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On special fiber rings of modules. (English) Zbl 1429.13008
The author studies properties concerning the multiplicity as well as the Cohen-Macaulay and Gorenstein properties of the special fiber ring of a finitely generated \(R\)-module over a Noetherian local ring \(R\) with infinite residue field. More precisely, the author deals with the special fiber of the Rees algebra, the so-called special fiber ring, of finitely generated modules over a Noetherian local ring \((R,\mathfrak{m})\) in order to study the Cohen-Macaulay and the Gorenstein properties of this blowup algebra. The case of modules of finite colength in a free \(R\)-module, where \(R\) is one-dimensional and Cohen-Macaulay, is considered. Regarding the Gorenstein property, he studies in a more general context where \(R\) is Cohen-Macaulay of arbitrary dimension and the module is not necessarily of finite colength. The investigation of certain numerical invariants, such as analytic spread, multiplicity, and reduction number has a crucial role in the study of the special fiber ring.
MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13H15 Multiplicity theory and related topics
13A02 Graded rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13E15 Commutative rings and modules of finite generation or presentation; number of generators
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