A test complex for Gorensteinness
HTML articles powered by AMS MathViewer
- by Lars Winther Christensen and Oana Veliche PDF
- Proc. Amer. Math. Soc. 136 (2008), 479-487 Request permission
Abstract:
Let $R$ be a commutative noetherian ring with a dualizing complex. By recent work of Iyengar and Krause (2006), the difference between the category of acyclic complexes and its subcategory of totally acyclic complexes measures how far $R$ is from being Gorenstein. In particular, $R$ is Gorenstein if and only if every acyclic complex is totally acyclic. In this note we exhibit a specific acyclic complex with the property that it is totally acyclic if and only if $R$ is Gorenstein.References
- Luchezar L. Avramov and Hans-Bjørn Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129–155. MR 1117631, DOI 10.1016/0022-4049(91)90144-Q
- Luchezar L. Avramov, Hans-Bjørn Foxby, and Stephen Halperin, Differential graded homological algebra, preprint (2004).
- Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866, DOI 10.1007/BFb0103980
- Lars Winther Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839–1883. MR 1813596, DOI 10.1090/S0002-9947-01-02627-7
- Lars Winther Christensen, Anders Frankild, and Henrik Holm, On Gorenstein projective, injective and flat dimensions—a functorial description with applications, J. Algebra 302 (2006), no. 1, 231–279. MR 2236602, DOI 10.1016/j.jalgebra.2005.12.007
- Edgar E. Enochs, Overtoun M. G. Jenda, and Blas Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993), no. 1, 1–9 (English, with Chinese summary). MR 1248299
- Hans-Bjørn Foxby, Hyperhomological algebra & commutative rings, notes in preparation.
- Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
- Srikanth Iyengar and Henning Krause, Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math. 11 (2006), 207–240. MR 2262932
Additional Information
- Lars Winther Christensen
- Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
- Address at time of publication: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409
- MR Author ID: 671759
- ORCID: 0000-0002-9360-123X
- Email: winther@math.unl.edu, lars.w.christensen@ttu.edu
- Oana Veliche
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Email: oveliche@math.utah.edu
- Received by editor(s): July 14, 2006
- Received by editor(s) in revised form: December 6, 2006, and January 17, 2007
- Published electronically: November 6, 2007
- Additional Notes: The first author was partly supported by a grant from the Carlsberg Foundation.
- Communicated by: Bernd Ulrich
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 479-487
- MSC (2000): Primary 13H10, 13D25
- DOI: https://doi.org/10.1090/S0002-9939-07-09129-0
- MathSciNet review: 2358487