Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Associated Primes of Graded Components of Local Cohomology Modules

Markus P. Brodmann, Mordechai Katzman and Rodney Y. Sharp
Transactions of the American Mathematical Society
Vol. 354, No. 11 (Nov., 2002), pp. 4261-4283
Stable URL: http://www.jstor.org/stable/3072898
Page Count: 23
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Associated Primes of Graded Components of Local Cohomology Modules
Preview not available

Abstract

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 R0, the component of R of degree 0. It is known that the n-th component HiR+(M)n of this local cohomology module HiR+(M) n is zero for all n >> 0. This paper is concerned with the asymptotic behaviour of AssR0(HiR+ (M) n)) as n→ -∞. 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 HjR+(M) is not finitely generated. Brodmann and Hellus have shown that AssR0(hfR+(M)n) is constant for all n << 0 (that is, in their terminology, AssR0(hfR+(M)n) is asymptotically stable for n→-∞. 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 R0 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 AssR0(h3R+(R)n) need not be asymptotically stable for n→ -∞. The second main aim of this paper is to determine, for Singh's example, AssR0(H3R+(R)n) quite precisely for every integer n, and, thereby, answer one of the questions raised by Brodmann and Hellus.

Page Thumbnails

  • Thumbnail: Page 
4261
    4261
  • Thumbnail: Page 
4262
    4262
  • Thumbnail: Page 
4263
    4263
  • Thumbnail: Page 
4264
    4264
  • Thumbnail: Page 
4265
    4265
  • Thumbnail: Page 
4266
    4266
  • Thumbnail: Page 
4267
    4267
  • Thumbnail: Page 
4268
    4268
  • Thumbnail: Page 
4269
    4269
  • Thumbnail: Page 
4270
    4270
  • Thumbnail: Page 
4271
    4271
  • Thumbnail: Page 
4272
    4272
  • Thumbnail: Page 
4273
    4273
  • Thumbnail: Page 
4274
    4274
  • Thumbnail: Page 
4275
    4275
  • Thumbnail: Page 
4276
    4276
  • Thumbnail: Page 
4277
    4277
  • Thumbnail: Page 
4278
    4278
  • Thumbnail: Page 
4279
    4279
  • Thumbnail: Page 
4280
    4280
  • Thumbnail: Page 
4281
    4281
  • Thumbnail: Page 
4282
    4282
  • Thumbnail: Page 
4283
    4283