Taiwanese Journal of Mathematics

EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES

Nemat Abazari and Kamal Bahmanpour

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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.

Article information

Source
Taiwanese J. Math., Volume 19, Number 1 (2015), 211-220.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133626

Digital Object Identifier
doi:10.11650/tjm.19.2015.4315

Mathematical Reviews number (MathSciNet)
MR3313413

Zentralblatt MATH identifier
1357.13020

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 14B15: Local cohomology [See also 13D45, 32C36] 13E05: Noetherian rings and modules

Keywords
associated prime ideal cofinite module complete local ring Krull dimension local cohomology

Citation

Abazari, Nemat; Bahmanpour, Kamal. EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES. Taiwanese J. Math. 19 (2015), no. 1, 211--220. doi:10.11650/tjm.19.2015.4315. https://projecteuclid.org/euclid.twjm/1499133626


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