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## Cofiniteness of torsion functors of cofinite modules

### Tom 136 / 2014

Colloquium Mathematicum 136 (2014), 221-230 MSC: 13D45, 14B15, 13E05. DOI: 10.4064/cm136-2-4

#### Streszczenie

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. Let $M$ be an $I$-cofinite and $N$ a finitely generated $R$-module. It is shown that the $R$-modules ${\rm Tor}_i^R(N,M)$ are $I$-cofinite for all $i\geq 0$ whenever $\dim\mathop {\rm Supp}(M)\leq 1$ or $\dim\mathop {\rm Supp}(N)\leq 2$. This immediately implies that if $I$ has dimension one (i.e., $\dim R/I=1$) then the $R$-modules ${\rm Tor}_i^R(N,H^{j}_{I}(M))$ are $I$-cofinite for all $i, j\geq 0$. Also, we prove that if $R$ is local, then the $R$-modules ${\rm Tor}_i^R(N,M)$ are $I$-weakly cofinite for all $i\geq 0$ whenever $\dim\mathop {\rm Supp}(M)\leq 2$ or $\dim\mathop{\rm Supp}(N)\leq 3$. Finally, it is shown that the $R$-modules ${\rm Tor}_i^R(N,H^{j}_{I}(M))$ are $I$-weakly cofinite for all $i, j\geq 0$ whenever $\dim R/I\leq 2$.

#### Autorzy

• Reza NaghipourDepartment of Mathematics
University of Tabriz
Tabriz, Iran
and
School of Mathematics
Institute for Research
in Fundamental Sciences (IPM)
P.O. Box 19395-5746
Tehran, Iran
e-mail
e-mail
• Kamal BahmanpourFaculty of Science
University of Mohaghegh Ardabili
Ardabil, Iran
and
School of Mathematics
Institute for Research
in Fundamental Sciences (IPM)
P.O. Box 19395-5746
Tehran, Iran
e-mail
• Imaneh Khalili GorjiDepartment of Basic Sciences
Imam Khomeini International University
P.O. Box 34149-1-6818
Qazvin, Iran
e-mail

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