On the coefficients of the Coxeter polynomial of an accessible algebra

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 chi(A)(T) of a one-point extension algebra A = B[M] and the polynomial of the extension p(T) = 1/T((1 + T))chi(B)(T) - chi(A)(T)). If M is exceptional then p(T) = 1 + p(1) T + ... + p(n-3)T(n-3) + Tn-2. In that case, we call s(A : B) = p(1) the linear index of the extension A = B[M]. We give conditions for s(A : B) >= 0. For a tower T = (k = A(1), A(2), ..., A(n) = A) of access to A, that is, A(i) is a one-point (co-)extension of A(i-1) by an exceptional module, the index s(T) = Sigma((n)(i=2) s(A(i) : A(i-1)) = n - 1 - a(2), is an invariant depending on the derived equivalence class of A, where a(2) is the quadratic coefficient of chi(A)(T). We show that, in the case A is piecewise hereditary, then a(2) = 1 if and only if A is derived equivalent to a quiver algebra of type A(n). (C) 2012 Elsevier Inc. All rights reserved.