A test complex for Gorensteinness

Authors:
Lars Winther Christensen and Oana Veliche

Journal:
Proc. Amer. Math. Soc. **136** (2008), 479-487

MSC (2000):
Primary 13H10, 13D25

Published electronically:
November 6, 2007

MathSciNet review:
2358487

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let 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 is from being Gorenstein. In particular, 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 is Gorenstein.

**1.**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**, 10.1016/0022-4049(91)90144-Q**2.**Luchezar L. Avramov, Hans-Bjørn Foxby, and Stephen Halperin,*Differential graded homological algebra*, preprint (2004).**3.**Lars Winther Christensen,*Gorenstein dimensions*, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR**1799866****4.**Lars Winther Christensen,*Semi-dualizing complexes and their Auslander categories*, Trans. Amer. Math. Soc.**353**(2001), no. 5, 1839–1883 (electronic). MR**1813596**, 10.1090/S0002-9947-01-02627-7**5.**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**, 10.1016/j.jalgebra.2005.12.007**6.**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****7.**Hans-Bjørn Foxby,*Hyperhomological algebra & commutative rings*, notes in preparation.**8.**Robin Hartshorne,*Residues and duality*, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR**0222093****9.**Srikanth Iyengar and Henning Krause,*Acyclicity versus total acyclicity for complexes over Noetherian rings*, Doc. Math.**11**(2006), 207–240 (electronic). MR**2262932**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
13H10,
13D25

Retrieve articles in all journals with MSC (2000): 13H10, 13D25

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

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

DOI:
https://doi.org/10.1090/S0002-9939-07-09129-0

Keywords:
Gorenstein rings,
dualizing complexes,
totally acyclic complexes

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

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.