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On Special Fiber Rings of Modules

Part of: Theory of modules and ideals Local rings and semilocal rings Chain conditions, finiteness conditions General commutative ring theory Commutative algebra: Homological methods

Published online by Cambridge University Press:  09 January 2019

Cleto B. Miranda-Neto
Cleto B. Miranda-Neto*
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, Paraíba, Brazil Email: cletoneto2011@hotmail.com
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Abstract

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

Keywords

special fiber ringRees algebrareductionreduction numberanalytic spreadHilbert-Samuel multiplicityCohen-MacaulayGorensteinBuchsbaum-Rim multiplicity

MSC classification

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13H15: Multiplicity theory and related topics 13A02: Graded rings 13C15: Dimension theory, depth, related rings (catenary, etc.) 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13E15: Rings and modules of finite generation or presentation; number of generators
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 1 , February 2020 , pp. 225 - 242
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

The author was partially supported by CAPES-Brazil (grant 88881.121012/2016-01), and by CNPq-Brazil (grant 421440/2016-3).

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