Some results on local cohomology modules (Q2505188)

Let \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. For a non-negative integer \(t\), \(H_{\mathfrak{a}}^t(M)\) denotes the local cohomology modules of \(M\) with respect to \(\mathfrak{a}\). The authors prove that \(H_{\mathfrak{a}}^t(M)\) is \(\mathfrak{a}\)-cofinite whenever \(H_{\mathfrak{a}}^t(M)\) is Artinian and \(H_{\mathfrak{a}}^i(M)\) is \(\mathfrak{a}\)-cofinite for all \(i<t\). In particular, this characterizes the \(\mathfrak{a}\)-cofiniteness property of local cohomology modules of certain regular local rings. Also, it is shown that for a local ring \((R,\mathfrak{m})\), \(f\)-depth\((\mathfrak{a},M)\) is the least integer \(i\) such that \(H_{\mathfrak{a}}^i(M)\ncong H_{\mathfrak{m}}^i(M)\). Finally, the authors extend Grothendieck's non-vanishing theorem to \(\mathfrak{a}\)-cofinite modules. Thus, if \((R,\mathfrak{m})\) is a local ring and \(M\) is a non-zero \(\mathfrak{a}\)-cofinite \(R\)-module of dimension \(n\), then \(H_{\mathfrak{m}}^n(M)\neq 0\).