Package deal theorems and splitting orders in dimension 1

Levy, Lawrence; Odenthal, Charles

doi:10.1090/S0002-9947-96-01620-0

Package deal theorems and splitting orders in dimension 1
HTML articles powered by AMS MathViewer

by Lawrence S. Levy and Charles J. Odenthal PDF

Trans. Amer. Math. Soc. 348 (1996), 3457-3503 Request permission

Abstract:

Let $\Lambda$ be a module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations $\Lambda _{\mathbf {m}}$, at maximal ideals of $R$, is the family of all localizations $M_{\mathbf {m}}$ of a finitely generated $\Lambda$-module $M$. When $R$ is semilocal we also determine which finitely generated modules over the $J(R)$-adic completion of $\Lambda$ are completions of finitely generated $\Lambda$-modules. If $\Lambda$ is an $R$-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a “splitting order” of $\Lambda$ that can replace maximal orders in many situations in which maximal orders do not exist.

References

Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
Robert M. Guralnick and Lawrence S. Levy, Presentations of modules when ideals need not be principal, Illinois J. Math. 32 (1988), no. 4, 593–653. MR 955382
Robert M. Guralnick and Lawrence S. Levy, Cancellation and direct summands in dimension $1$, J. Algebra 142 (1991), no. 2, 310–347. MR 1127066, DOI 10.1016/0021-8693(91)90310-5
Alex Heller, On group representations over a valuation ring, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1194–1197. MR 125163, DOI 10.1073/pnas.47.8.1194
Jeremy Haefner and Lawrence S. Levy, Commutative orders whose lattices are direct sums of ideals, J. Pure Appl. Algebra 50 (1988), no. 1, 1–20. MR 931905, DOI 10.1016/0022-4049(88)90002-3
M. Hochster, Non-openness of loci in Noetherian rings, Duke Math. J. 40 (1973), 215–219. MR 311653, DOI 10.1215/S0012-7094-73-04020-9
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Daniel Katz, On the number of minimal prime ideals in the completion of a local domain, Rocky Mountain J. Math. 16 (1986), no. 3, 575–578. MR 862283, DOI 10.1216/RMJ-1986-16-3-575
Lawrence S. Levy, Projectives of large uniform-rank, in Krull dimension 1, Bull. London Math. Soc. 21 (1989), no. 1, 57–64. MR 967791, DOI 10.1112/blms/21.1.57
L. S. Levy and C. J. Odenthal, “Krull-Schmidt Theorems in Dimension 1”, Trans. Amer. Math. Soc. 348 (1996), 3391–3455.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
Eben Matlis, $1$-dimensional Cohen-Macaulay rings, Lecture Notes in Mathematics, Vol. 327, Springer-Verlag, Berlin-New York, 1973. MR 0357391, DOI 10.1007/BFb0061666
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
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
Charles J. Odenthal, Presentations over HNP rings with enough invertible ideals and torsionfree cancellation over neoclassical orders, J. Algebra 127 (1989), no. 2, 290–319. MR 1028455, DOI 10.1016/0021-8693(89)90254-8
I. Reiner, Maximal orders, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0393100
J. C. Robson and Lance W. Small, Hereditary prime P.I. rings are classical hereditary orders, J. London Math. Soc. (2) 8 (1974), 499–503. MR 349743, DOI 10.1112/jlms/s2-8.3.499
Roger Wiegand, Noetherian rings of bounded representation type, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 497–516. MR 1015536, DOI 10.1007/978-1-4612-3660-3_{2}7