Parameterized Picard-Vessiot extensions and Atiyah extensions

Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard-Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations.Comment: 90 pages, minor correction

Multisorted modules and their model theory

Multisorted modules, equivalently representations of quivers, equivalently additive functors on preadditive categories, encompass a wide variety of additive structures. In addition, every module has a natural and useful multisorted extension by imaginaries. The model theory of multisorted modules works just as for the usual, 1-sorted modules. A number of examples are presented, some in considerable detail

Model companions of distributive p-algebras

Let B n , 0 ≤ n ≤ ω, be the equational classes of distributive p-algebras (precise definitions are given in §1). It has been known for some time that the elementary theories T n of B n possess model companions ; see, e.g., [6] and [14] and the references given there. However, no axiomatizations of were given, with the exception of n = 0 (Boolean case) and n= 1 (Stonian case). While the first case belongs to the folklore of the subject (see [6], also [11]), the second case presented considerable difficulties (see Schmitt [13]). Schmitt's use of methods characteristic for Stone algebras seems to prevent a ready adaptation of his results to the cases n ≥ 2. The natural way to get a hold on is to determine the class E( B n ) of existentially complete members of B n : Since exists, it equals the elementary theory of E( B n ). The present author succeeded [12] in solving the simpler problem of determining the classes A( B n ) of algebraically closed algebras in B n (exact definitions of A( B n ) and E( B n ) are given in §1) for all 0 > n < ω. A( B n ) is easier to handle since it contains sufficiently many "small” algebras-viz. finite direct products of certain subdirectly irreducibles-in terms of which the members of A( B n ) may be analyzed (in contrast, all members of E( B n ) are infinite and ℵ-homogeneous). As it turns out, A( B n ) is finitely axiomatizable for all n, and comparing the theories of A( B 0), A( B 1) with the explicitly known theories of E( B 0), E( B 1)-viz. , , a reasonable conjecture for , 2 ≤ n ≤ ω, is immediate. The main part of this paper is concerned with verifying that the conditions formalized by suffice to describe the algebras in E( B n ) (necessity is easy). This verification rests on the same combinatorial techniques as used in [12] to describe the members of A( B n

Model-theoretic aspects of the Gurarij operator system

We establish some of the basic model theoretic facts about the Gurarij operator system GS recently constructed by the second-named author. In particular, we show: (1) GS is the unique separable 1-exact existentially closed operator system; (2) GS is the unique separable nuclear model of its theory; (3) every embedding of GS into its ultrapower is elementary; (4) GS is the prime model of its theory; and (5) GS does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any q ∈ N, the theories of M_q -spaces and M_q -systems do have a model companion, namely the Fraïssé limit of the class of finite-dimensional M_q -spaces and M_q -systems respectively; moreover, we show that the model companion is separably categorical. We conclude the paper by showing that no C* algebra can be existentially closed as an operator system