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Given a Frobenius category   $\mathcal {F}$   satisfying certain finiteness conditions, we consider the localization of its Hall algebra   $\mathcal {H}(\mathcal {F})$   at the classes of all projective–injective objects. We call it the  “semi-derived Hall algebra”   $\mathcal {SDH}(\mathcal {F}, \mathcal {P}(\mathcal {F}))$  . We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group   $K_0(\mathcal {P}(\mathcal {F}))$   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   $\mathbb {Q}K_0(\mathcal {P}(\mathcal {F}))$   with the derived Hall algebra (in the sense of Toën 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 [ 2 ] who used it to categorify quantum groups.

Semi-Derived and Derived Hall Algebras for Stable Categories