1 results
On Special Fiber Rings of Modules
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 09 January 2019, pp. 225-242
- Print publication:
- February 2020
-
- Article
- Export citation
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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.
