Cofiniteness of torsion functors of cofinite modules

Reza Naghipour¹ ; Kamal Bahmanpour² ; Imaneh Khalili Gorji³

¹Department 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
²Faculty 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
³Department of Basic Sciences Imam Khomeini International University P.O. Box 34149-1-6818 Qazvin, Iran

Colloquium Mathematicum, Tome 136 (2014) no. 2, pp. 221-230

Résumé

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 $\dim R/I\leq 2$.

Détail

DOI : 10.4064/cm136-2-4

Keywords: noetherian ring ideal i cofinite finitely generated r module shown r modules tor m i cofinite geq whenever dim mathop supp leq dim mathop supp leq immediately implies has dimension dim r modules tor h i cofinite geq prove local r modules tor m i weakly cofinite geq whenever dim mathop supp leq dim mathop supp leq finally shown r modules tor h i weakly cofinite geq whenever dim leq