Constructing big indecomposable modules

Crabbe, Andrew; Striuli, Janet

doi:10.1090/S0002-9939-09-09760-3

Constructing big indecomposable modules
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by Andrew Crabbe and Janet Striuli PDF

Proc. Amer. Math. Soc. 137 (2009), 2181-2189 Request permission

Abstract:

Let $R$ be local Noetherian ring of depth at least two. We prove that there are indecomposable $R$-modules which are free on the punctured spectrum of constant, arbitrarily large, rank.

References

M. Auslander, Finite type implies isolated singularity, Orders and their applications (Oberwolfach, 1984) Lecture Notes in Math., vol. 1142, Springer, Berlin, 1985, pp. 1–4. MR 812487, DOI 10.1007/BFb0074789
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), no. 1, 165–182. MR 877011, DOI 10.1007/BF01405096
Andrew Crabbe, Daniel Katz, Janet Striuli, and Emanoil Theodorescu, Hilbert polynomials for the controvarient functor, preprint (2007).
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Anders J. Frankild, Sean Sather-Wagstaff, and Roger Wiegand, Ascent of module structures, vanishing of ext, and extended modules, Mich. J. Math., to appear.
W. Hassler, R. Karr, L. Klingler, and R. Wiegand, Large indecomposable modules over local rings, J. Algebra 303 (2006), no. 1, 202–215. MR 2253659, DOI 10.1016/j.jalgebra.2006.05.016
Wolfgang Hassler, Ryan Karr, Lee Klingler, and Roger Wiegand, Big indecomposable modules and direct-sum relations, Illinois J. Math. 51 (2007), no. 1, 99–122. MR 2346189
Wolfgang Hassler, Ryan Karr, Lee Klingler, and Roger Wiegand, Indecomposable modules of large rank over Cohen-Macaulay local rings, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1391–1406. MR 2357700, DOI 10.1090/S0002-9947-07-04226-2
Wolfgang Hassler and Roger Wiegand, Big indecomposable mixed modules over hypersurface singularities, Abelian groups, rings, modules, and homological algebra, Lect. Notes Pure Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 159–174. MR 2229110, DOI 10.1201/9781420010763.ch15
Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR 600654
Daniel Katz and Emanoil Theodorescu, On the degree of Hilbert polynomials associated to the torsion functor, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3073–3082. MR 2322736, DOI 10.1090/S0002-9939-07-08879-X
Lee Klingler and Lawrence S. Levy, Representation type of commutative Noetherian rings. I. Local wildness, Pacific J. Math. 200 (2001), no. 2, 345–386. MR 1868696, DOI 10.2140/pjm.2001.200.345
Lee Klingler and Lawrence S. Levy, Representation type of commutative Noetherian rings. I. Local wildness, Pacific J. Math. 200 (2001), no. 2, 345–386. MR 1868696, DOI 10.2140/pjm.2001.200.345
Lee Klingler and Lawrence S. Levy, Representation type of commutative Noetherian rings. III. Global wildness and tameness, Mem. Amer. Math. Soc. 176 (2005), no. 832, viii+170. MR 2147090, DOI 10.1090/memo/0832
Vijay Kodiyalam, Homological invariants of powers of an ideal, Proc. Amer. Math. Soc. 118 (1993), no. 3, 757–764. MR 1156471, DOI 10.1090/S0002-9939-1993-1156471-5
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169