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Hilbert-Samuel multiplicity and Northcott’s inequality relative to an Artinian module. (English) Zbl 1437.13041

Authors’ abstract: Let \((A, m)\) be a commutative quasi-local ring with identity, and let \(I\subset m\) be an \(A\)-ideal such that \(\ell(0 :_M I)<\infty\). For \(M\) an Artinian \(A\)-module of \(N\)-dimension \(d\), we introduce the notion of Hilbert-coefficients of \(I\) relative to \(M\) and give several properties. When \(M\) is a co-Cohen-Macaulay \(A\)-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D45 Local cohomology and commutative rings
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