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

Bruns, Winfried; Simis, Aron; Trung, Ngô Viêt

doi:10.1090/S0002-9947-1991-1005076-8

Blow-up of straightening-closed ideals in ordinal Hodge algebras
HTML articles powered by AMS MathViewer

by Winfried Bruns, Aron Simis and Ngô Viêt Trung

Trans. Amer. Math. Soc. 326 (1991), 507-528

DOI: https://doi.org/10.1090/S0002-9947-1991-1005076-8

PDF | Request permission

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

J. F. Andrade and A. Simis, On ideals of minors fixing a submatrix, J. Algebra 102 (1986), no. 1, 246–259. MR 853243, DOI 10.1016/0021-8693(86)90140-7
J. F. Andrade and A. Simis, Free resolutions of certain codimension three perfect radical ideals, Arch. Math. (Basel) 53 (1989), no. 5, 448–460. MR 1019159, DOI 10.1007/BF01324720
Winfried Bruns and Udo Vetter, Determinantal rings, Monografías de Matemática [Mathematical Monographs], vol. 45, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1988. MR 986492, DOI 10.1007/BFb0080378

Hodge algebras

91

David Eisenbud and Craig Huneke, Cohen-Macaulay Rees algebras and their specialization, J. Algebra 81 (1983), no. 1, 202–224. MR 696134, DOI 10.1016/0021-8693(83)90216-8
Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR 232780, DOI 10.2307/1970720
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
J. Herzog, A. Simis, and W. V. Vasconcelos, On the canonical module of the Rees algebra and the associated graded ring of an ideal, J. Algebra 105 (1987), no. 2, 285–302. MR 873664, DOI 10.1016/0021-8693(87)90194-3

The arithmetic of normal Rees algebras

Craig Huneke, On the symmetric and Rees algebra of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), no. 2, 268–275. MR 563225, DOI 10.1016/0021-8693(80)90179-9
Craig Huneke, Powers of ideals generated by weak $d$-sequences, J. Algebra 68 (1981), no. 2, 471–509. MR 608547, DOI 10.1016/0021-8693(81)90276-3
Craig Huneke, Aron Simis, and Wolmer Vasconcelos, Reduced normal cones are domains, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 95–101. MR 999985, DOI 10.1090/conm/088/999985
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
D. W. Sharpe, On certain polynomial ideals defined by matrices, Quart. J. Math. Oxford Ser. (2) 15 (1964), 155–175. MR 163927, DOI 10.1093/qmath/15.1.155
A. Simis and Ngô Việt Trung, The divisor class group of ordinary and symbolic blow-ups, Math. Z. 198 (1988), no. 4, 479–491. MR 950579, DOI 10.1007/BF01162869
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