Brodmann, Markus P., Katzman, Mordechai and Sharp, Rodney Y. (2002) Associated primes of graded components of local cohomology modules. Transactions of the American Mathematical Society, 354 (11). pp. 4261-4283. ISSN 0002-9947
The i-th local cohomology module of a finitely generated graded module M over a standard positively graded commutative Noetherian ring R with respect to the irrelevant ideal R+, is itself graded; all its graded components are finitely generated modules over R-0, the component of R of degree 0. It is known that the n-th component H-R+(i) (M)(n) of this local cohomology module H-R+(i) (M) is zero for all nmuch greater than0. This paper is concerned with the asymptotic behaviour of Ass(R0)(H-R+(i) (M)(n)) as n--> -infinity.
The smallest i for which such study is interesting is the finiteness dimension f of M relative to R+, defined as the least integer j for which H-R+(j) (M) is not finitely generated. Brodmann and Hellus have shown that AssR(0)(H-R+(f) (M)(n)) is constant for all nmuch less than0 ( that is in their terminology AssR(0)(H-R+(f) (M)(n)) is asymptotically stable for n--> -infinity). The first main aim of this paper is to identify the ultimate constant value ( under the mild assumption that R is a homomorphic image of a regular ring) : our answer is precisely the set of contractions to R-0 of certain relevant primes of R whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.
Brodmann and Hellus raised various questions about such asymptotic behaviour when i>f. They noted that Singh's study of a particular example ( in which f=2) shows that AssR(0)(H-R+(3) (R)(n)) need not be asymptotically stable for n--> -infinity. The second main aim of this paper is to determine, for Singh's example, AssR(0)(H-R+(3) (R)(n)) quite precisely for every integer n and, thereby answer one of the questions raised by Brodmann and Hellus.
|Copyright, Publisher and Additional Information:||© 2002 American Mathematical Society. Reproduced in accordance with the publisher's self-archiving policy.|
|Keywords:||graded commutative Noetherian ring; graded local cohomology module; associated prime ideal; ideal transform; regular ring; Grobner bases|
|Institution:||The University of Sheffield|
|Academic Units:||The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)|
|Depositing User:||Miss Anthea Tucker|
|Date Deposited:||18 Nov 2009 12:20|
|Last Modified:||17 Nov 2015 03:40|
|Publisher:||American Mathematical Society|