Semi-derived and derived Hall algebras for stable categories

الأصل · EN

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 K₀(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. We prove that it is isomorphic to an appropriately twisted tensor product of QK₀(P(F)) with the derived Hall algebra (in the sense of Toën and Xiao-Xu) of 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.