arXiv 2012.030562020-12-09math.ACmath.NT

交换环理想的等价生成对

Equivalent generating pairs of an ideal of a commutative ring

Luc Guyot

备注：: 26 pages, no figure. Minor changes: fix a couple of embarrassing typos

摘要：

设$R$是一个有单位元的交换环，$I$是$R$的一个二生理想.我们用$\operatorname{SL}_2(R)$表示$R$上具有行列式$1$的$2 \times 2$矩阵群。研究了$\operatorname{SL}_2(R)$通过矩阵右乘在$I$的生成对集合$\operatorname{V}_2(I)$上的作用.设$\operatorname{Fitt}_1(I)$是$I$的第二Fitting理想.我们的主要结果断言，$\operatorname{V}_2(I)/\operatorname{SL}_2(R)$认同一组单位的$R/\operatorname{Fitt}_1(I)$通过一个自然推广的行列式，如果$I$可以产生的两个正规元素。这一结果在几个Bass环中得到了说明，我们还证明了对于每个$n > 2$，$\operatorname{SL}_n(R)$传递地作用于$\operatorname{V}_n(I)$。作为应用，我们得到了二次阶上模群的尖点个数的一个公式。

英文摘要：

Let $R$ be a commutative ring with identity and let $I$ be a two-generated ideal of $R$. We denote by $\operatorname{SL}_2(R)$ the group of $2 \times 2$ matrices over $R$ with determinant $1$. We study the action of $\operatorname{SL}_2(R)$ by matrix right-multiplication on $\operatorname{V}_2(I)$, the set of generating pairs of $I$. Let $\operatorname{Fitt}_1(I)$ be the second Fitting ideal of $I$. Our main result asserts that $\operatorname{V}_2(I)/\operatorname{SL}_2(R)$ identifies with a group of units of $R/\operatorname{Fitt}_1(I)$ via a natural generalization of the determinant if $I$ can be generated by two regular elements. This result is illustrated in several Bass rings for which we also show that $\operatorname{SL}_n(R)$ acts transitively on $\operatorname{V}_n(I)$ for every $n > 2$. As an application, we derive a formula for the number of cusps of a modular group over a quadratic order.

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摘要：

英文摘要：

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