Turing Degrees of Isomorphism Types of Geometric Objects

Abstract

We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures may fail to have a least degree. a category, one must construct pathological examples, and all of the examples in this paper are, in some sense, pathological.A broader goal of this paper is to initiate work in model theory and computable model theory on these classes of objects, especially schemes. To date, while there has been work on varieties, compact complex manifolds, and Kähler manifolds [13,14,15], we have been able to find no work on schemes in model theory or computable model theory.We believe that this does not owe to a lack of interest in schemes, but rather to their reputation for difficulty. Since the calculation of degrees of isomorphism types is a central question of computable model theory, it is our hope that our success with this problem will encourage more work in this area. In any case, Section 2 provides the logical infrastructure that any such work would require: a determination of a language whose morphisms correspond exactly to the classical homomorphisms.More precisely, we study Turing degrees of isomorphism types of structures from some well-known classes. This is a natural way, introduced by Jockusch and Richter (see [16]), of expressing the algorithmic complexity of the structure. We consider only countable structures for computable languages. The universe A of an infinite countable structure A can be identified with the set ω of all natural numbers. Furthermore, we often use the same symbol for the structure and its universe. (For the definition of a language and a structure see p. 8 of [11], and for a definition of a computable language see p. 509 of [12].)The two motivating constructions of the paper are of a union of varieties (Section 4) and schemes (Section 5). In the remainder of the present section, we will describe some necessary algorithmic and geometric background. Section 2 will describe a language for treating varieties, schemes, and other ringed spaces as first order structures (in the sense of model theory). The remaining three sections (excluding the appendix) will each develop an example of a class of structures that admits (in a sense to be made precise in Section 1.1) the encoding of arbitrary Turing degrees and of a minimal pair of Turing degrees. Section 3 explores the class of unions of subspaces of a fixed variety, under a weak topology. Section 4 constructs a union of varieties, and section 5 constructs schemes.1.1. Definitions, and Statement of Main Results. We say that a set X is Turing reducible to (computable in) a set Y , in symbols X ≤ T Y , if X can be computed by an algorithm with Y in its oracle. Turing reducibility is the more basic notion, in terms of which Turing degree is defined. We say that the sets X and Y are Turing equivalent, or have the same Turing degree, if X ≤ T Y and Y ≤ T X. We use ≡ T for Turing equiv...