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 $<n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{{Ext}^j_R(N,H^i_I(M)):j\ge0 and 0\le i \le t\} \cup\{{Hom}_R(N,H^{t+1}_I(M)),{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $<n-2$ and so as a consequence, it follows that the set $$ {Ass}_R(L)\cap\{{\frak{p}} \in Spec(R): :\,\dim (R/{\frak{p}})\ge n − 2\} $$ is finite. In particular, the set $$ {Ass}_R(\oplus_{i=0}^{t+1}H^i_I(R))\cap \{{\frak{p}}\in \,{Spec}(R)\,:\,\dim(R/{\frak{p}})\geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules ${Ext}^j_R(N,H^i_I(M))$ are in dimension $<n-1$, 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.