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

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.

### MSC:

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

### References:

 [2] M. Brodmann, A particular class of regular domains , J. Algebra, 54 (1978), 366-373. · Zbl 0372.13009 [3] M. Brodmann and C. Rotthaus, Local domains with bad sets of formal prime divisors , J. Algebra, 75 (1982), 386-394. · Zbl 0483.13011 [4] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press (1998). · Zbl 0903.13006 [5] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press (1993). · Zbl 0788.13005 [6] S. Greco, Two theorems on excellent rings , Nagoya Math. J., 60 (1976), 139-149. · Zbl 0308.13010 [7] H. C. Hutchins, Examples of commutative rings, Polygonal, Passaic, New Jersey (1981). · Zbl 0492.13001 [8] D. Kirby, Artinian modules and Hilbert polynomials , Quart. J. Math. Oxford (2), 24 (1973), 47-57. · Zbl 0248.13020 [9] H. Matsumura, Commutative algebra, Benjamin, New York (1970). · Zbl 0211.06501 [10] —-, Commutative ring theory, Cambridge University Press (1986). · Zbl 0603.13001 [11] L. Melkersson and P. Schenzel, The co-localization of an Artinian module , Proc. Edinburgh Math. Soc., 38 (1995), 121-131. · Zbl 0824.13011 [12] R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings , Quart. J. Math. Oxford (2), 26 (1975), 269-273. · Zbl 0311.13006 [13] R. Y. Sharp, Some results on the vanishing of local cohomology modules , Proc. London Math. Soc. (3), 30 (1975), 177-195. · Zbl 0298.13011
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.