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)



Blow-up of straightening-closed ideals in ordinal Hodge algebras

Authors: Winfried Bruns, Aron Simis and Ngô Viêt Trung
Journal: Trans. Amer. Math. Soc. 326 (1991), 507-528
MSC: Primary 13C05; Secondary 13C13, 13C15, 13H10
MathSciNet review: 1005076
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of ideals $ I$ in graded ordinal Hodge algebras $ A$. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for $ A/{I^n}$ which, under certain conditions, is sharp for $ n$ large. Frequently one observes that $ {I^n}= {I^{(n)}}$. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".

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

  • [AS 1] J. F. Andrade and A. Simis, On ideals of minors fixing a submatrix, J. Algebra 102 (1986), 246-259. MR 853243 (87j:13028)
  • [AS 2] -, Free resolutions of certain codimension three perfect radical ideals, Arch. Math. 53 (1989), 448-460. MR 1019159 (91c:13008)
  • [BV] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math., vol. 1327 (Subseries: IMPA, Rio de Janeiro), Springer, Berlin, Heidelberg, and New York, 1988. MR 986492 (90f:14032)
  • [DEP] C. De Concini, D. Eisenbud and C. Procesi, Hodge algebras, Astérisque 91 (1982).
  • [EH] D. Eisenbud and C. Huneke, Cohen-Macaulay Rees algebras and their specializations, J. Algebra 81 (1983), 202-224. MR 696134 (84h:13030)
  • [Ha] R. Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. 88 (1968), 403-450. MR 0232780 (38:1103)
  • [HE] M. Hochster and J. Eagon, Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. MR 0302643 (46:1787)
  • [HSV 1] J. Herzog, A. Simis and W. Vasconcelos, On the canonical module of the Rees algebra and the associated graded ring of an ideal, J. Algebra 105 (1987), 285-302. MR 873664 (87m:13029)
  • [HSV 2] -, The arithmetic of normal Rees algebras, preprint.
  • [Hu 1] C. Huneke, On the symmetric and Rees algebra of an ideal generated by a $ d$-sequence, J. Algebra 62 (1980), 268-275. MR 563225 (81d:13016)
  • [Hu 2] -, Powers of ideals generated by weak $ d$-sequences, J. Algebra 68 (1981), 471-509. MR 608547 (82k:13003)
  • [HuSV] C. Huneke, A. Simis and W. Vasconcelos, Reduced normal cones are domains, Contemp. Math. (to appear). MR 999985 (90c:13010)
  • [Ma] H. Matsumura, Commutative algebra, 2nd ed., Benjamin/Cummings, Reading, Mass., 1980. MR 575344 (82i:13003)
  • [Sh] D. W. Sharpe, On certain polynomial ideals defined by matrices, Quart. J. Math. Oxford Ser. 15 (1964), 155-175. MR 0163927 (29:1226)
  • [ST] A. Simis and N. V. Trung, The divisor class group of ordinary and symbolic blow-ups, Math. Z. 198 (1988), 479-491. MR 950579 (89i:13015)
  • [Sv] T. Svanes, Coherent cohomology on Schubert subschemes of flag schemes and applications, Adv. in Math. 14 (1974), 369-453. MR 0419469 (54:7490)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13C05, 13C13, 13C15, 13H10

Retrieve articles in all journals with MSC: 13C05, 13C13, 13C15, 13H10

Additional Information

Keywords: Ordinal Hodge algebras, standard monomial, straightening-closed ideal, filtration of powers, Rees algebra, associated graded ring, normal, Cohen-Macaulay, Gorenstein, divisor class group, rank, arithmetical rank, generic matrix, virtual maximal minor
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society