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

DOI:
https://doi.org/10.1090/S0002-9939-03-07009-6

Published electronically:
April 21, 2003

MathSciNet review:
1993202

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Abstract | References | Similar Articles | Additional Information

Abstract: Let $R=\bigoplus _{n\geq 0}R_n$ 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 $\dim R_0\leq 1$, 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

MR Author ID:
41830

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

MR Author ID:
673912

Email:
roshan@iranpasargad.net

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