EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES

Abstract

Let $(R,m)$ be a complete Noetherian local ring, $I$ a proper ideal of R and $M$, $N$ two finitely generated R-modules such that Supp $(N)\subseteq V(I)$. Let $t\geq 0$ be an integer such that for each $0\leq i\leq t$, the $R$-module $H^i_I(M)$ is in dimension $\lt n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{\operatorname{Ext}^j_R(N,H^i_I(M)):j\ge0 \ and \ 0\le i \le t\} \\ \cup \{\operatorname{Hom}_R(N,H^{t+1}_I(M)), \operatorname{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $\lt n-2$ and so as a consequence, it follows that the set $$ \operatorname{Ass}_R(L) \cap \{\mathfrak{p} \in \operatorname{Spec}(R): \dim(R/\mathfrak{p}) \ge n − 2\} $$ is finite. In particular, the set $$ \operatorname{Ass}_R(\oplus_{i=0}^{t+1} H^i_I(R)) \cap \{\mathfrak{p} \in \operatorname{Spec}(R): \dim(R/\mathfrak{p}) \geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules $\operatorname{Ext}^j_R(N,H^i_I(M))$ are in dimension $\lt n-1$, for all integers $i,j \geq 0$, whenever $\dim(M/IM) = n$. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in thecase of complete local rings.