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.

我们证明关于有限生成的 $ R $- 模块 $ E \ subsetneq R ^ {e} $ 的特殊纤维环 $ \ mathscr {F}（E）$ 的多重性以及Cohen–Macaulay和Gorenstein性质的结果在具有无限残基场的Noetherian局部环 $ R $ 上。假定 $ R $ 是维恩的Cohen–Macaulay，并且 $ E $在 $ R ^ {e} $中 具有有限的长度，我们的主要结果为 $ \ mathscr {F}（E）的多重性建立一个渐近长度公式$ ，除了具有独立的兴趣外，它还使我们能够导出一个Cohen–Macaulayness准则，并在这种情况下检测与Buchsbaum–Rim多重性 $ E $ 的奇异关系。此外，在更一般的情况下，我们提供 $ \ mathscr {F}（E）$ 的Gorensteinness刻画，其中 $ R $ 是任意维度的Cohen–Macaulay，而 $ E $ 不一定是有限的长度，我们注意到一个约束如果模块的减少数量至少为3 ，则根据模块 $ E $ 的第二个分析偏差。