BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Sh. Payrovi, S. Babaei, I. Khalili-Gorji

Abstract: Let $R$ be a Noetherian ring, $M$ a finitely generated $R$-module and $N$ an arbitrary $R$-module. We consider the generalized local cohomology modules, with respect to an arbitrary ideal $I$ of $R$, and prove that, for all nonnegative integers $r,t$ and all $\frak p\in\operatorname{Spec}(R)$ the Bass number $\mu^r(\frak p,H^t_I(M,N))$ is bounded above by $\sum_{j=0}^t\mu^r\big(\frak p,\operatorname{Ext}^{t-j}_R(M, H^j_I(N))\big)$. A corollary is that $ \operatorname{Ass}\big(H_I^t(M,N)\big)\subseteq \bigcup_{j=0}^t\operatorname{Ass}\big(\operatorname{Ext}^{t-j}_R(M,H^j_I(N))\big). $ In a slightly different direction, we also present some well known results about generalized local cohomology modules.

Keywords: generalized local cohomology, Bass numbers

Classification (MSC2000): 13D45; 14B15

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