CM-triviality and Geometric Elimination of Imaginaries
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1 CM-triviality and Geometric Elimination of Imaginaries Ikuo Yoneda Department of Mathematics Tokai University 18/7/2007 1
2 1 Introduction Hrushovski gave a counterexample to the Zilber s conjecture on strongly minimal sets by Generic Relational Structures, i.e. relational countable structures constructed by amalgamating relational finite structures. Generic relational structures are usually CM-TRIVIAL. 2
3 To show the CM-triviality of generic structures, we needed two steps. 1st step: Show weak elimination of imaginaries. 2nd step: Working in the real sort, show CM-triviality. 3
4 Following these two steps, I proved CM-triviality of Herwig s weight ω small theory, and Baldwin-Shi s stable generic structures. 4
5 A question comes up : Is there a way to show CM-triviality without showing Weak Elimination of Imaginaries? I find the following answer. 5
6 THE MAIN RESULT In simple theories with elimination of hyperimaginaries, CM-triviality in the real sort (I will define) Geometric elimination of imaginaries + CM-triviality in the original sense, firstly introduced by Hrushovski. 6
7 2 Set-up From now on, let T be a simple theory with elimination of hyperimaginaries and M be a sufficiently saturated model of T. (Hyper-)imaginary elements are equivalence classes of (type-)definable equivalence relations. We work in M eq, the eq-structure, consisting of imaginary elements. 7
8 3 CM-triviality Hrushovski s Definition for CM-triviality T is CM-trivial, if for any a, A, B M eq with acl eq (aa) acl eq (B) = acl eq (A), Cb(stp(a/A)) acl eq (Cb(stp(a/B))). acl eq ( ) denotes algebraic closure in M eq 8
9 Equivalently, for any a, A = acl eq (A), B = acl eq (B) M eq, a B a B. A A acl eq (a,b) We are working in the eq-structure, not in the real sort. 9
10 My Definition for CM-triviality T is CM-trivial in the real sort, if, for any ā, A = acl(a), B = acl(b) M, ā B ā B. A A acl(ā,b) Notice that everything is in the real sort. 10
11 IND/I T has the independence over intersections (IND/I), if, for any ā, A = acl(a), B = acl(b) M ā A B, ā B A ā A B AB. Notice that everything is in the real sort. 11
12 Proposition A CM-triviality in the real sort IND/I. 12
13 The key point of the proof: Assume ā A B, ā B A, A = acl(a), B = acl(b). By ā B AB, acl(ā, B) AB = B follows. So we have B A acl(ā, B) A (acl(ā, B) AB) A B A. By CM-triviality in the real sort, we have ā B. acl(ā,b) A 13
14 Proposition B IND/I GEI. Geometric Elimination of Imaginaries means that for any i M eq, there exists ā M such that i acl eq (ā), ā acl eq (i). 14
15 The proof of IND/I GEI. Fix i= ā E. Take b, c such that b, c = tp(ā/i) and ā, b, c are independent over i. By ā b c, ā c b and IND/I, we have ā b, c. acl( b) acl( c) Let A= acl( b) acl( c). As i dcl eq (ā), we have i acl eq (A). By b i c, we see A acleq (i). 15
16 Under GEI, CM-triviality in the real sort=cm-triviality. Main Theorem CM-triviality in the real sort GEI+CM-triviality. 16
17 Two Remarks (1) Simple generic structures have the following NICE characterization of non-forking; A A B B A A B B = A B = cl M (A B) for any A = acl(a), B = acl(b) M. 17
18 From this, we can check that simple generic structures are CM-trivial in the real sort. Main Theorem directly shows the CM-triviality of simple generic structures. 18
19 (2) CM-triviality in the real sort CM-triviality in the original sense. In [E], D.Evans gave an ω-categorical SU = 1 CM-trivial structure C without WEI interpreted in an ω-categorical SU = 2 generic binary graph. I checked C does not have GEI. 19
20 Remark on IND/I In pregeometric surgical theories, IND/I GEI. In O-minimal case, IND/I EI only holds. 20
21 ENDING : 4 Problems on CM-triviality (1) In stable theories, does CM-triviality imply CM-triviality in the real sort? 21
22 (2) Is any superstable CM-trivial theory ω-stable? This is a generalization of Baldwin s Problem: Is any superstable ω-saturated generic structure ω-stable? 22
23 (3) Recall that n-ampleness is defined in the eq-structures. non-1-amlpleness One-basedness non-2-ampleness CM-triviality non-3-ampleness non-4-ampleness Define non-3-ampleness in the real sort. And does it imply GEI? 23
24 (4) In Zariski geometries, local modularity is equivalent to CM-triviality. In O-minimal theories, is local modularity equivalent to CM-triviality? 24
25 References [E] D.M.Evans, ℵ 0 -categorical structures with a predimension, Annals of Pure and Applied Logic 116 (2002), [G] Jerry Gagelman, Stability in geometric theories, Annals of Pure and Applied Logic 132 (2005),
26 [H] Ehud Hrushovski, A new strongly minimal sets, Annals of Pure and Applied Logic 88 (1993), [Y] Ikuo Yoneda, Forking and some eliminations of imaginaries, submitted, October
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