Complexity classes of modules over finite dimensional algebras

Directed graphs called “syzygy quivers” are used to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finite dimensional syzygy-finite representation (e.g. any representation of a monomial algebra), we show that this growth rate is poly-exponential, i.e. the product of a polynomial and an exponential function, and give a procedure for computing the corresponding degree and base from a syzygy quiver. We characterize the growth rates arising in this context. The bases of the occurring exponential functions are the Perron numbers: real, nonnegative algebraic integers b   whose irreducible polynomial over QQ has no root with modulus larger than b. Modulo this restriction, arbitrary degrees and bases occur. Moreover, we show that these growth rates are invariant under stable derived equivalences.