ALGEBRAIC ENTROPY OF ENDOMORPHISMS OVER LOCAL ONE-DIMENSIONAL DOMAINS

Abstract

Let R be a local one-dimensional integral domain, with maximal ideal 𝔐 and field of fractions Q. Here, a local ring is not necessarily Noetherian. We consider the algebraic entropy ent g, defined using the invariant gen, where, for M a finitely generated R-module, gen (M) is its minimal number of generators. We relate some natural properties of R with the algebraic entropies ent g(ϕ) of the elements ϕ ∈ Q, regarded as endomorphisms in End R(Q). Specifically, let R be dominated by an Archimedean valuation domain V, with maximal ideal P. We examine the uniqueness of V, the transcendency of the residue field extension V/P over R/𝔐, and the condition for R to be a pseudo-valuation domain. We get mutual information between these properties and the behavior of ent g, focusing on the conditions ent g(ϕ) = 0 for every ϕ ∈ Q, ent g(ψ) = ∞ for some ψ ∈ Q, and ent g(ϕ) < ∞ for every ϕ ∈ Q.