Book Review
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MathSciNet review:
1568170
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Book Information:
Author:
Winfried Bruns and J\"urgen Herzog
Title:
Cohen-Macaulay rings
Additional book information:
Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993, xi + 403~pp., US$79.95. ISBN 0-521-41068-1.
Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. MR 179211
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
Jean-François Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), no. 1, 65–68 (French). MR 877006, DOI 10.1007/BF01405091
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
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
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, DOI 10.1007/978-3-662-02421-8
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Raymond C. Heitmann, A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 94–97. MR 1197425, DOI 10.1090/S0273-0979-1993-00410-5
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
Melvin Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), no. 2, 503–553. MR 723406, DOI 10.1016/0021-8693(83)90092-3
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 Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, DOI 10.1090/S0894-0347-1990-1017784-6
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, DOI 10.2307/2946563
Melvin Hochster and Craig Huneke, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math. 113 (1995), no. 1, 45–117. MR 1332808, DOI 10.1006/aima.1995.1035
[HH4] -, Tight closure in equal characteristic zero, in preparation.
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
George Kempf, The Hochster-Roberts theorem of invariant theory, Michigan Math. J. 26 (1979), no. 1, 19–32. MR 514958
F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612
[M2] -, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
James R. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1984), no. 1, 113–128. MR 736476, DOI 10.1307/mmj/1029002969
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
Christian Peskine and Lucien Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1421–1424 (French). MR 349659
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, Le théorème d’intersection, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 7, 177–180 (French, with English summary). MR 880574
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, DOI 10.1007/978-1-4612-3660-3_{2}3
[S] Serre J.-P., Algèbre locale. Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965.
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
Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
- [AK]
- Altman, S. and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math., vol. 146, Springer-Verlag, Berlin, Heidelberg, and New York, 1970. MR 0274461 (43:224)
- [Au]
- Auslander, M., Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631-645. MR 0179211 (31:3460)
- [B]
- Bass, H., On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 0153708 (27:3669)
- [Bou]
- Boutot, J.-F., Singularités rationelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65-68. MR 877006 (88a:14005)
- [C]
- Cohen, I.S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106. MR 0016094 (7:509h)
- [F1]
- Fulton, W., Algebraic curves, Benjamin, New York, 1969. MR 0313252 (47:1807)
- [F2]
- -, Intersection theory, Springer-Verlag, Berlin, Heidelberg, and New York, 1984. MR 732620 (85k:14004)
- [Ha]
- Hartshorne, R., Algebraic geometry, Springer-Verlag, New York, Heidelberg, and Berlin, 1977. MR 0463157 (57:3116)
- [He]
- Heitmann, R., A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 94-97. MR 1197425 (93m:13007)
- [Ho1]
- Hochster, M., Topics in the homological theory of modules over commutative rings, CBMS Regional Conf. Ser. in Math., vol. 24, Amer. Math. Soc., Providence, RI, 1975. MR 0371879 (51:8096)
- [Ho2]
- -, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), 503-553. MR 723406 (85j:13021)
- [HoE]
- Hochster, M., 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)
- [HH1]
- -, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116. MR 1017784 (91g:13010)
- [HH2]
- -, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. 135 (1992), 53-89. MR 1147957 (92m:13023)
- [HH3]
- -, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math. (to appear). MR 1332808 (96d:13014)
- [HH4]
- -, Tight closure in equal characteristic zero, in preparation.
- [HR]
- Hochster, M. and J.L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math. 13 (1974), 115-175. MR 0347810 (50:311)
- [Ke]
- Kempf, G., The Hochster-Roberts theorem of invariant theory, Michigan Math. J. 26 (1979), 19-32. MR 514958 (80g:14040)
- [M1]
- Macaulay, F.S., The algebraic theory of modular systems, Cambridge Univ. Press, Cambridge, 1916. MR 1281612 (95i:13001)
- [M2]
- -, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.
- [Mum]
- Mumford, D., Algebraic geometry I complex projective varieties, Springer-Verlag, Berlin, Heidelberg, and New York, 1976. MR 0453732 (56:11992)
- [Mun]
- Munkres, J., Topological results in combinatorics, Michigan Math. J. 31 (1984), 113-128. MR 736476 (85k:13022)
- [PS1]
- Peskine, C. 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)
- [Re]
- Reisner, G., Cohen-Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), 30-49. MR 0407036 (53:10819)
- [Ro1]
- Roberts, P., Le théorème d'intersection, C. R. Acad. Sc. Paris Sér. I Math. 304 (1987), 177-180. MR 880574 (89b:14008)
- [Ro2]
- -, Intersection theorems, Commutative Algebra, Math. Sci. Res. Inst. Publ., vol. 15, Springer-Verlag, New York, Berlin, and Heidelberg, 1989, pp. 417-436. MR 1015532 (90j:13024)
- [S]
- Serre J.-P., Algèbre locale. Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin, Heidelberg, and New York, 1965.
- [St1]
- Stanley, R., Cohen-Macaulay rings and constructible polytopes, Bull. Amer. Math. Soc. 81 (1975), 135-142. MR 0364231 (51:486)
- [St2]
- -, The upper bound conjecture and Cohen-Macaulay rings, Stud. Appl. Math. 54 (1975), 133-142. MR 0458437 (56:16640)
- [St3]
- -, The number of faces of a simplicial convex polytope, Adv. Math. 35 (1980), 236-238. MR 563925 (81f:52014)
Review Information:
Reviewer:
Melvin Hochster
Journal:
Bull. Amer. Math. Soc.
32 (1995), 265-275
DOI:
https://doi.org/10.1090/S0273-0979-1995-00577-X