Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T13:41:12.497Z Has data issue: false hasContentIssue false
  • English
  • Français

GORENSTEIN DIMENSIONS MODULO A REGULAR ELEMENT

Part of: Homological algebra Commutative algebra: Homological methods

Published online by Cambridge University Press:  03 June 2015

SHAHAB RAJABI  and
SIAMAK YASSEMI
SHAHAB RAJABI*
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran email [email protected]
SIAMAK YASSEMI
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $R$ be a commutative ring. In this paper we study the behavior of Gorenstein homological dimensions of a homologically bounded $R$-complex under special base changes to the rings $R_{x}$ and $R/xR$, where $x$ is a regular element in $R$. Our main results refine some known formulae for the classical homological dimensions. In particular, we provide the Gorenstein counterpart of a criterion for projectivity of finitely generated modules, due to Vasconcelos.

Keywords

Gorenstein homological dimensionderived category

MSC classification

Primary: 13D05: Homological dimension
Secondary: 18G20: Homological dimension 18G25: Relative homological algebra, projective classes 18G35: Chain complexes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 2 , October 2015 , pp. 260 - 266
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Christensen, L. W., Gorenstein Dimensions, Lecture Notes in Mathematics, 1747 (Springer, Berlin, 2000).CrossRefGoogle Scholar
Christensen, L. W., ‘Semi-dualizing complexes and their Auslander categories’, Trans. Amer. Math. Soc. 353(5) (2001), 1839–1883.CrossRefGoogle Scholar
Christensen, L. W., Foxby, H. B. and Frankild, A., ‘Restricted homological dimensions and Cohen–Macaulayness’, J. Algebra 251 (2002), 479–502.CrossRefGoogle Scholar
Christensen, L. W., Frankild, A. and Holm, H., ‘On Gorenstein projective, injective and flat dimensions—a functorial description with applications’, J. Algebra 302 (2006), 231–279.CrossRefGoogle Scholar
Christensen, L. W. and Sather-Wagstaff, S., ‘Transfer of Gorenstein dimensions along ring homomorphisms’, J. Pure Appl. Algebra 214 (2010), 982–989.CrossRefGoogle Scholar
Holm, H., ‘Gorenstein homological dimensions’, J. Pure Appl. Algebra 189 (2004), 167–193.CrossRefGoogle Scholar
Rajabi, S., Torrecillas, B. and Yassemi, S., ‘Homological dimensions and special base changes’, J. Pure Appl. Algebra 219(3) (2015), 646–651.CrossRefGoogle Scholar
Vasconcelos, W. V., ‘On finitely generated flat modules’, Trans. Amer. Math. Soc. 138 (1969), 505–512.CrossRefGoogle Scholar