zbMATH — the first resource for mathematics

Singularities of certain ladder determinantal varieties. (English) Zbl 0842.13008
A noetherian local ring \(R\) of prime characteristic is said to be \(F\)-rational if every parameter ideal is tighly closed. In an earlier paper, the second author [“\(F\)-rational rings have rational singularities” (preprint)] showed that \(F\)-rational type implies rational singularities. A. Conca and J. Herzog [“Determinantal rings of one-sided ladders are \(F\)-rational” (preprint)] used this result to prove that one-sided ladder determinantal ideals have rational singularities. This can be deduced also from works of Mulay and Ramanathan on Schubert varieties. Since the classical determinantal varieties have rational singularities, one may ask whether this holds for the larger class of ladder determinantal varieties. The main result of this paper confirms this question for complete intersection ladder determinantal varieties by showing that all ideals of their coordinate rings are tightly closed.

13C40 Linkage, complete intersections and determinantal ideals
14B05 Singularities in algebraic geometry
14M12 Determinantal varieties
13B22 Integral closure of commutative rings and ideals
Full Text: DOI
[1] Abhyankar, S., Enumerative combinatorics of Young tableaux, (1988), Marcel Dekker New York · Zbl 0643.05001
[2] Boutôt, J.-F., Singularités rationelles et quotients par LES groupes réductifs, Invent. math., 88, 65-68, (1987) · Zbl 0619.14029
[3] Conca, A., Ladder determinantal rings, J. pure appl. algebra, 98, 119-134, (1995) · Zbl 0842.13007
[4] Conca, A.; Herzog, J., Determinantal rings of one-sided ladders are F-rational, (1993), preprint
[5] Fedder, R., F-purity and rational singularity, Trans. amer. math. soc., 278, 461-480, (1983) · Zbl 0519.13017
[6] Fedder, R.; Watanabe, K., A characterization of F-regularity in terms of F-purity, (), 227-245 · Zbl 0738.13004
[7] Herzog, J., Ladder determinantal ideals, (), (reporting on joint work with Trung and Conca), Special Talk
[8] Hibi, T., Distributive lattices, affine semigroup rings, and algebras with straightening laws, (), 93-109
[9] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of math., 2, 96, 318-337, (1972) · Zbl 0233.14010
[10] Hochster, M.; Huneke, C., Tight closure and strong F-regularity, Mémoires de la soc. math. de France, 38, 119-133, (1989) · Zbl 0699.13003
[11] Hochster, M.; Huneke, C., Tight closure, invariant theory and the briançon-skoda theorem, J. amer. math. soc., 3, 31-116, (1990) · Zbl 0701.13002
[12] M. Hochster and C. Huneke, Tight closures of parameter ideals and splitting in module finite extensions, preprint. · Zbl 0832.13007
[13] M. Hochster and C. Huneke, F-regularity, test elements, and smooth base change, preprint. · Zbl 0844.13002
[14] Hochster, M.; Huneke, C., Tightly closed ideals, Bull. amer. math. soc., 18, 45-48, (1988) · Zbl 0674.13003
[15] Hochster, M.; Roberts, J.L., The purity of Frobenius and local cohomology, Adv. in math., 21, 117-172, (1976) · Zbl 0348.13007
[16] Herzog, J.; Trung, N.V., Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. in math, 96, 1-37, (1992) · Zbl 0778.13022
[17] Huneke, C., An algebraist commuting in Berkeley, Math. intelligencer, 11, 40-55, (1989) · Zbl 0697.13001
[18] Mulay, S.B., Determinantal loci and the flag variety, Adv. in math., 74, 1-30, (1989) · Zbl 0693.14021
[19] Narasimhan, H., The irreducibility of ladder determinantal varieties, J. algebra, 102, 162-185, (1986) · Zbl 0604.14045
[20] Ramanathan, A., Schubert varieties are arithmetically Cohen-Macaulay, Invent. math., 80, 283-294, (1985) · Zbl 0541.14039
[21] K.E. Smith, F-rational rings have rational singularities, preprint. · Zbl 0910.13004
[22] Stanley, R., Some combinatorial aspects of the Schubert calculus, (), 217-251
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.