Finiteness of associated primes of extension and torsion functors (Q6188423)

‎‎Let \(R\) be a commutative Noetherian ring with identity‎, ‎and ‎let \(\mathfrak{a}\) be an‎ ‎ideal of \(R\)‎.‎‎‎‎‎‎‎‎‎‎‎ ‎In the paper under review‎, ‎the author ‎studied the structure of ‎‎the functors ‎\(‎\operatorname{Ext}‎\)‎ and ‎\(‎\operatorname{Tor}‎\), ‎and‎ obtained some results on the finiteness of associated primes of these functors ‎‎‎f‎or modules that are not necessarily finitely generated. ‎‎ ‎Recall that an ‎\(‎R‎\)‎-module ‎\(‎M‎\)‎ is said to be ‎\(‎\mathfrak{a}‎\)‎-cofinite if ‎\(‎\Supp(M)‎\subseteq {\rm ‎V}(\mathfrak{a})‎\)‎ and ‎‎\(‎\operatorname{Ext}^i_R(R/{\mathfrak{a}}, M)‎\)‎ is finitely generated for all ‎\(‎i‎\geq‎0‎\)‎‎ [\textit{R. Hartshorne}, Invent. Math. 9, 145--164 (1970; Zbl 0196.24301)]. Also, an ‎\(‎R‎\)‎-module ‎\(‎M‎\)‎ is called minimax, if there is a finitely generated submodule ‎\(‎N‎\)‎ of ‎\(‎M‎\)‎ such that ‎\(‎M/N‎\)‎ is Artinian [\textit{H. Zöschinger}, J. Algebra 102, 1--32 (1986; Zbl 0593.13012)]. ‎ Suppose that ‎‎‎‎\(‎M‎\) ‎is a ‎finitely ‎generated ‎\(‎R‎\)‎-module, ‎and ‎‎\(‎N‎\) ‎is a nonzero ‎\(‎\mathfrak{a}‎\)‎-cofinite ‎\(‎R‎\)‎-module‎.‎‎ It is shown that \(‎\operatorname{Ass}_R(\operatorname{Tor}^R_i(M,N))‎\) ‎is ‎finite ‎for ‎all ‎‎\(‎i‎\geq0‎\)‎, ‎whenever ‎one ‎of ‎the ‎following ‎is ‎valid:‎ \begin{itemize}‎ ‎\item[(i)] \(‎\dim_R M=1, 2‎\)‎ or ‎\(‎\dim_R N‎\leq ‎1‎‎\)‎; ‎\item[(ii)] ‎\(‎R\) is a local ring, \(‎\dim_R M=3‎\)‎ or ‎\(‎\dim_R N‎\leq 2‎‎\)‎. ‎\end{itemize}‎‎‎ ‎‎‎Also, it is proved that ‎\(‎\operatorname{Ass}_R(\operatorname{Ext}_R^i(N,H))‎\) ‎is ‎finite ‎for ‎all ‎‎\(‎i‎\geq0‎\)‎, whenever ‎‎\(‎R‎\) ‎‎is ‎a ‎‎local ring, ‎\(N‎\)‎ is a minimax ‎\(‎R‎\)‎-module, and ‎\(H‎\)‎ is an Artinian and ‎\(‎\mathfrak{a}‎\)‎-cofinite ‎\(‎R‎\)‎-module. ‎