zbMATH — the first resource for mathematics

On annihilators and associated primes of local cohomology modules. (English) Zbl 0968.13010
Let \(R\) be a commutative Noetherian ring, \(\mathfrak{a}\) and ideal of \(R\) and \(M\) a finitely generated \(R\)-module, then \(H_{\mathfrak a }^i(M)\) is the local cohomology module of \(M\) with respect to \({\mathfrak a}\). G. Faltings proved [Math. Ann. 255, 45-56 (1981; Zbl 0451.13008)] a ‘local-global’ principle: If \(r\) is a positive integer, then the \(R_{\mathfrak p}\)-module \(H^i_{\mathfrak a} {R}_{\mathfrak p}(M_{\mathfrak p})\) is finitely generated for all \(i \leq r\) and all \({\mathfrak p}\in \text{Spec}(R)\) if and only if \(H_{\mathfrak a}^i(M)\) is finitely generated for all \(i \leq r\). This leads to a notion of finiteness dimension \(f_{\mathfrak a}(M)\), being the infimum of those \(i\) for which \(H_{\mathfrak a}^i(M)\) is not finitely generated. If \(\mathfrak{b}\) is a second ideal, a variant of this finiteness dimension is the \(\mathfrak{b}\)-finiteness dimension, \(f^{\mathfrak b}_{\mathfrak a}(M)\), defined to be \(\inf\{i\in\mathbb{N}_0 : {\mathfrak b}^{nHi}_{\mathfrak a}(M) \neq 0\) for all \(n \in \mathbb{N} \}\). This paper continues the study of analogues of Faltings’ local-global principle for this \(\mathfrak{b}\)-finiteness dimension, establishing it for arbitrary \(R\) ‘at level 2’, and for \(R\) with \(\dim R \leq 4\) at all levels.
Reviewer: T.Porter (Bangor)

13D45 Local cohomology and commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13E05 Commutative Noetherian rings and modules
14B15 Local cohomology and algebraic geometry
Zbl 0451.13008
Full Text: DOI
[1] C. Bănică, O. Stănăşilă, Algebraic Methods in the Global Theory of Complex Spaces, Wiley, London, 1976.
[2] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82, 8-28, (1963) · Zbl 0112.26604
[3] Brodmann, M., Asymptotic stablity of ass(M/inm), Proc. amer. math. soc., 74, 16-18, (1979) · Zbl 0395.13008
[4] M.P. Brodmann, R.Y. Sharp, Local Cohomology: an Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998. · Zbl 0903.13006
[5] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993.
[6] Faltings, G., Über die annulatoren lokaler kohomologiegruppen, Arch. math., 30, 473-476, (1978) · Zbl 0368.14004
[7] Faltings, G., Der endlichkeitssatz in der lokalen kohomologie, Math. ann., 255, 45-56, (1981) · Zbl 0451.13008
[8] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), North-Holland, Amsterdam, 1968.
[9] R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer, Berlin, 1966.
[10] Katzman, M., Finiteness of ∪_{e}ass fe(M) and its connections to tight closure, Illinois J. math., 40, 330-337, (1996) · Zbl 0852.13003
[11] T. Kawasaki, On Macaulayfication of quasi-projective schemes, preprint, Tokyo Metropolitan University, 1997.
[12] Levin, G.; Vasconcelos, W.V., Homological dimensions and Macaulay rings, Pacific J. math., 25, 315-324, (1968) · Zbl 0161.03903
[13] H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986. · Zbl 0603.13001
[14] Raghavan, K.N., Local-global principle for annihilation of local cohomology, Contemporary math., 159, 329-331, (1994) · Zbl 0818.13009
[15] Ratliff, L.J., Characterizations of catenary rings, American J. math., 93, 1070-1108, (1971) · Zbl 0225.13008
[16] Sharp, R.Y., The Euler characteristic of a finitely generated module of finite injective dimension, Math. Z., 130, 79-93, (1973) · Zbl 0237.13009
[17] Sharp, R.Y., Acceptable rings and homomorphic images of Gorenstein rings, J. algebra, 44, 246-261, (1977) · Zbl 0345.13011
[18] R.Y. Sharp, Necessary conditions for the existence of dualizing complexes in commutative algebra, Séminaire d’algèbre Paul Dubreil, Proceedings, Paris 1977-78, Lecture Notes in Mathematics, vol. 740, Springer, Berlin, 1979, pp. 213-229.
[19] Sharp, R.Y.; Tousi, M., A characterization of generalized Hughes complexes, Math. proc. Cambridge philos. soc., 120, 71-85, (1996) · Zbl 0880.13009
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.