Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Existence of unliftable modules

Author: David A. Jorgensen
Journal: Proc. Amer. Math. Soc. 127 (1999), 1575-1582
MSC (1991): Primary 13D25, 13H99
Published electronically: February 5, 1999
MathSciNet review: 1476141
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $(Q,\operatorname{\mathfrak{n}})$ be a commutative Noetherian local ring, and let $R=Q/(x)$ where $x$ is a non-zerodivisor of $Q$ contained in $\operatorname{\mathfrak{n}}$. Then a finitely generated $R$-module $M$ is said to lift to $Q$ if there exists a finitely generated $Q$-module $M'$ such that $x$ is $M'$-regular and $M \cong M'/xM'$. In this paper we give a general construction of finitely generated $R$-modules of finite projective dimension over $R$ which fail to lift to $Q$ provided $x \in \operatorname{\mathfrak{n}}^{2}$ and the depth of $R$ is at least 2.

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

  • [B] W. Bruns, ``Jede'' endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals, J. Algebra 39 (1976), 429 - 439. MR 53:2925
  • [BE1] D. Buchsbaum and D. Eisenbud, Some structure theorems for finite free resolutions, Advances in math. 12 (1974), 84-139. MR 49:4995
  • [BE2] -, Lifting modules and a theorem on finite free resolutions, Ring Theory (Robert Gordon, ed.), Academic Press, New York, London, 1972, pp. 63-74. MR 49:5098
  • [BH] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993. MR 95h:13020
  • [H] M. Hochster, An obstruction to lifting cyclic modules, Pacific J. Math. 61 (1975), 457 - 463. MR 54:296
  • [Ho] J. W. Hoffman, Counterexamples to the lifting problem for singularities, Comm. Algebra 11 (1983), 523 - 549. MR 84g:13027
  • [Mac] D. Bayer and M. Stillman, Macaulay, a computer algebra system for computing in algebraic geometry and commutative algebra, 1990.
  • [PS] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. Paris, 42 (1973), 323-395. MR 51:10330

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13D25, 13H99

Retrieve articles in all journals with MSC (1991): 13D25, 13H99

Additional Information

David A. Jorgensen
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Address at time of publication: Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019

Keywords: Lifting of modules, free resolution, lifting of complexes, dualized complex
Received by editor(s): June 10, 1997
Received by editor(s) in revised form: September 4, 1997
Published electronically: February 5, 1999
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society