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

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

MathSciNet review:
1005076

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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), 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*-*sequence*, J. Algebra**62**(1980), 268-275. MR**563225 (81d:13016)****[Hu 2]**-,*Powers of ideals generated by weak*-*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)**

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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