AI 中文总结
本文针对Cramer关于双Hall代数与导出等价的定理，修正了其证明中依赖的特征三角计数公式错误，给出保留原策略的新证明。
AI 中文摘要
设𝕜为有限域，𝒜、ℬ为𝕜线性的Ext有限遗传阿贝尔范畴。Cramer定理断言，在适当假设下，此类范畴间的导出等价𝒟ᵇ(𝒜)→𝒟ᵇ(ℬ)会诱导对应双Hall代数DH𝒜→DHℬ的代数同构。但研究发现，Cramer证明所依赖的𝒟ᵇ(𝒜)中特定特征三角的计数公式通常不成立。我们给出了Cramer定理的修正证明，该证明保留了其方法的整体策略。
英文摘要
Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A}) \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such categories induces an algebra isomorphism of the corresponding double Hall algebras $\mathsf{DH}_\mathcal{A} \!\longrightarrow\! \mathsf{DH}_\mathcal{B}$. It turns out that a counting formula for certain distinguished triangles in $\mathcal{D}^b(\mathcal{A})$, on which Cramer's proof relies, is incorrect in general. We give a corrected proof of Cramer's theorem which preserves the overall strategy of his approach.
