Determinantal varieties, monomial semigroups, and algebras associated with ideals

Barshay, Jacob

doi:10.1090/S0002-9939-1973-0318137-8

Determinantal varieties, monomial semigroups, and algebras associated with ideals
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Proc. Amer. Math. Soc. 40 (1973), 16-22 Request permission

Abstract:

This paper is concerned with the Rees and symmetric algebras of powers of ideals generated by $A$-sequences. These algebras are represented as quotients of polynomial rings over $A$ by ideals defined by minors of matrices. Their Krull dimensions are computed when $A$ is a finite domain over a field. When $A$ is a polynomial ring over a Cohen-Macaulay ring ${A_0}$ and the $A$-sequence consists of indeterminates, the Rees algebra is shown to be Cohen-Macaulay. If furthermore ${A_0}$ is a finite domain over a field, the symmetric algebra is shown to be Cohen-Macaulay only for $A$-sequences and squares of $A$-sequences of length two. Connections with algebras generated by monomials and Veronese varieties are pointed out.

References

Jacob Barshay, Graded algebras of powers of ideals generated by $A$-sequences, J. Algebra 25 (1973), 90–99. MR 332748, DOI 10.1016/0021-8693(73)90076-8
J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
L. S. Goddard, Bases for the prime ideals associated with certain classes of algebraic varieties, Proc. Cambridge Philos. Soc. 39 (1943), 35–48. MR 7619
L. S. Goddard, Prime ideals and postulation formulae, Proc. Cambridge Philos. Soc. 44 (1948), 43–49. MR 23094
W. Gröbner, Über Veronesesche Varietäten und deren Projektionen, Arch. Math. (Basel) 16 (1965), 257–264 (German). MR 183712, DOI 10.1007/BF01220031
M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. MR 304376, DOI 10.2307/1970791
M. Hochster and John A. Eagon, A class of perfect determinantal ideals, Bull. Amer. Math. Soc. 76 (1970), 1026–1029. MR 266912, DOI 10.1090/S0002-9904-1970-12543-5
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
D. G. Northcott, Semi-regular rings and semi-regular ideals, Quart. J. Math. Oxford Ser. (2) 11 (1960), 81–104. MR 114835, DOI 10.1093/qmath/11.1.81
D. G. Northcott, Some remarks on the theory of ideals defined by matrices, Quart. J. Math. Oxford Ser. (2) 14 (1963), 193–204. MR 151482, DOI 10.1093/qmath/14.1.193
D. Rees, The grade of an ideal or module, Proc. Cambridge Philos. Soc. 53 (1957), 28–42. MR 82967, DOI 10.1017/s0305004100031960
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
D. W. Sharpe, The syzygies and semi-regularity of certain ideals defined by matrices, Proc. London Math. Soc. (3) 15 (1965), 645–679. MR 188245, DOI 10.1112/plms/s3-15.1.645

Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups