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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of ideals in graded ordinal Hodge algebras . 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 which, under certain conditions, is sharp for large. Frequently one observes that . 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".

**[AS 1]**J. F. Andrade and A. Simis,*On ideals of minors fixing a submatrix*, J. Algebra**102**(1986), no. 1, 246–259. MR**853243**, 10.1016/0021-8693(86)90140-7**[AS 2]**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**, 10.1007/BF01324720**[BV]**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****[DEP]**C. De Concini, D. Eisenbud and C. Procesi,*Hodge algebras*, Astérisque**91**(1982).**[EH]**David Eisenbud and Craig Huneke,*Cohen-Macaulay Rees algebras and their specialization*, J. Algebra**81**(1983), no. 1, 202–224. MR**696134**, 10.1016/0021-8693(83)90216-8**[Ha]**Robin Hartshorne,*Cohomological dimension of algebraic varieties*, Ann. of Math. (2)**88**(1968), 403–450. MR**0232780****[HE]**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**0302643****[HSV 1]**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**, 10.1016/0021-8693(87)90194-3**[HSV 2]**-,*The arithmetic of normal Rees algebras*, preprint.**[Hu 1]**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**, 10.1016/0021-8693(80)90179-9**[Hu 2]**Craig Huneke,*Powers of ideals generated by weak 𝑑-sequences*, J. Algebra**68**(1981), no. 2, 471–509. MR**608547**, 10.1016/0021-8693(81)90276-3**[HuSV]**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**, 10.1090/conm/088/999985**[Ma]**Hideyuki Matsumura,*Commutative algebra*, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR**575344****[Sh]**D. W. Sharpe,*On certain polynomial ideals defined by matrices*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 155–175. MR**0163927****[ST]**A. Simis and Ngô Vi\cfudot{e}t Trung,*The divisor class group of ordinary and symbolic blow-ups*, Math. Z.**198**(1988), no. 4, 479–491. MR**950579**, 10.1007/BF01162869**[Sv]**Torgny Svanes,*Coherent cohomology on Schubert subschemes of flag schemes and applications*, Advances in Math.**14**(1974), 369–453. MR**0419469**

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

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

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