Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cohomological dimension of certain algebraic varieties


Authors: K. Divaani-Aazar, R. Naghipour and M. Tousi
Journal: Proc. Amer. Math. Soc. 130 (2002), 3537-3544
MSC (2000): Primary 13D45, 14B15
Published electronically: May 14, 2002
MathSciNet review: 1918830
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$. For finitely generated $R$-modules $M$ and $N$ with $\operatorname{Supp} N\subseteq\operatorname{Supp} M$, it is shown that $\mathrm{cd}(\mathfrak {a},N)\leq \mathrm{cd}(\mathfrak {a},M)$. Let $N$ be a finitely generated module over a local ring $(R,\mathfrak m)$ such that $\operatorname{Min}_{\hat{R}}\hat{N}=\operatorname{Assh}_{\hat{R}}\hat{N}$. Using the above result and the notion of connectedness dimension, it is proved that $\mathrm{cd}(\mathfrak {a},N)\geq\dim N-c(N/\mathfrak {a} N)-1.$ Here $c(N)$ denotes the connectedness dimension of the topological space $\operatorname{Supp} N$. Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13D45, 14B15

Retrieve articles in all journals with MSC (2000): 13D45, 14B15


Additional Information

K. Divaani-Aazar
Affiliation: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran – and – Department of Mathematics, Az-Zahra University, Tehran, Iran
Email: kdivaani@ipm.ir

R. Naghipour
Affiliation: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran – and – Department of Mathematics, University of Tabriz, Tabriz, Iran
Email: naghipour@tabrizu.ac.ir

M. Tousi
Affiliation: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran – and – Department of Mathematics, Shahid Beheshti University, Tehran, Iran
Email: mtousi@vax.ipm.ac.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06500-0
PII: S 0002-9939(02)06500-0
Keywords: Cohomological dimension, connectedness dimension, subdimension, canonical module
Received by editor(s): October 17, 2000
Received by editor(s) in revised form: August 3, 2001
Published electronically: May 14, 2002
Additional Notes: This research was supported in part by a grant from IPM
Dedicated: Dedicated to Professor Hossein Zakeri
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society