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On formal local cohomology modules with respect to a pair of ideals. (English) Zbl 1348.13026
Let \(R\) be a Noetherian commutative ring with identity. Let \(\mathfrak a\), \(I\) and \(J\) be ideals of \(R\) and \(M\) an \(R\)-module. The notion of formal local cohomology was introduced by P. Schenzel [J. Algebra 315, No. 2, 894–923 (2007; Zbl 1131.13018)].
In the present paper, the authors introduce and study two generalizations of the notion of formal local cohomology. They denote these generalizations by \(\mathfrak {F}_{\mathfrak a,I,J}^i(M)\) and \(\check{\mathfrak {F}}_{\mathfrak a,I,J}^i(M)\). We bring the definitions of these notions in the sequel.
Let \[ W(I,J):=\left\{\mathfrak{p}\in \mathrm{Spec }R\mid I^{n} \subseteq \mathfrak{p}+J\text{ for some } n\geq 1\right\}. \] The endofunctor \(\Gamma_{I,J}(-)\) on the category of \(R\)-modules is defined by setting \[ \Gamma_{I,J}(M):= \left\{x\in M \mid\mathrm{Supp}_{R}(Rx) \subseteq W(I,J) \right\}, \] for an \(R\)-module \(M\), and \(\Gamma_{I,J}(f):= f|_{\Gamma_{I,J}(M)}\) for an \(R\)-homomorphism \(f:M\rightarrow N\). For each integer \(i\geq 0\), the \(i\)th local cohomology module of \(M\) with respect to the pair \((I,J)\) is defined to be \(H^{i}_{I,J}(M): = R^{i}\Gamma_{I,J}(M)\). This notion was introduced by R. Takahashi et al. [J. Pure Appl. Algebra 213, No. 4, 582–600 (2009; Zbl 1160.13013)]. Let \(\underline{x}=x_1,\dots ,x_s\) be a set of generators of \(I\). In the same paper, the authors introduced the notion, \(\check{C}_{\underline{x},J}\), the Ćech complex of \(R\) with respect to \((I,J)\).
For each nonnegative integer \(i\), the authors define \[ \check{\mathfrak {F}}_{\mathfrak a,I,J}^i(M):=H^i\big(\underset{n}{\varprojlim}\big (\check{C}_{\underline{x},J}\otimes_R\frac{M}{{\mathfrak a}^nM}\big)\big) \] and \[ \mathfrak {F}_{\mathfrak a,I,J}^i(M):=\underset{n}{\varprojlim} \big(H^{i}_{I,J}\big(\frac{M}{{\mathfrak a}^nM}\big)\big). \] The authors establish several upper and lower vanishing bounds for these cohomology modules. In particular, they show that if \((R,\mathfrak m)\) is local, \(M\) is finitely generated and \(I+J\) is \(\mathfrak m\)-primary, then \[ \dim_R(M/(\mathfrak a+J)M)=\sup\{i\in\mathbb{Z}\mid \mathfrak {F}_{\mathfrak a,I,J}^i(M)\neq 0\}. \]

MSC:
13D45 Local cohomology and commutative rings
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