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Finiteness properties of local cohomology modules (an application of \(D\)- modules to commutative algebra). (English) Zbl 0795.13004

Let \(R\) be a commutative Noetherian ring, \(M\) an \(R\)-module and \(Y\) a locally closed subscheme of \(\text{Spec} (R)\), and write \(H_ Y^ i (M)\) for the \(i\)-th local cohomology module of \(M\) with support in \(Y\). Also, write \(H_ I^ i (M)\) for \(H^ i_ Y (M)\) for closed \(Y\subset \text{Spec} (R)\) defined by the ideal \(I \subset R\). One is interested in finiteness properties of \(\text{Hom}_ R (R/I, H^ i_ I (M))\). For two families \(\varphi\) and \(\psi\) of supports such that \(\varphi \subset \psi\), let \(\Gamma_{\psi/ \varphi} (X,S)\), \(S\) an abelian sheaf on \(X=\text{Spec} (R)\), denote \(\Gamma_ \psi (X,S)/ \Gamma_ \varphi (X,S)\), and denote the derived functors of \(\Gamma_ \psi\) and \(\Gamma_{\psi/ \varphi}\) by \(H^ i_ \psi\) and \(H^ i_{ \psi/ \varphi}\), respectively. One has a long exact sequence \[ \cdots \to H^ i_ \varphi (X,-) \to H^ i_ \psi (X,-) \to H^ i_{\psi/ \varphi} (X,-) \to H^{i+1}_ \varphi (X,-) \to \cdots \] Let \({\mathcal T}\) denote a composite functor of the form \({\mathcal T} = {\mathcal T}_ 1 \circ {\mathcal T}_ 2 \circ \cdots \circ {\mathcal T}_ t\), where each \({\mathcal T}_ j\) is either \(H^{i_ j}_{\psi_ j/ \varphi_ j} (X,-)\) or the kernel of any arrow in the above sequence with \(\psi=\psi_ j\) and \(\varphi = \varphi_ j\) for two families \(\varphi_ j \subset \psi_ j\) of supports on \(X\). In case \(Y \subset X\) is locally closed, i.e. \(Y\) is the difference of two closed subsets \(Y''\) and \(Y'\) of \(X\), an analogous construction leads to an exact sequence for the local cohomology \(H^ \bullet_ Y\), \(H^ \bullet_{Y'}\) and \(H^ \bullet_ {Y''}\), and corresponding functor \(T\). Here the subscripts \(Y\), \(Y'\) and \(Y''\) mean families \(\varphi_ Y\), \(\varphi_{Y'}\) and \(\varphi_{Y''}\) of all closed subsetsof \(Y\), \(Y'\) and \(Y''\), respectively. The functors \({\mathcal T}\) and \(T\) are additive and covariant functors on the category of \(R\)- modules. The main result of the paper can be stated:
Let \(K\) be a field of characteristic zero and let \(R\) be a regular \(K\)- algebra. Let \(m\) be a maximal ideal of \(R\). Then
(a) \(H^ i_ m ({\mathcal T} (R))\) is an injective \(R\)-module.
(b) \(\text{inj}\dim_ R ({\mathcal T} (R)) \leq \dim_ R ({\mathcal T} (R))\). In particular, if \(\dim_ R ({\mathcal T} (R))=0\), then \({\mathcal T} (R)\) is injective.
(c) For every maximal ideal \(m\) of \(R\) the set of associated primes of \(T(R)\) contained in \(m\) is finite.
(d) All the Bass numbers of \(T(R)\) are finite.
Similar results in characteristic \(p>0\) were obtained by Huneke and Sharp. As a corollary one obtains for an ideal \(I\) of \(R\): Let \(i\) be an integer bigger than the height of all the minimal primes of \(I\). Then \(\text{Hom}_ R (R/I, H^ i_ I (R))\) is finitely generated if and only if \(H^ i_ I(R)=0\). The proof of the main result surprisingly relies on a result on \(D=D (R,K)\)-modules, where \(R=K[[X_ 1, X_ 2, \dots, X_ n]]\), the ring of formal power series in \(n\) variables over \(K\). The finiteness of the Bass numbers leads to a new numerical invariant for local rings.

MSC:

13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
13D05 Homological dimension and commutative rings
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References:

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