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Associated primes of graded components of local cohomology modules

Authors:	Markus P Brodmann Mordechai Katzman Rodney Y Sharp

Institution:	Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland ; Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom ; Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Abstract:	The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component $H^i_{R_+}(M)_n$ of this local cohomology module $H^i_{R_+}(M)$ is zero for all . This paper is concerned with the asymptotic behaviour of $\operatorname{Ass}_{R_0}(H^i_{R_+}(M)_n)$ as $n \rightarrow -\infty$ . The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which $H^j_{R_+}(M)$ is not finitely generated. Brodmann and Hellus have shown that $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is constant for all (that is, in their terminology, $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is asymptotically stable for $n \rightarrow -\infty$ ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when . They noted that Singh's study of a particular example (in which ) shows that $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ need not be asymptotically stable for $n \rightarrow -\infty$ . The second main aim of this paper is to determine, for Singh's example, $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.

Keywords:	Graded commutative Noetherian ring graded local cohomology module associated prime ideal ideal transform regular ring Gr\"{o}bner bases

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