
On special fiber rings of modules
We prove results concerning the multiplicity as well as the CohenMacaulay
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 CohenMacaulay 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 CohenMacaulayness
criterion and to detect a curious relation to the BuchsbaumRim
multiplicity of $E$ in this setting. Further, we provide a Gorensteinness
characterization for $\mathscr{F}(E)$ in the more general situation where
$R$ is CohenMacaulay 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 ring, Rees algebra, reduction, reduction number, analytic spread, HilbertSamuel multiplicity, CohenMacaulay, Gorenstein, BuchsbaumRim multiplicity Categories:13A30, 13H10, 13H15, 13A02, 13C15, 13D40, , 13E15 