Functions Holomorphic over Finite-Dimensional Commutative Associative Algebras 1: One-Variable Local Theory I

Abstract

We study in detail the one-variable local theory of functions holomorphic over a finite-dimensional commutative associative unital $\mathbb{C}$-algebra $\mathcal{A}$, showing that it shares a multitude of features with the classical one-variable Complex Analysis, including the validity of the Jacobian conjecture for $\mathcal{A}$-holomorphic regular maps and a generalized Homological Cauchy's Integral Formula. In fact, in doing so we replace $\mathcal{A}$ by a morphism $\varphi: \mathcal{A} \to \mathcal{B}$ in the category of finite-dimensional commutative associative unital $\mathbb{C}$-algebras in a natural manner, paving a way to establishing an appropriate category of Funktionentheorien (ger. function theories). We also treat the very instructive case of non-unital finite-dimensional commutative associative $\mathbb{R}$-algebras as far as it serves above agenda.