论文标题
方程的扰动何时保留正常锥体
When does a perturbation of the equations preserve the normal cone
论文作者
论文摘要
令$（R，\ Mathfrak M）$为本地戒指，$ i，J $两个任意的理想$ r $。令$ \ operatotorname {gr} _j（r/i）$表示相对于$ j $的$ r/i $的环，这对应于几何形状的普通锥。本文的主要结果表明，如果$ i =（f_1，...，f_r）$，其中$ f_1，...，f_r $是$ j $ -filter的常规序列，则存在一个数字$ n $，因此，如果$ f_i'\ equiv f_i \ equiv f_i \ equiv f_i \ mod j^n $ and $ i'=（f_1'=（f_1'=（f_1''，... $ \ operatoTorname {gr} _j（r/i）\ cong \ permatatorName {gr} _j（r/i'）$。如果$ j $是$ \ mathfrak m $ primary的理想，则此结果意味着长期存在的srinivas和trivedi在小型扰动下的希尔伯特 - 塞缪尔功能的不变性，这是MA，Quy and Quy and Quy and Smirnov最近解决的。作为副产品，相对于$ j $的$ i $和$ i'$的artin-rees数量相同。此外，我们用上述属性给出了最小数字$ n $的明确上限。这些结果解决了MA，Quy和Smirnov提出的两个问题。在阿基里斯 - 曼纳雷斯（Achilles-Manaresi）功能，关系类型，Castelnuovo-Mumford的规律性，Cohen-Macaulayness和REES代数的Gorensteiness $ r/i $相对于$ J $ $ j $的情况下，还有其他有趣的后果。我们还证明了主要结果的相反，表明条件$ i $由$ j $ filter常规序列产生是其有效性的最佳选择。关于理想过滤的主要结果也可以扩展到扰动。结果，如果$ r $是电源系列戒指，$ f_1，...，f_r $是过滤器的常规序列，而$ f_i'$是$ n $ f_i $ of $ f_i $ for $ n \ gg 0 $，那么$ i $ y $ and $ i $ and $ i'$具有与任何NOYERENOTERIAN MONOMIAL MONOMIAL SORMOMIAL SORMIAMIAL MONEMILIAL MONEMIAL ORDER相同的最初理想。这种后果的一种特殊情况是对分析完整交点奇异性的近似值的Adamus和Seyedinejad的猜想。
Let $(R,\mathfrak m)$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $\operatorname{gr}_J(R/I)$ denote the associated ring of $R/I$ with respect to $J$, which corresponds to the normal cone in geometry. The main result of this paper shows that if $I = (f_1,...,f_r)$, where $f_1,...,f_r$ is a $J$-filter regular sequence, there exists a number $N$ such that if $f_i' \equiv f_i \mod J^N$ and $I' = (f_1',...,f_r')$, then $\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')$. If $J$ is an $\mathfrak m$-primary ideal, this result implies a long standing conjecture of Srinivas and Trivedi on the invariance of the Hilbert-Samuel function under small perturbations, which has been solved recently by Ma, Quy and Smirnov. As a byproduct, the Artin-Rees number of $I$ and $I'$ with respect to $J$ are the same. Furthermore, we give explicit upper bounds for the smallest number $N$ with the above property. These results solve two problems raised by Ma, Quy and Smirnov. There are other interesting consequences on the invariance of the Achilles-Manaresi function, the relation type, the Castelnuovo-Mumford regularity, the Cohen-Macaulayness and the Gorensteiness of the Rees algebra of $R/I$ with respect to $J$ under small perturbation of $I$. We also prove a converse of the main result showing that the condition $I$ being generated by a $J$-filter regular sequence is the best possible for its validity. The main result can be also extended to perturbations with respect to filtrations of ideals. As a consequence, if $R$ is a power series ring, $f_1,...,f_r$ is a filter regular sequence, and $f_i'$ is the $n$-jet of $f_i$ for $n \gg 0$, then $I$ and $I'$ have the same initial ideal with respect to any Noetherian monomial order. A special case of this consequence was a conjecture of Adamus and Seyedinejad on approximations of analytic complete intersection singularities.
