Introduction to Algebraic Analysis on Ore Extensions

Where can Ore extensions and homological algebra be applied? One possibility is in algebraic analysis: algebraic analysis studies functional linear systems (FLSs) of (ordinary or partial) differential equations, difference equations, differential equations with delay, etc., by means of techniques of homological algebra which use finitely presented modules over noncommutative rings of polynomial type, such as rings of differential operators, rings of polynomials with delay, iterated skew polynomial rings, Ore extensions, or more generally, domains. The structural properties of the FLSs can be described by homological-matrix constructive methods, and their study is one of the central goals of the present work. We have divided the content into four sections. In the first one, we provide enough examples of Ore extensions [28]. We also define a closely related class of rings that generalizes a quite wide class of Ore extensions, the so-called Poincaré-Birkhoff-Witt skew extensions. In the second section,we review some basic topics of homological algebra and define FLSs. The third section studies the decomposition and factorization of FLSs. We will see the relationship between homological properties of modules and triangular decompositions of FLSs, as well as the relationship between idempotent matrices and diagonal block decompositions. In the last section, we present the main ingredients of the noncommutative theory of Gröbner bases for skew PBW extensions and their application in some of the calculations involved in the results of Sects. 2. We illustrate with concrete examples of Ore algebras some of the results by means of Gröbner bases and computational tools.