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.
Reviewer: Monica La Barbiera (Messina)
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 |
Keywords:
fiber ring; Rees algebra; analytic spread; Hilbert-Samuel multiplicity; Cohen-Macaulay, Gorenstein, Buchsbaum-Rim multiplicityReferences:
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