On finite imaginaries
Ehud Hrushovski, S. Barry Cooper, Herman Geuvers, Anand Pillay, Jouko Vaananen
Logic Colloquium 2006
Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T . We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T σ , and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite
... ds. The questions this manuscript addresses arose in the course of an investigation of the imaginary sorts in ultraproducts of p-adic fields. These were shown to be understandable given the imaginary sorts of certain finite-dimensional vector spaces over the residue field. The residue field is pseudo-finite, and the imaginary elements there were previously studied, and shown in fact to be eliminable over an appropriate base. It remains therefore to describe the imaginaries of finite-dimensional vector spaces over a field F , given those of F . I expected this step to be rather easy; but it turned out to become easy only after a number of issues, of interest in themselves, are made clear. Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T . Internal covers were recognized as central in the study of totally categorical structures, but nevertheless remained mysterious; it was not clear how to describe the possible T from the point of view of T . We give an account of this here, in terms of groupoids in place of equivalence relations. This description permits the view of the cover as a generalized imaginary sort. This seems to be a useful language even for finite covers, though there the situation is rather wellunderstood; cf. ,  . We concentrate on finite generalized imaginaries, and describe a a connection between elimination of imaginaries and higher amalgamation principles within the algebraic closure of an independent n-tuple. The familiar imaginaries of T eq correspond to 3-amalgamation, as was understood for some time for stable and simple theories, and finite generalized imaginaries correspond to 4-amalgamation. This brings out ideas present in some form in , , ,  . In particular, 4-amalgamation always holds for stable theory T , if "algebraic closure" is taken to include generalized imaginaries. We also relate uniqueness of n-amalgamation to existence of n + 1 -amalgamation; using "all" finite imaginaries (not necessarily arising from groupoids) we show that n-amalgamation exists and is unique for all n. Thanks to Yad Hanadiv, and to the Israel Science Foundation, grant 1048/07. Thanks to the referee for many comments. MSC 03c99, 18B40. HRUSHOVSKI Adding an automorphism to the language to obtain a Robinson theory T σ has the effect of shifting the amalgamation dimension by one; n-amalgamation in the expanded language corresponds to n+1 -amalgamation for T . Thus ordinary imaginaries of T σ can be understood, given generalized imaginaries of T . We thus find a strong relation between four things: covers, failure of uniqueness for 3-amalgamation, imaginaries of T σ , and definable groupoids. A clear continuation to n = 4 would be interesting. Returning to the original motivation, we use these ideas to determine the imaginaries for systems of finite-dimensional vector spaces over fields, and especially over pseudo-finite fields (Theorem 5.10). Preliminaries Let T be a first-order theory, with universal domain U. Def(U) is the category of U-definable sets (with parameters) and maps between them. 1 Let A, B be small subsets of U. For each b ∈ B , we provide a new constant symbol c b ; and for each a ∈ A, a new variable x a . We write tp(A/B) for the set of all formulas with these new variables and constants, true in U under the eponymous interpretation of constant symbols and assignment of variables. This is useful in expressions such as tp(A/B) |= tp(A/B ). An ∞ -definable set is the solution set of a partial type (of bounded size; say bounded by the cardinality of the language.) Morphisms between ∞ -definable sets are still induced by ordinary definable maps. If the partial type is allowed to have infinitely many variables, the set is called -definable instead. -definable sets can also be viewed as projective systems of definable sets and maps. Dually, a -definable set is the complement of an ∞ -definable set. When we say a set P is definable, we mean: without parameters. If we wish to speak about a set definable with parameters a, we will exhibit these parameters in the notation: P a . We will often consider two languages L ⊂ L . The language L may have more sorts than L. Let T be a complete theory for L , T = T |L. We say T is embedded if any relation of L is T -equivalent to a formula of L . We say the sorts of L are stably embedded if in any model M |= T , any M -definable subset of S 1 × ×S k (where the S i are L-sorts) is also definable with parameters from ∪S i (M ). This basic notion has various equivalent forms, see appendix to  and also  . Let D be a definable set of L . We say D is internal to L if in some (or any) model M of T , there exist sorts S 1 , . . . , S k of L and an M -definable map f whose domain is a subset of S 1 × ×S k , and whose image is D . See  , appendix, where it is shown that internality is associated with definable automorphism groups; indeed, assuming T is embedded and stably embedded in T , and L \ L is finite for simplicity, and letting M denote the L -sorts of M , there exists a definable group G such that G(M ) can be identified with Aut(D(M )/M ). G is called the liaison group, a term due to Poizat. It is also shown in  that G is M -isomorphic to an M -definable group. In §2 we will prove a more precise, parameter-free version, using the notion of a definable groupoid. We can immediately introduce one of the main notions of the paper. 1 More generally we can work with a "Robinson theory", a universal theory with the amalgamation property for substructure; one then works with substructures of a universal domain, and takes "definable" to mean: quantifier-free definable. This was one of the "contexts" of ; I dubbed it "Robinson" when unaware of this reference, and the name stuck.