Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lattices over orders: finitely presented functors and preprojective partitions

Authors: M. Auslander and S. O. Smalø
Journal: Trans. Amer. Math. Soc. 273 (1982), 433-446
MSC: Primary 16A64
MathSciNet review: 667155
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ R$ is a commutative noetherian equidimensional Gorenstein ring and $ \Lambda $ an $ R$-algebra which is finitely generated as an $ R$-module. A $ \Lambda $-module $ M$ is a lattice if $ {M_{\underline{\underline p} }}$ is $ {\Lambda _{\underline{\underline p} }}$-projective and $ {\text{Ho}}{{\text{m}}_R}{(M,R)_{\underline{\underline p} }}$ is $ \Lambda _{\underline{\underline p} }^{{\text{op}}}$-projective for all nonmaximal prime ideals $ \underline{\underline p} $ in $ R$. We assume that $ \Lambda $ is an $ R$-order in the sense that $ \Lambda $ is a lattice when viewed as a $ \Lambda $-module. The first main result is to show that simple contravariant functors from lattices to abelian groups are finitely presented. This is then applied to showing that if $ R$ is also local and complete, then the category of lattices has a preprojective partition. This generalizes previous results of the authors in the cases $ R$ is artinian or a discrete valuation ring.

References [Enhancements On Off] (What's this?)

  • [1] M. Auslander, Existence theorems for almost split sequences, Ring Theory. II, Proc. Second Oklahoma Conf., Marcel Dekker, New York and Basel, 1977. MR 0439883 (55:12764)
  • [2] -, Functors and morphisms determined by objects, Representation Theory of Algebras (Proc. Conf. Temple Univ., Philadelphia, Pennsylvania, 1976), Lecture Notes in Pure and Appl. Math., vol. 37, Marcel Dekker, New York, 1978. MR 0480688 (58:844)
  • [3] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc., No. 94 (1969). MR 0269685 (42:4580)
  • [4] M. Auslander and I. Reiten, Representation theory of Artin algebras. III, Comm. Algebra 3 (1975), 239-294. MR 0379599 (52:504)
  • [5] -, Representation theory of Artin algebras. IV, Comm. Algebra 5 (1977), 443-518. MR 0439881 (55:12762)
  • [6] M. Auslander and S. O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61-122. MR 591246 (83a:16039)
  • [7] -, Preprojective partitions of lattices over classical orders, Lecture Notes in Math., vol. 822, Springer-Verlag, Berlin and New York, 1981, pp. 326-344.
  • [8] H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 0153708 (27:3669)
  • [9] E. Cartan and S. Eilenberg, Homological algebras, Princeton Univ. Press, Princeton, N. J., 1956.
  • [10] J. Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Math., vol. 238, Springer-Verlag, New York, 1971. MR 0412177 (54:304)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A64

Retrieve articles in all journals with MSC: 16A64

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society