Cofiniteness of torsion functors of cofinite modules
Tom 136 / 2014
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 $\dim R/I\leq 2$.
