Cofiniteness of local cohomology modules (Q2928444)

The authors study cofiniteness of local cohomology modules and applied it to finiteness problem of local cohomology.NEWLINENEWLINELet \(R\) be a Noetherian ring, \(I \subset R\) an ideal and \(M \neq 0\) a finitely generated \(R\)-module. \textit{C. Huneke} [in: Free resolutions in commutative algebra and algebraic geometry: Sundance 90. Proceedings of a conference, held in May, 1990 in Sundance, UT, USA. 93--108 (1992; Zbl 0782.13015)] posed a conjecture: Let \(W = \{\text{depth} M_{\mathfrak p} + \text{height} (I + \mathfrak p)/\mathfrak p \mid \mathfrak p\) is a prime ideal such that \(I \not\subset \mathfrak p\}\). Then \(t \notin W\) if and only if \(H^t_I(M)\) is finitely generated. The authors give a counter example of this conjecture. On the other hand, they show a variant. Assume that \((R, \mathfrak m)\) is complete local and let \(W = \{i + \dim R/\mathfrak p \mid H^i_{\mathfrak p R_{\mathfrak p}}(M_{\mathfrak p}) \neq 0\) and \(\mathfrak m \not\subset \mathfrak p\}\), which is larger than Huneke's \(W\) in general. Then they show that \(t \notin W\) if and only if \(H^t_{\mathfrak m}(M)\) is finitely generated.NEWLINENEWLINEThis paper contains one more interesting result. \textit{G. Faltings} [Arch. Math. 30, 473--476 (1978; Zbl 0368.14004)] showed that the following are equivalent: {\parindent=6mm \begin{itemize} \item[(1)] \(H_I^i(M)\) is finitely generated for \(i = 0\), \dots, \(t\);\item [(2)] \(H_I^i(M)\) is annihilated for some power of \(I\) for \(i = 0\), \dots, \(n\);\item [(3)] \(H_{IR_{\mathfrak p}}^i(M_{\mathfrak p})\) is a finitely generated \(R_{\mathfrak p}\)-module for any prime ideal \(\mathfrak p\) and for \(i = 0\), \dots, \(t\).NEWLINENEWLINE\end{itemize}} The authors generalize this result. They show that the following are equivalent whenever \(\dim R/I = 1\): {\parindent=6mm \begin{itemize} \item[(1)] \(H_I^t(M)\) is finitely generated;\item [(2)] \(H_I^t(M)\) is annihilated by some power of \(I\);\item [(3)] \(H_{IR_{\mathfrak p}}^t(M_{\mathfrak p})\) is a finitely generated \(R_{\mathfrak p}\)-module.NEWLINENEWLINE\end{itemize}}