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On Special Fiber Rings of Modules
Canadian Journal of Mathematics ( IF 0.6 ) Pub Date : 2019-01-09 , DOI: 10.4153/cjm-2018-031-6
Cleto B. Miranda-Neto

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.



中文翻译:

在模块的特殊光纤环上

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 $ $ 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。

更新日期:2019-01-09
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