The center of an order with finite global dimension

Ramras, Mark

doi:10.1090/S0002-9947-1975-0374191-5

The center of an order with finite global dimension
HTML articles powered by AMS MathViewer

by Mark Ramras PDF

Trans. Amer. Math. Soc. 210 (1975), 249-257 Request permission

Abstract:

Let $\Lambda$ be a quasi-local ring of global dimension $n < \infty$. Assume that its center $R$ is a noetherian domain, that $\Lambda$ is finitely generated torsion-free as an $R$-module, and that $R$ is an $R$-direct summand of $\Lambda$. Then $R$ is integrally closed in its quotient field $K$ and Macauley of dimension $n$. Furthermore, when $n = 2,\Lambda$ is a maximal $R$-order in the central simple $K$-algebra $\Lambda { \otimes _R}K$. This extends an earlier result of the author, in which $R$ was assumed to have global dimension 2. Examples are given to show that in the above situation $R$ can have infinite global dimension.

References

M. Auslander, Remarks on a theorem of Bourbaki, Nagoya Math. J. 27 (1966), 361–369. MR 193109, DOI 10.1017/S0027763000012113
Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI 10.2307/1970159
Maurice Auslander and Oscar Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24. MR 117252, DOI 10.1090/S0002-9947-1960-0117252-7
Marie-José Bertin, Anneaux d’invariants d’anneaux de polynomes, en caractéristique $p$, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A653–A656 (French). MR 215826
David Eisenbud, Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970), 247–249. MR 262275, DOI 10.1007/BF01350264
K. L. Fields, On the global dimension of skew polynomial rings, J. Algebra 13 (1969), 1–4. MR 245623, DOI 10.1016/0021-8693(69)90002-7
Manabu Harada, Hereditary orders, Trans. Amer. Math. Soc. 107 (1963), 273–290. MR 151489, DOI 10.1090/S0002-9947-1963-0151489-9
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
Christian U. Jensen and Søren Jøndrup, Centres and fixed-point rings of Artinian rings, Math. Z. 130 (1973), 189–197. MR 318222, DOI 10.1007/BF01214261
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
K. R. Nagarajan, Groups acting on Noetherian rings, Nieuw Arch. Wisk. (3) 16 (1968), 25–29. MR 229628
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Mark Ramras, Maximal orders over regular local rings of dimension two, Trans. Amer. Math. Soc. 142 (1969), 457–479. MR 245572, DOI 10.1090/S0002-9947-1969-0245572-2
Mark Ramras, Maximal orders over regular local rings, Trans. Amer. Math. Soc. 155 (1971), 345–352. MR 272808, DOI 10.1090/S0002-9947-1971-0272808-3