Modified Ringel–Hall algebras, Green's formula and derived Hall algebras

摘要

In this paper we define the modified Ringel-Hall algebra$mathcal{M}mathcal{H}(mathcal{A})$ of a hereditary abelian category$mathcal{A}$ from the category $C^b(mathcal{A})$ of bounded$mathbb{Z}$-graded complexes, and prove that in certain twisted cases thederived Hall algebra can be embedded in the modified Ringel-Hall algebra. As aconsequence, we get that the modified Ringel-Hall algebra is isomorphic to thetensor algebra of the derived Hall algebra and the torus of acyclic complexesand so the modified Ringel-Hall algebra is invariant under derivedequivalences.