正则分母的Mayer-Vietoris演算
A Mayer-Vietoris calculus for regular denominators
AI总结：
本文研究交换环上理想的正则分母，探讨其与理想交、和的关系，在诺特环上实现分母等价类分类，证明平坦基变换等性质，为具体情形的primary分解计算提供替代方案。
AI中文摘要：
设A为交换环，I为理想，若A中元素a在A/I上的乘法是单射，则称a为I的正则分母，所有这类元素构成的集合记为S_I。我们研究两个理想I与J的公共正则分母如何与I∩J、I+J相关联，当I与J互极大时，这一关系有特别简洁的描述。在诺特环上，同一视角通过素理想的有限非空反链对分母等价类分类，给出交的相伴素理想的界，并导出局部长度恒等式。此外，我们证明平坦基变换保持正则分母，忠实平坦性反映正则分母，有限局部自由商的纤维正则轨迹由满足乘性Mayer-Vietoris公式的行列式定义。我们的结果在许多具体情形下提供了 primary分解计算的替代方案。
英文摘要：
Let $A$ be a commutative ring and let $I$ be an ideal. An element $a\in A$ is a regular denominator for $I$ when multiplication by $a$ on $A/I$ is injective; we denote the set of all such elements by $S_I$. We study how the common regular denominators for two ideals $I$ and $J$ are related to $I\cap J$ and $I+J$. This yields a particularly simple description when $I$ and $J$ are comaximal. Over Noetherian rings, the same viewpoint classifies denominator-equivalence classes by finite nonempty antichains of prime ideals, gives bounds for the associated primes of an intersection and leads to a local length identity. Furthermore, we show that flat base change preserves regular denominators, faithful flatness reflects them, and finite locally free quotients have fiberwise regular loci defined by determinants satisfying a multiplicative Mayer-Vietoris formula. Our results provide alternatives to primary-decomposition computations in many concrete situations.
