Semi-derived and derived Hall algebras for stable categories

Abstract: Given a Frobenius category $\mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $\mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {\it "semi-derived Hall algebra"} $\mathcal{SDH(F, P(F))}.$ We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group $K_0(\mathcal{P(F)})$ of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category $\underline{\mathcal{F}}$. We prove that it is isomorphic to an appropriately twisted tensor product of $\mathbb{Q}K_0(\mathcal{P(F)})$ with the derived Hall algebra (in the sense of To\"{e}n and Xiao-Xu) of $\underline{\mathcal{F}},$ when both of them are well-defined. We discuss some situations where the semi-derived Hall algebra is defined while the derived Hall algebra is not. The main example is the case of $2-$periodic derived category of an abelian category with enough projectives, where the semi-derived Hall algebra was first considered by Bridgeland who used it to categorify quantum groups.