Semi-Derived and Derived Hall Algebras for Stable Categories

Given a Frobenius category F satisfying certain finiteness conditions, we consider the localization of its Hall algebra H(F) at the classes of all projective-injective objects. We call it the "semi-derived Hall algebra" 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(P(F)) of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category (F) under bar. We prove that it is isomorphic to an appropriately twisted tensor product of QK(0)(P(F)) with the derived Hall algebra (in the sense of Toen and Xiao-Xu) of (F) under bar, 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 [2] who used it to categorify quantum groups.