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Extension functors of local cohomology modules and Serre categories of modules. (English) Zbl 1357.13020
Summary: 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 \(\mathrm{Supp}(N)\subseteq V(I)\). 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: \[ \{\mathrm{Ext}^j_R(N,H^i_I(M)):j\geq0 \text{ and } 0\leq i \leq t\}\cup\{\operatorname{Hom}_R(N,H^{t+1}_I(M)),\mathrm{Ext}^1_R(N,H^{t+1}_I(M))\} \] is in dimension \(<n-2\) and so as a consequence, it follows that the set \[ \mathrm{Ass}_R(L)\cap\{{\mathfrak{p}} \in \mathrm{Spec}(R)::\,\dim(R/{\mathfrak{p}})\geq n - 2\} \] is finite. In particular, the set \[ \mathrm{Ass}_R(\bigoplus_{i=0}^{t+1}H^i_I(R))\cap \{{\mathfrak{p}}\in \,\mathrm{Spec}(R)\,:\,\dim(R/{\mathfrak{p}})\geq n-2\} \] is finite. Also, as an immediately consequence of this result it follows that the \(R\)-modules \(\mathrm{Ext}^j_R(N,H^i_I(M))\) are in dimension \(<n-1\), for all integers \(i,j\geq 0\), whenever \(\dim(M/IM)=n\). These results generalizes the main results of C. Huneke and J. Koh [Math. Proc. Camb. Philos. Soc. 110, No. 3, 421–429 (1991; Zbl 0749.13007)], D. Delfino [Math. Proc. Camb. Philos. Soc. 115, No. 1, 79–84 (1994; Zbl 0806.13005)], G. Chiriacescu [Bull. Lond. Math. Soc. 32, No. 1, 1–7 (2000; Zbl 1018.13009)], D. Asadollahi and R. Naghipour [Commun. Algebra 43, No. 3, 953–958 (2015; Zbl 1318.13024)], P. H. Quy [Proc. Am. Math. Soc. 138, No. 6, 1965–1968 (2010; Zbl 1190.13010)], M. P. Brodmann and A. Lashgari Faghani [Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007)], K. Bahmanpour and R. Naghipour [J. Algebra 321, No. 7, 1997–2011 (2009; Zbl 1168.13016)] and K. Bahmanpour et al. [Commun. Algebra 41, No. 8, 2799–2814 (2013; Zbl 1273.13025)] in the case of complete local rings.

MSC:
13D45 Local cohomology and commutative rings
13C60 Module categories and commutative rings
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[1] D. Asadollahi and R. Naghipour, Faltings’ local-global principle for the finiteness of local cohomology modules, Comm. Algebra, to appear. · Zbl 1306.13009
[2] K. Bahmanpour, On the category of weakly Laskerian cofinite modules, Math. Scand., to appear. 220 · Zbl 1306.13010
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