具有任意弱整体维数的全商环

中文摘要 AI 辅助

对于每个非负整数 $n$，我们构造一个弱整体维数恰好为 $n$ 的交换全商环。对于 $n\geq1$，这些例子是约化且非凝聚的。该构造适用于任何非零局部环 $(A,\m)$：我们添加可数个剩余域坐标，并要求它们最终与 $A$ 坐标的剩余一致。所得环 $\T(A)$ 满足 $Q(\T(A))=\T(A)$ 且 $\wgd\T(A)=\wgd A$。我们显式描述了它的素谱、极大局部化、幂零根、Jacobson 根和幂等元。对于每对模，$\T(A)$ 上的正次数 Tor 在除以由正交幂等元生成的投射理想后，自然等同于 $A$ 上的 Tor。我们证明 $\T(A)$ 是凝聚的当且仅当 $A$ 是域。特化到正则局部多项式环可得到指定的维数，并给出下界的显式 Koszul 见证。我们还考察了极小谱、整体与局部 Prüfer 条件的区别，以及一个平坦维数为一且投射维数为二的有限生成循环模。

英文摘要

For every nonnegative integer $n$, we construct a commutative total ring of quotients of weak global dimension exactly $n$. For $n\geq1$ the examples are reduced and non-coherent. The construction applies to any nonzero local ring $(A,\m)$: one adjoins countably many residue-field coordinates subject to eventual agreement with the residue of the $A$-coordinate. The resulting ring $\T(A)$ satisfies $Q(\T(A))=\T(A)$ and $\wgd\T(A)=\wgd A$. Its prime spectrum, maximal localizations, nilradical, Jacobson radical, and idempotents are described explicitly. For every pair of modules, positive-degree Tor over $\T(A)$ is naturally identified with Tor over $A$ after quotienting by a projective ideal generated by orthogonal idempotents. We prove that $\T(A)$ is coherent if and only if $A$ is a field. Specializing to regular local polynomial rings yields the prescribed dimensions, with explicit Koszul witnesses for the lower bounds. We also examine the minimal spectrum, the distinction between global and local Prüfer conditions, and a finitely presented cyclic module of flat dimension one and projective dimension two.

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）

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