A note on the vanishing of certain local cohomology modules (Q2888784)

Let \((R,\mathfrak m)\) be a commutative noetherian local ring, \(\mathfrak a\) an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. The author investigates the vanishing of the local cohomology modules NEWLINE\[NEWLINEH_{\mathfrak a}^i(M):={\varinjlim}_nExt^i_R(R/{\mathfrak{a}} ^n,M).NEWLINE\]NEWLINE It is known that \(H_{\mathfrak a}^i(M)=0\) for all \(i>\dim_RM\). On the other hand, Grothendieck's non-vanishing theorem asserts that \(H_{\mathfrak m}^{\dim_RM}(M) \neq 0\). When \(R\) contains a field, the author gives an easy proof of Grothendieck's non-vanishing theorem.NEWLINENEWLINENext, assume that \((R,\mathfrak m)\) is a complete local domain contains a field \(k\) and \(x_1,x_2,\dots ,x_n\in R\). Let \(Q(R)\) denote the quotient field of the domain \(R_0:=k[[x_1,x_2,\dots ,x_n]]\). The author shows that if \(H_{(x_1,x_2,\dots ,x_n)} ^n(R)\neq 0\), then \(\dim R_0=n\) and \(R\cap Q(R_0)=R_0\).