Holomorphy over Finite-dimensional Commutative Associative C-algebras: Local Theory

Abstract

To a morphism $$\mathcal{A} \xrightarrow{\varphi} \mathcal{B}$$ of finite-dimensional commutative associative unital Banach $$\mathbb{C}$$-algebras one can associate a sheaf $$\mathcal{O}_\varphi$$ on the underlying topological space $$|\mathcal{A}|$$ of $$\mathcal{A}$$ consisting of $$\mathcal{B}$$-valued differentiable functions $$f$$ with $$\mathcal{A}$$-linear differential $$Df$$. It turns out that this class of functions exhibits a theory very similar to the classical complex analysis of one variable. In this article we give only an overview of some new results concerning the fundamentals of the corresponding local theory while at the same time also strengthen various already existing results scattered throughout the literature.