Representations of monomial algebras have poly-exponential complexities

TL;DR

This paper introduces the use of syzygy quivers to analyze the asymptotic growth of syzygies in monomial algebra representations, revealing they grow poly-exponentially with computable parameters and invariance under stable derived equivalences.

Contribution

It provides a novel method to compute and characterize poly-exponential growth rates of syzygies in monomial algebra representations using syzygy quivers.

Findings

Growth rates are poly-exponential, combining polynomial and exponential functions.

The exponential bases are nonnegative algebraic integers with specific polynomial root properties.

Growth rates are invariant under stable derived equivalences.

Abstract

We use directed graphs called "syzygy quivers" to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finitely generated 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 real, nonnegative algebraic integers $b$ whose irreducible polynomial over $\mathbb{Q}$ has no root with with modulus larger than $b$. Moreover, we show that these growth rates are invariant under stable derived equivalences.