Resumen de Complexity classes of modules over finite dimensional algebras

Tom Howard

• 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 Q 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.