On Special Fiber Rings of Modules

Miranda-Neto, Cleto B.

Canadian journal of mathematics, Tome 72 (2020) no. 1, pp. 225-242

Résumé

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.

Détail

DOI : 10.4153/CJM-2018-031-6

Mots-clés : special fiber ring, Rees algebra, reduction, reduction number, analytic spread, Hilbert-Samuel multiplicity, Cohen-Macaulay, Gorenstein, Buchsbaum-Rim multiplicity