The eventual stability of depth, associated primes and cohomology of a graded module. (English) Zbl 1275.13014
There is an account of results on asymptotic behavior of several homological invariants of graded pieces of finitely generated graded modules over Noetherian graded rings. In the interesting paper under review, the authors establish some new asymptotic results. Also, they extend some already established asymptotic results by relaxing the finiteness assumptions.
Let \(R\) be a commutative ring with identity. Let \(I=(a_1, a_2,\dots, a_r)\) be a finitely generated ideal of \(R\) and \(N\) an \(R\)-module. By the definition, the \(i\)th local cohomology module of \(N\) with respect to \(I\) is the \(i\)th homology module of the complex \(\mathcal{C}_{\underline{a}} ^\bullet(N),\) where \(\mathcal{C}_{\underline{a}}^\bullet(N)\) denote the Čech complex of \(N\) with respect to \(\underline{a}:=a_1,a_2,\dots, a_r.\) It can be seen that if \(R\), \(I\) and \(N\) are graded, then each local cohomology module \(H_I^i(N)\) is a graded \(R\)-module. The cohomological dimension of \(N\) with respect to \(I\) is defined by \[ \mathrm{cd}_I(M):=\sup\{i\in \mathbb{Z}|H_I^i(M) \neq 0\}. \] (Keep in mind the convention that \(\sup \emptyset=-\infty.\))
Let \(S\) be a finitely generated standard graded algebra over \(R\) and \(M=\oplus_{\mu\in \mathbb{Z}} M_\mu\) be a finitely generated graded \(S\)-module. The Castelnuovo-Mumford regularity of \(M\) is defined by \[ \mathrm{reg}(M):= \sup\{i\in \mathbb{Z}|\exists j, H_{S_+}^j(M)_{i-j} \neq 0\}. \] Here \(S_+\) is the irrelevant ideal of \(S\).
One of the main result of this paper, asserts that if \(S=R[X_1,\dots, X_n]\) is a polynomial ring over \(R\), then \(\mathrm{cd}_I(M_\mu)\) is constant for \(\mu\geq \mathrm{reg}(M).\)
Assume that \(S\) is Noetherian and \(R\) has dimension at most two. Another main result of this paper, shows that if either \(R\) is an epimorphic image of a Gorenstein ring or \(R\) is local, then there exists an integer \(\gamma_0\) such that, for any \(i\), \(H_{S_+}^i(M)_{\gamma}=0\) for all \(\gamma < \gamma_0\) or \(H_{S_+}^i(M)_{\gamma}\neq 0\) for all \(\gamma < \gamma_0.\)
Let \(R\) be a commutative ring with identity. Let \(I=(a_1, a_2,\dots, a_r)\) be a finitely generated ideal of \(R\) and \(N\) an \(R\)-module. By the definition, the \(i\)th local cohomology module of \(N\) with respect to \(I\) is the \(i\)th homology module of the complex \(\mathcal{C}_{\underline{a}} ^\bullet(N),\) where \(\mathcal{C}_{\underline{a}}^\bullet(N)\) denote the Čech complex of \(N\) with respect to \(\underline{a}:=a_1,a_2,\dots, a_r.\) It can be seen that if \(R\), \(I\) and \(N\) are graded, then each local cohomology module \(H_I^i(N)\) is a graded \(R\)-module. The cohomological dimension of \(N\) with respect to \(I\) is defined by \[ \mathrm{cd}_I(M):=\sup\{i\in \mathbb{Z}|H_I^i(M) \neq 0\}. \] (Keep in mind the convention that \(\sup \emptyset=-\infty.\))
Let \(S\) be a finitely generated standard graded algebra over \(R\) and \(M=\oplus_{\mu\in \mathbb{Z}} M_\mu\) be a finitely generated graded \(S\)-module. The Castelnuovo-Mumford regularity of \(M\) is defined by \[ \mathrm{reg}(M):= \sup\{i\in \mathbb{Z}|\exists j, H_{S_+}^j(M)_{i-j} \neq 0\}. \] Here \(S_+\) is the irrelevant ideal of \(S\).
One of the main result of this paper, asserts that if \(S=R[X_1,\dots, X_n]\) is a polynomial ring over \(R\), then \(\mathrm{cd}_I(M_\mu)\) is constant for \(\mu\geq \mathrm{reg}(M).\)
Assume that \(S\) is Noetherian and \(R\) has dimension at most two. Another main result of this paper, shows that if either \(R\) is an epimorphic image of a Gorenstein ring or \(R\) is local, then there exists an integer \(\gamma_0\) such that, for any \(i\), \(H_{S_+}^i(M)_{\gamma}=0\) for all \(\gamma < \gamma_0\) or \(H_{S_+}^i(M)_{\gamma}\neq 0\) for all \(\gamma < \gamma_0.\)
Reviewer: Kamran Divaani-Aazar (Tehran)
MSC:
| 13D45 | Local cohomology and commutative rings |
