论文标题
部分可观测时空混沌系统的无模型预测
Homological dimensions of Burch ideals, submodules and quotients
论文作者
论文摘要
Dao-Kobayashi-Takahashi在2020年和Dey-Kobayashi于2022年分别引入了（并研究）Burch理想和Burch子模型的概念。本文的目的是根据Burch理想，Burch子模型或相应商的各种局部戒指来表征各种本地环。我们结果的特定应用程序包括以下内容：让$（R，\ Mathfrak {M}）$为可交换的Noetherian本地环。令$ m = i $是$ r $的一个整体封闭的理想，使得$ {\ rm depth}（r/i）= 0 $，或$ m = \ mathfrak {m} n \ neq 0 $对于某些supperated $ r $ -module $ l $ l $ l $ lm $ n $的subsodule $ n $，既可以$ lmeve pee {结果表明：（1）$ i $具有最大投影$（$ resp。，Impotive $）$复杂性和曲率。 （2）$ r $是gorenstein时，仅当$ {\ rm ext} _r^n（m，r）= 0 $的任何三个连续值的$ n \ ge \ ge \ max \ {{\ rm depth}（r）（r）-1,1,0 \} $。 （3）$ r $是CM（Cohen-Macaulay），并且仅当CM-$ \ dim_r（m）$是有限的时。
The notion of Burch ideals and Burch submodules were introduced (and studied) by Dao-Kobayashi-Takahashi in 2020 and Dey-Kobayashi in 2022 respectively. The aim of this article is to characterize various local rings in terms of homological invariants of Burch ideals, Burch submodules, or that of the corresponding quotients. Specific applications of our results include the following: Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $M=I$ be an integrally closed ideal of $R$ such that ${\rm depth}(R/I)=0$, or $M = \mathfrak{m} N \neq 0$ for some submodule $N$ of a finitely generated $R$-module $L$ such that either ${\rm depth}(N)\ge 1$ or $L$ is free. It is shown that: (1) $I$ has maximal projective $($resp., injective$)$ complexity and curvature. (2) $R$ is Gorenstein if and only if ${\rm Ext}_R^n(M,R)=0$ for any three consecutive values of $n \ge \max\{{\rm depth}(R)-1,0\}$. (3) $R$ is CM (Cohen-Macaulay) if and only if CM-$\dim_R(M)$ is finite.
