Mathematics > Commutative Algebra
[Submitted on 15 Mar 2014 (v1), last revised 3 Oct 2014 (this version, v2)]
Title:Tannakian categories with semigroup actions
View PDFAbstract:Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.
Submission history
From: Alexey Ovchinnikov [view email][v1] Sat, 15 Mar 2014 20:39:45 UTC (32 KB)
[v2] Fri, 3 Oct 2014 19:51:40 UTC (30 KB)
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