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

Abstract

In this paper we define the modified Ringel-Hall algebra MH(A) of a hereditary abelian category A from the category C b (A) of bounded Z-graded complexes. Two main results have been 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 so the modified Ringel-Hall algebra is invariant under derived equivalences.

Toën's formula and Green's formula

“…Actually, if A is a hereditary abelian category with enough projectives, it is similar to [4] that we can also give a simple proof of Green's formula via the associativity of Bridgeland's Hall algebra of bounded complexes or m-cyclic (m > 2) complexes over projective objects, which is defined in [2] and further studied in [16].…”

“…This work is inspired by Ji Lin and Liangang Peng's work in [4]. The author is pleasantly informed that Jie Xiao and Fan Xu have also known this idea after the paper of Lin and Peng, but they do not write out.…”

“…In order to generalise Bridgeland's construction to any hereditary abelian category which may not have enough projectives, Lu and Peng [5] introduced the modified Ringel-Hall algebra. Recently, Lin and Peng [4] used the associativity of the modified Ringel-Hall algebra of bounded complexes of a hereditary abelian category to give a simple proof of Green's formula.…”

“…In this note, inspired by the work in [4], we want to disclose the relation between Green's formula and Toën's formula, which are two significant formulas in Hall algebra and derived Hall algebra, respectively. Explicitly, we will prove that Toën's formula "implies" Green's formula.…”

Hall algebras associated to complexes of fixed size

“…By Proposition 5.5 and [13, Proposition 5.3], we obtain that the Hall algebra MH (A ) is isomorphic to the modified Ringel-Hall algebra MH tw (A ) defined in[13]. Explicitly, there exists an isomorphism ϕ : MH (A ) −→ MH tw (A ) defined on generators by E M,r → U M,−r and K α,r → K α,−r .For simplicity, we recall the twisted derived Hall algebra DH(A ) of A in the form of generators and relations in the following Proposition 5.7.…”