On the dimension and multiplicity of local cohomology modules. (English) Zbl 1044.13007

Brodmann, Markus P.; Sharp, Rodney Y.

Nagoya Math. J. 167, 217-233 (2002).

Let \(A\) be a Noetherian local ring with maximal ideal \(\mathfrak m\) and \(N\) an Artinian \(A\)-module. We can define the Hilbert-Samuel polynomial, the dimension and the multiplicity of \(N\) which coincide to ones of \(M\) if \(N\) is the Matlis dual of a Noetherian module \(M\). If \(M\) is a finitely generated \(A\)-module, then the local cohomology module \(H_{\mathfrak m}^p(M)\) is Artinian. In the present paper, the authors study the multiplicity of \(H_{\mathfrak m}^p(M)\). If \(A\) is a homomorphic image of a Gorenstein local ring, then \(H_{\mathfrak m}^p(M)\) is the Matlis dual of a Noetherian module. Hence there is an equation on the multiplicity of \(H_{\mathfrak m}^p(M)\) like the Bézout theorem.
The main theorem of this paper is a generalization of such an equation when \(A\) is universally catenary and all the formal fibers of \(A\) are Cohen-Macaulay. We need both condition above. The author gives some counter-examples.

Reviewer: Takesi Kawasaki (Tokyo)

MSC:

13D45	Local cohomology and commutative rings
13C15	Dimension theory, depth, related commutative rings (catenary, etc.)
13H15	Multiplicity theory and related topics

Keywords:

Artinian module; multiplicity of local cohomology module; Cohen-Macaulay fibers; universally catenary module; Matlis dual; Noetherian local ring

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