Abstract

Let $R$ be a commutative Noetherian ring, $\frak{b}$ be an ideal of $R$, $M$ an $R$-module and let $d$ be a non-negative integer. We introduce a general $d$-transform functor $T_{d,\frak{b}}(M,-)$ and its right derived functors $T^i_{d,\frak{b}}(M,-)$, $i\in\mathbb{N}_0$, on the category of $R$-modules and study their various properties. The connection of these functors with some kind of generalized local cohomology functors $H^i_{d,\frak{b}}(M,-)$ is discussed. When both $M$ and $N$ are finitely generated, some finiteness results on $T^i_{d,\frak{b}}(M,N)$ and $H^i_{d,\frak{b}}(M,N)$ are concluded. Then, we study how the depth and dimension of certain subsets of $\mathrm{Spec}(R)$ affect the behavior and vanishing of these modules.

Keywords: Local cohomology; generalized local cohomology; projective dimension; $d$-transform; injective modules.

MSC: 13C05, 13C12, 16D70

DOI: 10.57016/MV-EqWL2965

Pages:  1--12