Abstract

In this paper we define the modified Ringel–Hall algebra MH(A) of a hereditary abelian category A from the category Cb(A) of bounded Z-graded complexes. Two main results are obtained. One is to give a new proof of Green's formula on Ringel–Hall numbers by using the associative multiplication of the modified Ringel–Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel–Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel–Hall algebra is isomorphic to the tensor algebra of the derived Hall algebra and the torus of acyclic complexes and therefore the modified Ringel–Hall algebra is invariant under derived equivalences.

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