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

*(English)*Zbl 0795.13004Let \(R\) 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.

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.

Reviewer: W.W.J.Hulsbergen (Breda)

##### MSC:

13D45 | Local cohomology and commutative rings |

14B15 | Local cohomology and algebraic geometry |

13D05 | Homological dimension and commutative rings |

##### Keywords:

characteristic \(p\); \(D\)-module; injective dimension; local cohomology module; finiteness of the Bass numbers**OpenURL**

##### References:

[1] | [Ba] Bass, H.: On the Ubiquity of Gorenstein Rings. Math. Z.82, 8-28 (1963). · Zbl 0112.26604 |

[2] | [Bj] Bjork, J.-E.: Rings of Differential Operators. Amsterdam North-Holland 1979 |

[3] | [F1] Faltings, G.: Über die Annulatoren lokaler Kohomologiegruppen. Arch. Math.30, 473-476 (1978) · Zbl 0368.14004 |

[4] | [F2] Faltings, G.: Über lokale Kohomologiegruppen hoher Ordnung. J. Riene Angew. Math.313, 43-51 (1980) · Zbl 0411.13010 |

[5] | [G1] Gorthendieck, A.: Local Cohomology (Lect. Notes Math., Vol. 41) Berlin Heidelberg New York: Springer 1966 |

[6] | [G2] Grothendieck, A.: Cohomologie Locale de Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux (SGA2). Amsterdam: North-Holland 1968 |

[7] | [Ha1] Hartshorne, R.: Lectures on the Grothendieck Duality Theory, (Lect. Notes Math., vol. 20) Berlin Heidelberg New York: Springer 1966 |

[8] | [Ha2] Hartshorne, R.: Affine Duality and Cofiniteness. Invent Math.9, 145-164 (1970) · Zbl 0196.24301 |

[9] | [Ha3] Hartshorne, R.: Cohomological Dimension of Algebraic Varieties. Ann. Math.88, 403-450 (1968) · Zbl 0169.23302 |

[10] | [Ha4] Hartshorne, R.: On the DeRham Cohomology of Algebraic Varieties. Publ. Math. Inst. Hautes Étud. Sci.45, 5-99 (1975) · Zbl 0326.14004 |

[11] | [Ha-Sp] Hartshorne, R., Speiser, R.: Local Cohomological Dimension in Characteristic p. Ann. Math.105, 45-79 (1977) · Zbl 0362.14002 |

[12] | [Ho-R] Hochster, M., Roberts, J.: The Purity of the Frobenius and Local Cohomology. Adv. Math.21 (no. 2), 117-172 (1976) · Zbl 0348.13007 |

[13] | [Hu-K] Huneke, C., Koh, J.: Cofiniteness and Vanishing of Local Cohomology Modules. Math. Proc. Camb. Philos. Soc.110, 421-429 (1991) · Zbl 0749.13007 |

[14] | [Hu-Ly] Huneke, C., Lyubeznik, G.: On the Vanishing of Local Cohomology Modules. Invent. Math.102, 73-93 (1990) · Zbl 0717.13011 |

[15] | [Hu-Sh] Huneke, C., Sharp, R.: Bass Numbers of Local Cohomology Modules (to appear) · Zbl 0785.13005 |

[16] | [Ly] Lyubeznik, G.: (in preparation) |

[17] | [O] Ogus, A.: LOcal Cohomological Dimension of Algebraic Varieties. Ann. Math.98, 1-34 (1973) · Zbl 0308.14003 |

[18] | [P-Sz] Peskine, C., Szpiro, L.: Dimension Projective Finie et Cohomologie Locale. Publ. Math., Inst. Hautes Étud. Sci.42, 323-395 (1973) |

[19] | [Sh] Sharp, R.: The Frobenius Homomorphism and Local Cohomology in Regular Local Rings of Positive Characteristic. J. Pure Appl. Algebra71 (no. 2-3), 313-317 (1991) · Zbl 0736.13008 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.