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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 BĂ©zout 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.

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