On the associated primes of generalized local cohomology modules
Abstract
Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the sets $\Ass_R(\Ext_R^i(M,H_{\fa}^{t-i}(N)))$, where $0\leq i\leq t$. As an immediate consequence, it follows that if either $H_{\fa}^i(N)$ is finitely generated for all $i<t$ or $\Supp_R(H_{\fa}^i(N))$ is finite for all $i<t$, then $\Ass_R(H_{\fa}^t(M,N))$ is finite. Also, we prove that if $d=\pd(M)$ and $n=\dim(N)$ are finite, then $H_{\fa}^{d+n}(M,N)$ is Artinian. In particular, $\Ass_R(H_{\fa}^{d+n}(M,N))$ is a finite set consisting of maximal ideals.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- October 2005
- DOI:
- 10.48550/arXiv.math/0510274
- arXiv:
- arXiv:math/0510274
- Bibcode:
- 2005math.....10274M
- Keywords:
-
- Mathematics - Commutative Algebra;
- 13D45;
- 13E99
- E-Print:
- 7 pages, to appear in Communications in Algebra