Zariski Geometries, Lecture 1
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1 Zariski Geometries, Lecture 1 Masanori Itai Dept of Math Sci, Tokai University, Japan August 30, 2011 at Kobe university
2 References HZ96 Ehud Hrushovski, Boris Zilber, Zariski geometries, J of the AMS, 1996 Z10 B. Zilber, Zariski Geometries, London Math. Soc. Lect Note Ser. 360, Cambridge, 2010
3 Table of Contents
4 Zilber Conjecture Strongly minimal trivial (no structure) linear (locally modular) : vector spaces non-linear (non-locally modular) : alg. closed fields
5 Zilber Conjecture Strongly minimal trivial (no structure) linear (locally modular) : vector spaces non-linear (non-locally modular) : alg. closed fields Zilber conjectured that any non-locally modular strongly minimal set interprets an acf!
6 Zilber Conjecture Strongly minimal trivial (no structure) linear (locally modular) : vector spaces non-linear (non-locally modular) : alg. closed fields Zilber conjectured that any non-locally modular strongly minimal set interprets an acf! Counter example was constructed by Hrushovski using generic model construction.
7 ZC is true for Zariski geometries In algebraic geometry; algebraically closed field Zariski topology, dimension notion
8 ZC is true for Zariski geometries In algebraic geometry; algebraically closed field Zariski topology, dimension notion Model theory of Zariski structures; Noetherian topology, dmension notion algebraically closed field
9 ZC is true for Zariski geometries In algebraic geometry; algebraically closed field Zariski topology, dimension notion Model theory of Zariski structures; Noetherian topology, dmension notion algebraically closed field Ample (non-linear, non-locally modular) Zariski geometry interprets an algebraically closed field.
10 Topological structures, p. 12 Consider a collection of topological spaces {M n : n N}. each M n is Noetherian proj is continuous graph of equality is closed fibers of closed sets are closed Cartesian products of closed sets is closed etc
11 Good dimension In subsection 3.1.1, we have a list of postulates for the dimesion notion; (DP) dimesion of a point (DU) dimension of unions (SI) strong irreducibility (AF) addition formula (FC) fiber condition
12 Zariski structures Definition (Def 3.1.3) {M n : n N} is Zariski structures if Noetherian topology dimension notion semi-proper : Let S cl M n be irreducible, then there exists a proper closed subset F prs such that. prs F prs
13 Zariski geometry Definition (Def 3.5.2) Zariski geometry = Zariski structure + (sps) + (EU) (sps) : strongly Pre-Smooth (EU) : Essentially Uncountable
14 Theorem (Thm ) A one-dimensional, uncountable, pre-smooth, irreducible Zariski structure M is a Zariski geometry.
15 Basic Examples (1) Algebraic varaity M over an acf. (Thm 3.4.1) Zariski structure is complete (ie. projection of closed is closed) if M is. (PS), pre-smooth if M is. (EU), essentially uncountable, if M.
16 Basic Examples (1) Algebraic varaity M over an acf. (Thm 3.4.1) Zariski structure is complete (ie. projection of closed is closed) if M is. (PS), pre-smooth if M is. (EU), essentially uncountable, if M. Compact complex manifold M with the notion of analytic dimension. It satisfies (PS) and (EU). (Thm 3.4.3)
17 Basic Examples (2) Proper varieties of rigid anlaytic geometry (Thm 3.4.7)
18 Basic Examples (2) Proper varieties of rigid anlaytic geometry (Thm 3.4.7) Definable sets of finite Morley rank and Morley degree 1 in differentially closed fields. (Thm 3.4.9, Pillay)
19 Semi-properness M is a Zariski structure. We want the theory of the structure M to have the elimination of quantifiers. For this, semi-properness axiom is needed: Definition (SP) Let S cl M n be irreducible, then there exists a proper closed subset F prs such that. prs F prs
20 QE for Zariski structures Theorem (Thm 3.2.1) A Zariski structure M admits elimination of quantifiers; that is any definable ssubset Q M n is constructible, i.e., boolean conbination of closed sets.
21 Proof of QE (1) We must show that the projection of a constructible subset is constructible. For this let Q = S P where both S, P closed. We show the theorem by induction on dim S. Let d S = min{dim S(a, M) : a prs}, F = {b prs : dim P(b, M) d S }.
22 Proof of QE (2) Let F be the closure of F.
23 Proof of QE (2) Let F be the closure of F. F is a proper closed subset of prs, by (FC).
24 Proof of QE (2) Let F be the closure of F. F is a proper closed subset of prs, by (FC). Since prs is irreducible, we have dim F < dim prs.
25 Proof of QE (2) Let F be the closure of F. F is a proper closed subset of prs, by (FC). Since prs is irreducible, we have dim F < dim prs. Let S = S pr 1 (F).
26 Proof of QE (2) Let F be the closure of F. F is a proper closed subset of prs, by (FC). Since prs is irreducible, we have dim F < dim prs. Let S = S pr 1 (F). Since F pr(s) pr(s), we have S S.
27 Proof of QE (2) Let F be the closure of F. F is a proper closed subset of prs, by (FC). Since prs is irreducible, we have dim F < dim prs. Let S = S pr 1 (F). Since F pr(s) pr(s), we have S S. Thus, dim S < dim S.
28 Proof of QE (3) Notice that pr(q) = pr(s P) pr(s P) pr(s F).
29 Proof of QE (3) Notice that pr(q) = pr(s P) pr(s P) pr(s F). If b pr(s P), then P(b, M) S(b, M). Thus (prs F) prq.
30 Proof of QE (3) Notice that pr(q) = pr(s P) pr(s P) pr(s F). If b pr(s P), then P(b, M) S(b, M). Thus (prs F) prq. Therefore, pr(q) = pr(s P) = pr(s P) (pr(s) F).
31 Proof of QE (3) Notice that pr(q) = pr(s P) pr(s P) pr(s F). If b pr(s P), then P(b, M) S(b, M). Thus (prs F) prq. Therefore, pr(q) = pr(s P) = pr(s P) (pr(s) F). Notice that pr(s) F is already in the desired form by induction hypothesis.
32 Proof of QE (3) Notice that pr(q) = pr(s P) pr(s P) pr(s F). If b pr(s P), then P(b, M) S(b, M). Thus (prs F) prq. Therefore, pr(q) = pr(s P) = pr(s P) (pr(s) F). Notice that pr(s) F is already in the desired form by induction hypothesis. Apply induction to S P to finish the proof.
33 QE for Zariski geometry Proposition (Prop 3.3.7) M : one-dimensional Zariski Geometry. Then the theory of M admits quantifier elimination.
34 Axioms for one-dimensional Zariski geometry, p. 31 (Z1) (QE) prs prs F, for some proper closed F cl prs (Z2) (SM) For S cl M n+1, there is m such that for all a M n S(a) = M or S(a) m (Z3) dim M n n. Given a closed irreducible S M n. Every component of the diagonal F {x i = x j } is of dimension dim S 1. Remark These are the axioms given in the paper [HZ].
35 strong minimality Corollary (Cor 3.3.8) M is strongly minimal.
36 Proof of S.M. with (Z2) Let E M n M be definable.
37 Proof of S.M. with (Z2) Let E M n M be definable. Show E(a) is finite or co-finite, with a uniform bound for all a M n.
38 Proof of S.M. with (Z2) Let E M n M be definable. Show E(a) is finite or co-finite, with a uniform bound for all a M n. WLOG, E = S F, with S, F closed.
39 Proof of S.M. with (Z2) Let E M n M be definable. Show E(a) is finite or co-finite, with a uniform bound for all a M n. WLOG, E = S F, with S, F closed. If S(a) is finite, then E(a) is finite with the same bound.
40 Proof of S.M. with (Z2) Let E M n M be definable. Show E(a) is finite or co-finite, with a uniform bound for all a M n. WLOG, E = S F, with S, F closed. If S(a) is finite, then E(a) is finite with the same bound. Apply (Z2) to both S, F to finish the proof.
41 Language M : Zariski structure C M n is closed, introduce a predicate symbol for each closed C M n
42 elementary extensions (1) Definition (Def ) a-closed : Let S(x, y) be l + m-ary closed set. S M l. Postulate that each S(a, M m ) is closed in M m. This gives the topology on each M. Dimension dim S(a, M ) = max{k N : a P(S, k)} + 1 where P(S, k) = {a prs : dim S pr 1 (a) > k}.
43 elementary extensions (2) Theorem (Thm ) M : Zariski structure satisfying (EU) M M M : Zariski structure
44 Summary Start with M : Noerthrian topology + dimension notion
45 Summary Start with M : Noerthrian topology + dimension notion Introduce language for M, hence model theory of M is possible
46 Summary Start with M : Noerthrian topology + dimension notion Introduce language for M, hence model theory of M is possible Theory of M admits quantifier elimination
47 Summary Start with M : Noerthrian topology + dimension notion Introduce language for M, hence model theory of M is possible Theory of M admits quantifier elimination M is strongly minimal
48 Summary Start with M : Noerthrian topology + dimension notion Introduce language for M, hence model theory of M is possible Theory of M admits quantifier elimination M is strongly minimal Elementary extension of M is Zariski structure (geometry)
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