On the coefficients of the Coxeter polynomial of an accessible algebra

Abstract

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter polynomial χA(T) of a one-point extension algebra A=B[M] and the polynomial of the extensionp(T)=1T((1+T)χB(T)−χA(T)). If M is exceptional then p(T)=1+p1T+⋯+pn−3Tn−3+Tn−2. In that case, we call s(A:B)=p1 the linear index of the extension A=B[M]. We give conditions for s(A:B)⩾0. For a tower T=(k=A1,A2,…,An=A) of access to A, that is, Ai is a one-point (co-)extension of Ai−1 by an exceptional module, the index s(T)=∑i=2ns(Ai:Ai−1)=n−1−a2, is an invariant depending on the derived equivalence class of A, where a2 is the quadratic coefficient of χA(T). We show that, in the case A is piecewise hereditary, then a2=1 if and only if A is derived equivalent to a quiver algebra of type An.