Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras

Abstract

Inspired by recent work of Bridgeland, from the category C^b(E) of bounded complexes over an exact category E satisfying certain finiteness conditions, we construct an associative unital "semi-derived Hall algebra" SDH(E). This algebra is an object sitting, in some sense, between the usual Hall algebra H(C^b(E)) and the Hall algebra of the bounded derived category D^b(E), introduced by Toen and further generalized by Xiao and Xu. It has the structure of a free module over a suitably defined quantum torus of acyclic complexes, with a basis given by the isomorphism classes of objects in the bounded derived category D^b(E). We prove the invariance of SDH(E) under derived equivalences induced by exact functors between exact categories. For E having enough projectives and such that each object has a finite projective resolution, we describe a similar construction for the category of Z/2-graded complexes, with similar properties of associativity, freeness over the quantum torus and derived invariance. In particular, we obtain that this Z/2-graded semi-derived Hall algebra is isomorphic to the two-periodic Hall algebra recently introduced by Bridgeland. We deduce that Bridgeland's Hall algebra is preserved under tilting. When E is hereditary and has enough projectives, we show that the multiplication in SDH(E) is given by the same formula as the Ringel-Hall multiplication, and SDH(E) is isomorphic to a certain quotient of the classical Hall algebra H(C^b(E)) localized at the classes of acyclic complexes. We also prove the same result in the Z/2-graded case.