Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The arithmetic perfection of Buchsbaum-Eisenbud varieties and generic modules of projective dimension two


Author: Craig Huneke
Journal: Trans. Amer. Math. Soc. 265 (1981), 211-233
MSC: Primary 13D25; Secondary 13H10, 14M12
DOI: https://doi.org/10.1090/S0002-9947-1981-0607117-X
MathSciNet review: 607117
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the ideals associated with the construction of generic complexes are prime and arithmetically perfect. This is used to construct the generic resolution for modules of projective dimension two.


References [Enhancements On Off] (What's this?)

  • [1] S. Abeasis and A. Del Fra, Young diagrams and ideals of Pfaffians, preprint, Universita Degli Studi di Roma, 1979-1979. MR 560133 (83f:14040)
  • [2] M. Auslander and D. Buchsbaum, Codimension and multiplicity, Ann. of Math. 68 (1958), 625-657. MR 0099978 (20:6414)
  • [3] S. Barger, Generic perfection and the theory of grade, Thesis, Univ. of Minnesota, 1970.
  • [4] J. Barshay, Graded algebras of powers of ideals generated by $ A$-sequences, J. Algebra 25 (1973), 90-99. MR 0332748 (48:11074)
  • [5] Boutot, Pure subrings of rings with rational singularities have rational singularities, preprint.
  • [6] M. Brodmann, The asymptotic nature of the analytic spread, preprint. MR 530808 (81e:13003)
  • [7] D. Buchsbaum and D. Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259-268. MR 0314819 (47:3369)
  • [8] -, Some structure theorems for finite free resolutions, Advances in Math. 12 (1974), 84-139. MR 0340240 (49:4995)
  • [9] D. Buchsbaum and D. S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197-224. MR 0159860 (28:3076)
  • [10] L. Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369-373. MR 0304377 (46:3512)
  • [11] R. Cowsik and M. Nori, On the fibers of blowing up, preprint.
  • [12] C. De Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), 333-354. MR 0422314 (54:10305)
  • [13] C. De Concini, D. Eisenbud and C. Procesi, Algebras with straightening law (in preparation).
  • [14] -, Young diagrams and determinantal ideals, preprint.
  • [15] J. Eagon and M. Hochster, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. MR 0302643 (46:1787)
  • [16] J. Eagon and D. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188-204. MR 0142592 (26:161)
  • [17] -, Generically acyclic complexes and generically perfect ideals, Proc. Roy. Soc. London Ser. A 299 (1967), 147-172. MR 0214586 (35:5435)
  • [18] D. Eisenbud and C. Huneke, On the Cohen-Macaulayness and torsion freeness of $ {\text{g}}{{\text{r}}_I}(R)$, preprint.
  • [19] T. Gulliksen and O. Negard, Un complexe résolvent pour certains idéaux déterminentiels, C. R. Acad. Sci. Paris Ser. A 274 (1972), A16-A19. MR 0296063 (45:5124)
  • [20] M. Hochster, Generically perfect modules are strongly generically perfect, Proc. London Math. Soc. 23 (1971), 477-488. MR 0301002 (46:162)
  • [21] -, Grassmannians and their Schubert subvarieties are arithmetically Cohen-Macaulay, J. Algebra 25 (1973), 40-57. MR 0314833 (47:3383)
  • [22] -, 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. MR 0371879 (51:8096)
  • [23] -, Criteria for equality of ordinary and symbolic powers of primes, Math. Z. 133 (1973), 53-65. MR 0323771 (48:2127)
  • [24] 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. MR 0347810 (50:311)
  • [25] W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, vol. 2, Cambridge Univ. Press, Cambridge, Mass., 1968.
  • [26] C. Huneke, Determinantal ideals and questions related to factoriality, Thesis, Yale Univ., 1978.
  • [27] -, On the symmetric and Rees algebras of an ideal generated by a $ d$-sequence, J. Algebra 62 (1980), 267-275.
  • [28] -, Powers of ideals generated by weak $ d$-sequences, J. Algebra (to appear).
  • [29] -, The theory of $ d$-sequences and powers of ideals, Advances in Math. (to appear).
  • [30] -, Symbolic powers of primes and special graded algebras (to appear).
  • [31] J. Igusa, On the arithmetic normality of the Grassmann variety, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 309-313. MR 0061423 (15:824a)
  • [32] G. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229-239. MR 0424841 (54:12799)
  • [33] -, Images of homogeneous vector bundles and varieties of complexes, Bull. Amer. Math. Soc. 81 (1975), 900-901. MR 0384817 (52:5689)
  • [34] H. Kleppe and D. Laksov, The algebraic structure and deformation of Pfaffian schemes, Institute Mittag-Leffler, Report No. 3, 1979. MR 575789 (82b:14027)
  • [35] R. Kutz, Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups, Trans. Amer. Math. Soc. 194 (1974), 115-129. MR 0352082 (50:4570)
  • [36] D. Laksov, The arithmetic Cohen-Macaulay character of Schubert schemes, Acta Math. 129 (1972), 1-9. MR 0382297 (52:3182)
  • [37] A. Lascoux, Syzygies des variétés déterminantales, Advances in Math. 30 (1978), 202-237. MR 520233 (80j:14043)
  • [38] H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 0266911 (42:1813)
  • [39] C. Musili, Postulation formula for Schubert varieties, J. Indian Math. Soc. 36 (1972), 143-171. MR 0330177 (48:8515)
  • [40] D. G. Northcott, Additional properties of generically acyclic complexes, Quart. J. Math. Oxford Ser. 20 (1969), 65-80. MR 0240086 (39:1440)
  • [41] -, Grade sensitivity and generic perfection, Proc. London Math. Soc. 20 (1970), 597-618. MR 0272771 (42:7652)
  • [42] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 0059889 (15:596a)
  • [43] C. Peskine and L. Szpiro, Dimension projectif finie et cohomologie locale, Inst. Hautes Étude Sci. Publ. Math. 42 (1973), 323-395. MR 0374130 (51:10330)
  • [44] L. Robbiano and G. Valla, Primary powers of a prime ideal, Pacific J. Math. 63 (1976), 491-498. MR 0409492 (53:13247)
  • [45] -, On normal flatness and normal torsion freeness, J. Algebra 43 (1976), 222-229. MR 0427315 (55:349)
  • [46] D. Sharpe, On certain polynomial ideals defined by matrices, Quart. J. Math. Oxford Ser. 15 (1964), 155-175. MR 0163927 (29:1226)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13D25, 13H10, 14M12

Retrieve articles in all journals with MSC: 13D25, 13H10, 14M12


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0607117-X
Keywords: Cohen-Macaulay, determinantal, complex
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society