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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Local cohomology over homogeneous rings with one-dimensional local base ring


Authors: M. Brodmann, S. Fumasoli and R. Tajarod
Journal: Proc. Amer. Math. Soc. 131 (2003), 2977-2985
MSC (2000): Primary 13D45, 13E10
Published electronically: April 21, 2003
MathSciNet review: 1993202
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Abstract: Let $R=\bigoplus_{n\geq0}R_n$ be a homogeneous Noetherian ring with local base ring $(R_0,\mathfrak{m}_0)$ and let $M$ be a finitely generated graded $R$-module. Let $H^i_{R_+}(M)$ be the $i$-th local cohomology module of $M$ with respect to $R_+:=\bigoplus_{n>0}R_n$. If $\dim R_0\leq1$, the $R$-modules $\Gamma_{\mathfrak{m}_0R}(H^i_{R_+}(M))$, $(0:_{H_{R_+}^i(M)}\mathfrak{m}_0)$and $H^i_{R_+}(M)/\mathfrak{m}_0H^i_{R_+}(M)$ are Artinian for all $i\in\mathbb{N} _0$. As a consequence, much can be said on the asymptotic behaviour of the $R_0$-modules $H^i_{R_+}(M)_n$ for $n\rightarrow -\infty$.


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Additional Information

M. Brodmann
Affiliation: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH – 8057 Zürich, Switzerland
Email: brodmann@math.unizh.ch

S. Fumasoli
Affiliation: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH – 8057 Zürich, Switzerland
Email: fumasoli@math.unizh.ch

R. Tajarod
Affiliation: Institute of Mathematics, University for Teacher Education, 599 Taleghani Avenue, Tehran 15614, Iran
Email: roshan@iranpasargad.net

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07009-6
PII: S 0002-9939(03)07009-6
Keywords: Local cohomology modules, Artinian modules, graded components
Received by editor(s): April 16, 2002
Published electronically: April 21, 2003
Additional Notes: The third author thanks the University of Zürich for the hospitality offered during the preparation of this paper.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society