Double Hall algebras and derived equivalences revisited

Abstract

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 ${\mathcal{D}}^b(\mathcal{A})⟶{\mathcal{D}}^b(\mathcal{B})$ between two such categories induces an algebra isomorphism of the corresponding double Hall algebras ${\mathsf{DH}}_{\mathcal{A}}⟶{\mathsf{DH}}_{\mathcal{B}}$. 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.