Goldman 素理想与高次循环条件在小有限维数中的应用

中文摘要 AI 辅助

设 $R$ 为交换环，$fPD(R)$ 为其在 Glaz 意义下的小有限维数。我们研究了 Koszul 级数在受限同调检验中可能逃脱的两种方式。对于多项式扩张，我们证明若 $R$ 为 Noether 环，则 $R[X]$ 的极大理想恰好检测 $R$ 的 Goldman 素理想，并由此得到公式 $$ fPD(R[X]) =1+\sup\{\mathrm{depth}\\,R_{\mathfrak p}\mid \mathfrak p\in G\mathrm{Spec}(R)\}.$$ 我们构造了 Noether 局部环 $R_m$，满足 $fPD(R_m)=0$ 且 $fPD(R_m[X])=m+1$，并证明 Koszul 级数沿相邻素理想对可增加 2。随后我们考虑循环弱 $(n,d)$-条件。对于每个 $n\geq2$ 和 $h\geq1$，我们构造一个局部理想化环 $T$，使得每个 $n$-呈现的循环 $T$-模都是投射的，而 $fPD(T)=h$。环 $T$ 没有非零真有限呈现理想，但它有一个有限生成理想 $I$，满足 $\mathrm{Ext}_T^i(T/I,T)=0$（$i<h$）且 $\mathrm{Ext}_T^h(T/I,T)\neq0$。因此，循环条件在呈现层级一上控制 $fPD$，但在更高层级上则不能。

英文摘要

Let $R$ be a commutative ring and let $fPD(R)$ be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if $R$ is Noetherian, the maximal ideals of $R[X]$ detect exactly the Goldman primes of $R$, and this gives the formula $$ fPD(R[X]) =1+\sup\{depth R_{\mathfrak p}\mid \mathfrak p\in GSpec(R)\}.$$ We construct Noetherian local rings $R_m$ with $fPD(R_m)=0$ and $fPD(R_m[X])=m+1$, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak $(n,d)$-conditions. For every $n\geq2$ and $h\geq1$, we construct a local idealization $T$ such that every $n$-presented cyclic $T$-module is projective, while $fPD(T)=h$. The ring $T$ has no nonzero proper finitely presented ideals, but it has a finitely generated ideal $I$ with $Ext_T^i(T/I,T)=0$ for $i<h$ and $Ext_T^h(T/I,T)\neq0$. Thus the cyclic condition controls $fPD$ at presentation level one, but at no higher level.

发表机构

• School of Mathematics and Statistics, Tianshui Normal University（天水师范学院数学与统计学院）
• Division of Computer Engineering, Hoseo University（湖西大学计算机工程学科）

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