Strongly étale difference algebras and Babbitt's decomposition

Abstract

We introduce a class of stronglyétale difference algebras, whose role in the study of difference equations is analogous to the role ofétale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's decomposition theorem and we present applications to difference algebraic groups and the compatibility problem.

Algebraic groups as difference Galois groups of linear differential equations

“…We will only introduce the definitions and results necessary for proving Theorem 6.1. Most of the required difference algebraic results appeared in [TW18].…”

“…Recall that a group scheme G of finite type over k is étale if k[G] is an étale k-algebra, i.e., k[G] ⊗ k k is a a finite direct product of copies of k. The group π 0 (G) of connected components of G can be defined through the following universal property: There exists a morphism G → π 0 (G) of affine group schemes over k such that π 0 (G) is étale and for every étale group scheme H with a morphism G → H, there exists a unique morphism π 0 (G) → H such that To follow a similar path for difference algebraic groups we first need to define an appropriate difference analog of étale algebras. Following [TW18] we make the following definition. Definition 6.2.…”

“…The following proposition was essentially proved in [TW18], but there the results were formulated in an algebraic manner. Here we give a more geometric interpretation.…”

“…2.3]). By Theorem 3.2 of [TW18] the k-σ-subalgebra π σ 0 (k{G}) of k{G} is a k-σ-Hopf-subalgebra and by Theorem 4.5 of [Wiba] every k-σ-Hopf-subalgebra of a finitely σ-generated k-σ-Hopf algebra is finitely σ-generated. It follows that the σ-algebraic group π σ 0 (G) represented by π σ 0 (k{G}) is strongly σ-étale.…”

Étale difference algebraic groups

“…The σ-identity component was already used in [BW22] to derive a necessary condition on a σ-algebraic group to be a σ-Galois group over the difference-differential field C(x) with derivation δ = d dx and endomorphism σ given by σ(f (x)) = f (x + 1). Some difference algebraic results necessary to define the difference identity component also already appeared in [TW18].…”