invariants, Proc. Sympos. Pure Math., vol. 28 (Part 2) Amer. Math. Soc., Providence, R. I., 1976, pp. 431-444.
Zentralblatt MATH: 0341.14019
Mathematical Reviews (MathSciNet): MR435076
[Mu4] D. Mumford, Algebraic geometry. I: Complex projective varieties, Die Grundlehren der math. Wissenschaften, Band 221, Springer-Verlag, Berlin and New York, 1976.
Zentralblatt MATH: 0356.14002
Mathematical Reviews (MathSciNet): MR453732
[Mus] C. Musili, Postulation formula for Schubert varieties, J. Indian Math. Soc. 36 (1972), 143-171.
Zentralblatt MATH: 0277.14021
Mathematical Reviews (MathSciNet): MR330177
[N1] M. Nagata, Complete reducibility of rational representation of a matric group, J. Math. Kyoto Univ. 1 (1961), 87-99.
Zentralblatt MATH: 0106.25201
Mathematical Reviews (MathSciNet): MR142667
Project Euclid: euclid.kjm/1250525107
[N2] M. Nagata, Local rings, Interscience, New York, 1962.
Zentralblatt MATH: 0123.03402
Mathematical Reviews (MathSciNet): MR155856
[N3] M. Nagata, Lectures on the fourteenth problem of Hilbert, Lecture Notes in Math., vol. 31, Tata Institute, Bombay, 1965.
Zentralblatt MATH: 0182.54101
Mathematical Reviews (MathSciNet): MR215828
[Nas] H.-J. Nastold, Zur Serreschen Multiplizitätstheorie in der arithmetischen Geometrie, Math. Ann. 143 (1973), 323-395.
Zentralblatt MATH: 0097.02303
Mathematical Reviews (MathSciNet): MR125858
Digital Object Identifier: doi:10.1007/BF01470614
[PS1] C. Peskine and L. Szpiro, Notes sur un air de H. Bass, Unpublished preprint (Brandeis University).
[PS2] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 323-395.
Zentralblatt MATH: 0268.13008
Mathematical Reviews (MathSciNet): MR374130
Digital Object Identifier: doi:10.1007/BF02685877
[PS3] C. Peskine and L. Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A-B 278 (1974), 1421-1424.
Zentralblatt MATH: 0281.13004
Mathematical Reviews (MathSciNet): MR349659
[PP] G. Pfister and D. Popescu, Die strenge Approximationseigenschaft lokaler Ring, Invent. Math. 30 (1975), 145-174.
Zentralblatt MATH: 0293.13011
Mathematical Reviews (MathSciNet): MR379490
Digital Object Identifier: doi:10.1007/BF01425506
[Po] K. Y. Poon, A resolution of certain perfect ideals defined by some matrices, Thesis, University of Minnesota, 1973.
Mathematical Reviews (MathSciNet): MR2623237
[Pr] C. Procesi, The invariant theory of n × n matrices, Advances in Math. 19 (1976), 306-381.
Zentralblatt MATH: 0331.15021
Mathematical Reviews (MathSciNet): MR419491
Digital Object Identifier: doi:10.1016/0001-8708(76)90027-X
[R] M. Raynaud, Anneaux locaux Henséliens, Lecture Notes in Math., vol. 169, New York, 1970.
Zentralblatt MATH: 0203.05102
Mathematical Reviews (MathSciNet): MR277519
[Rees] D. Rees, The grade of an ideal or module, Proc. Cambridge Philos. Soc. 53 (1957), 28-42.
Zentralblatt MATH: 0079.26602
Mathematical Reviews (MathSciNet): MR82967
Digital Object Identifier: doi:10.1017/S0305004100031960
[Rei] G. Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), 30-49.
Zentralblatt MATH: 0345.13017
Mathematical Reviews (MathSciNet): MR407036
Digital Object Identifier: doi:10.1016/0001-8708(76)90114-6
[Ro] P. Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. École Norm. Sup. (4) 9 (1976), 103-106.
Zentralblatt MATH: 0326.13004
Mathematical Reviews (MathSciNet): MR399075
[Sam] P. Samuel, Lectures on unique factorization domains, Lecture Notes in Math., vol. 30, Tata Institute, Bombay, 1964.
Zentralblatt MATH: 0184.06601
Mathematical Reviews (MathSciNet): MR214579
[Sei1] A. Seidenberg, A note on the dimension theory of rings, Pacific J. Math. 3 (1953), 505-512.
Zentralblatt MATH: 0052.26902
Mathematical Reviews (MathSciNet): MR54571
Project Euclid: euclid.pjm/1103051409
[Sei2] A. Seidenberg, On the dimension theory of rings. II, Pacific J. Math. 4 (1954), 603-614.
Zentralblatt MATH: 0057.26802
Mathematical Reviews (MathSciNet): MR65540
Project Euclid: euclid.pjm/1103044693
[S1] J.-P. Serre, Sur la dimension homologique des anneaux et des modules Noethériens, Proc. Internat. Sympos. Algebraic Number Theory, Tokyo, 1955, pp. 175-189.
Zentralblatt MATH: 0073.26004
Mathematical Reviews (MathSciNet): MR86071
[S2] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955), 197-278.
Zentralblatt MATH: 0067.16201
Mathematical Reviews (MathSciNet): MR68874
Digital Object Identifier: doi:10.2307/1969915
[S3] J.-P. Serre, Exemples de variétés projectives en caractéristique p non relèvables en caractéristique zéro, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 108-109.
Zentralblatt MATH: 0100.16701
Mathematical Reviews (MathSciNet): MR132067
Digital Object Identifier: doi:10.1073/pnas.47.1.108
[S4] J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin and New York, 1965.
Zentralblatt MATH: 0142.28603
Mathematical Reviews (MathSciNet): MR201468
[Sh] R. Y. Sharp, Local cohomology theory in commutative algebra, Quart. J. Math. Oxford Ser. (2) 21 (1970), 425-434.
Zentralblatt MATH: 0204.06003
Mathematical Reviews (MathSciNet): MR276217
Digital Object Identifier: doi:10.1093/qmath/21.4.425
[ST] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304.
Zentralblatt MATH: 0055.14305
Mathematical Reviews (MathSciNet): MR59914
Digital Object Identifier: doi:10.4153/CJM-1954-028-3
[St1] R. Stanley, Cohen-Macaulay rings and constructible polytopes, Bull. Amer. Math. Soc. 81 (1975), 133-135.
Zentralblatt MATH: 0304.52005
Mathematical Reviews (MathSciNet): MR364231
Digital Object Identifier: doi:10.1090/S0002-9904-1975-13670-6
Project Euclid: euclid.bams/1183536251
[St2] R. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), 135-142.
Zentralblatt MATH: 0308.52009
Mathematical Reviews (MathSciNet): MR458437
[Sv] T. Svanes, Coherent cohomology on Schubert subschemes of flag schemes and applications, Advances in Math. 14 (1974), 369-453.
Zentralblatt MATH: 0308.14008
Mathematical Reviews (MathSciNet): MR419469
Digital Object Identifier: doi:10.1016/0001-8708(74)90039-5
[T] D. Taylor, Ideals generated by monomials in an R-sequence, Thesis, University of Chicago, 1966.
Mathematical Reviews (MathSciNet): MR2611561
[Weil] A. Weil, Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Publ., vol. 29, Amer. Math. Soc., Providence, R. I., 1946. Revised ed. 1962.
Zentralblatt MATH: 0063.08198
Mathematical Reviews (MathSciNet): MR144898
[Weyl] H. Weyl, The classical groups, Princeton Univ. Press, Princeton, N. J., 1946.
Zentralblatt MATH: 1024.20502
Mathematical Reviews (MathSciNet): MR1488158
[Z1] O. Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. 40 (1939), 639-689.
Zentralblatt MATH: 0021.25303
Mathematical Reviews (MathSciNet): MR159
Digital Object Identifier: doi:10.2307/1968949
[Z2] O. Zariski, Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc. 53 (1943), 490-542.
Zentralblatt MATH: 0061.33004
Mathematical Reviews (MathSciNet): MR8468
Digital Object Identifier: doi:10.1090/S0002-9947-1943-0008468-9
[Z3] O. Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947), 1-52.
Zentralblatt MATH: 0031.26101
Mathematical Reviews (MathSciNet): MR21694
Digital Object Identifier: doi:10.1090/S0002-9947-1947-0021694-1
[ZS] O. Zariski and P. Samuel, Commutative algebra. Vols. 1, 2, Van Nostrand, Princeton, N. J., 1958, 1960.
Zentralblatt MATH: 0121.27801
Bulletin of the American Mathematical Society