AI 中文总结
本文构造了一个对称数值半群对应的Gorenstein局部环，其非主理想的张量积无挠，从而给出Huneke-Wiegand猜想在一般Gorenstein形式下的反例，并附有显式计算与验证。
AI 中文摘要
我们构造了一个重数56、嵌入维数26、Frobenius数181的对称数值半群Γ，使得对任意域k，一维Gorenstein局部环R=k[t^Γ]_m具有非主理想I=(t^56,t^70)R，且I⊗_R Hom_R(I,R)是无挠的。这给出了Huneke-Wiegand猜想在一般Gorenstein形式下的反例。证明归结为一个显式的有限和集恒等式，通过Huneke-Iyengar-Wiegand冒号准则和Leamer的张量-挠公式进行解释。我们还将该张量积与I的迹理想等同，计算了End_R(I)=R[t^101,t^107]及其传导子，并证明了该反例在完备化下仍然成立。论文附带了一个独立的梯度张量图计算和精确的标准库Python验证器。我们解释了该例子与Christensen、Gerko和Iyengar的域扩张构造的不同之处。
英文摘要
We construct a symmetric numerical semigroup $Γ$ of multiplicity $56$, embedding dimension $26$, and Frobenius number $181$ such that, over every field $k$, the one-dimensional Gorenstein local domain $R=k[t^Γ]_{\mathfrak{m}}$ has a nonprincipal ideal $I=(t^{56},t^{70})R$ for which $I\otimes_R\operatorname{Hom}_R(I,R)$ is torsion-free. This gives a counterexample to the Huneke-Wiegand conjecture in its general Gorenstein formulation. The proof reduces to an explicit finite sumset identity, interpreted through the Huneke-Iyengar-Wiegand colon criterion and Leamer's tensor-torsion formula. We also identify the tensor product with the trace ideal of $I$, compute $\operatorname{End}_R(I)=R[t^{101},t^{107}]$ and its conductor, and show that the counterexample persists under completion. A separate graded tensor-graph calculation and exact, standard-library Python verifiers accompany the paper. We explain how the example differs from the field-extension construction of Christensen, Gerko, and Iyengar.
Comments12 pages; exact Python verification code and deterministic outputs included as ancillary files
