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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
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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.

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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

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

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

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