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Model-Theoretic Galois Theory

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Model-Theoretic Galois Theory Model-theoretic Galois theory Jesse Han Introduction Poizat's imaginary Galois theory The Lascar group Grothendieck's Model-theoretic Galois theory Galois theory Internal covers and the Tannakian formalism Jesse Han Homology and cohomology GSCL 2016 Model-theoretic Galois theory What is Galois theory? Jesse Han Introduction § In the classical, algebraic sense: the classification of Poizat's imaginary intermediate field extensions in a field-theoretic Galois theory algebraic closure according to the structure of a The Lascar group profinite automorphism group. Grothendieck's § In the classical, model-theoretic sense: the classification Galois theory of definably closed subsets of a model-theoretic Internal covers and the Tannakian algebraic closure according to the structure of a formalism profinite automorphism group. (Poizat, 1983) Homology and cohomology § In the modern, algebraic sense: the classification of locally constant sheaves on a site according to the structure of a category of finite G-sets, where G is some flavor of fundamental group. (Grothendieck, SGA) § In the modern, model-theoretic sense: Lascar groups, (bounded) hyperimaginaries, generalized imaginaries and definable groupoids, internal covers? Model-theoretic Galois theory Elimination of imaginaries Jesse Han Recall: Introduction § A first-order theory T (uniformly) eliminates imaginaries Poizat's imaginary Galois theory if every 0-definable equivalence relation E arises as the kernel relation |ù kerpf qpa; bq ðñ |ù f paq “ f pbq of a The Lascar group 0-definable function f . Grothendieck's Galois theory § T codes definable sets if for every X “ 'pM; bq, there Internal covers and exists a tuple c and a formula px; cq such that the Tannakian formalism X “ pM; cq, with c unique. We usually suppress and say that c codes X . Homology and cohomology § EI always implies coding; the converse holds with some mild conditions on T (satisfied by ACFq. § In the monster M, codes are characterized up to interdefinability by the following property: c codes X precisely when AutpMq fixes X setwise ðñ AutpMq fixes c pointwise. Model-theoretic Galois theory Recovering the Galois correspondence Jesse Han Introduction Poizat's imaginary Definition. Galois theory § Let T be a first-order theory, and M |ù T a monster. If The Lascar group A Ď M is a small parameter set, AutpM{Aq acts on Grothendieck's Galois theory aclpAq. The image of the action Internal covers and the Tannakian Autp {Aq Ñ SympaclpAq{Aq formalism M Homology and cohomology is the absolute Galois group of A, and we denote it by GpaclpAq{Aq. § AutpM{Aq similarly induces a finite GpB{Aq for any finite A-definable set B, and in fact we have (as one might hope) GpaclpAq{Aq » lim GpB{Aq, so ÐÝ GpaclpAq{Aq is naturally a profinite group. Model-theoretic Galois theory Recovering the Galois correspondence Jesse Han Theorem. (Poizat) Let T code definable sets, M |ù T be a Introduction monster, and A Ď M small. Then Poizat's imaginary Galois theory Fixp´q The Lascar group Grothendieck's acl A acl A Galois theory Subpro-closed G p q {A Subdcl-closed p q {A Internal covers and ˆ ˙ ˆ ˙ the Tannakian ´ ¯ formalism GpaclpAq{´q Homology and cohomology is an inclusion-reversing bijective correspondence between the subgroups of GpaclpAq{Aq closed in the profinite topology and definably-closed intermediate extensions of A, where Fixp´q is given by H ÞÑ tb P aclpAq σpbq “ b; @h P Hu ˇ and GpaclpAq{´q by ˇ B ÞÑ tσ P GpaclpAq{Aq hpbq “ b; @b P Bu: ˇ ˇ Model-theoretic Galois theory Recovering the Galois correspondence Jesse Han Proof sketch. Introduction 1. Use coding of definable sets and the fact that algebraic Poizat's imaginary elements correspond to certain strong types to establish Galois theory a bijection GpSFpAq{Aq » GpaclpAq{Aq, where SFpAq is The Lascar group the space of strong types over A. Grothendieck's Galois theory 2. Using the fact that stppa{Aq “ stppb{Aq ðñ a „E b Internal covers and the Tannakian for every A-definable finite equivalence relation E, formalism establish that GpSFpAq{Aq » inverse limit of GpE{Aq, Homology and where GpE{Aq is the image of the action of AutpM{Aq cohomology on E-classes. 3. To every finite A-definable equivalence relation E with classes C1;:::; Cn and every subgroup Γ Ď GpE{Aq, associate a definable Γ-invariant relation rΓpx1;:::; xnq ðñ σPΓ i¤n xi P σpCi q: Denote the subgroup of such σ PŽGpSFŹpAq{Aq as StabprΓq. Model-theoretic Galois theory Recovering the Galois correspondence Jesse Han Introduction Poizat's imaginary 4. Since cylinder sets (hence stabilizers) are clopen in the Galois theory profinite topology, GpaclpAq{Bq is closed. By The Lascar group homogeneity of the monster, FixpHq is definably closed. Grothendieck's So the maps are well-defined. Galois theory Internal covers and 5. That Fixp´q is left-inverse to GpaclpAq{´q is also a the Tannakian formalism direct consequence of homogeneity in the monster. Homology and cohomology 6. To see that GpaclpAq{´q is left-inverse to Fixp´q, write a closed subgroup H as an intersection of open subgroups, each of the form StabprΓi q. Since each rΓi as a set has only finitely many A-conjugates, any of its codes ci is A-algebraic. Let B be the set of all codes for all rΓi . Since we're in a monster, this is definably closed, and we see that GpaclpAq{Bq “ H. By (5), FixpHq “ B, so GpaclpAq{´q left-inverts Fixp´q. X Model-theoretic Galois theory Recovering the Galois correspondence Jesse Han § Introduction Fact. ACF eliminates imaginaries. Poizat's imaginary Galois theory § Corollary. The same idea (with considerably less effort) The Lascar group works to recover the correspondence for Grothendieck's finitely-generated definably-closed normal extensions, Galois theory where \normal over A" means \closed under taking Internal covers and Autp {Aq-orbits". the Tannakian M formalism Homology and § Corollary. Classical Galois theory, at least in cohomology characteristic 0. § In characteristic p, due to the definability of Frobenius, definable closures are perfect hulls, and model-theoretic algebraic closures are field-theoretic algebraic closures of perfect hulls. So while our absolute Galois groups coincide with classical absolute Galois groups, they require the ground field to be perfect. Model-theoretic Galois theory Hyperimaginaries, ultraimaginaries Jesse Han Introduction Definition. Poizat's imaginary § Galois theory A subset X Ď M is type-definable if it is a The Lascar group possibly-infinite conjunction of definable sets. Grothendieck's § A hyperimaginary is an E-class (of possibly infinite Galois theory tuples) where E is a type-definable equivalence relation. Internal covers and the Tannakian formalism § An ultraimaginary is an E-class (of possibly infinite Homology and cohomology tuples) where E is an AutpMq-invariant equivalence relation. § If α is a flavor of imaginary, α is said to be bounded if its orbit under AutpMq is small (equivalently, if its parent equivalence relation E has only a small number of classes.) α is said to be (in)finitary if its parent equivalence relation E relates (in)finite tuples. Model-theoretic Galois theory What is the Lascar group? Jesse Han § Rather than classifying dcleq-closed parameter sets in Introduction eq M , the closed subgroups of the Lascar group will be Poizat's imaginary in Galois correspondence with finitary bounded Galois theory eq hyperimaginaries (so, certain small quotients of M The Lascar group instead of certain small subsets.) Grothendieck's Galois theory § But let's actually define this thing. Internal covers and the Tannakian formalism § First, we need to define: the group of Lascar strong Homology and automorphisms of M over a small parameter set A, cohomology which is the normal subgroup AutfpM{Aq of AutpM{Aq generated by AutpM{Mq: AĎMă ¤ M § This fixes bounded ultraimaginaries (in fact is the stabilizer of bounded ultraimaginaries). Model-theoretic Galois theory What is the Lascar group? Jesse Han Introduction § Definition. The Lascar group of T “ ThpMq over a Poizat's imaginary Galois theory small parameter set A is the quotient The Lascar group Autp {Aq ; Grothendieck's M AutfpM{Aq Galois theory Internal covers and M the Tannakian denoted GalLpT {Aq. formalism § Homology and This acts faithfully on bounded ultraimaginaries. cohomology § Topologize it as follows: for each E a parent equivalence relation of a bounded ultraimaginary, define a topology on the quotient (for some appropriate power of M) M{E by setting X Ď M{E closed ðñ ´1 πE pX q is an intersection of definable sets. GalLpT q acts on each M{E, and we take its topology to be the coarsest one making all the actions continuous. Model-theoretic Galois theory How does GalLpT {Aq relate to the absolute Jesse Han Galois group? Introduction Poizat's imaginary Galois theory § When aclpAq ă M, AutfpM{Aq “ AutpM{ aclpAqq, so The Lascar group that GalLpT {Aq » GpaclpAq{Aq. Grothendieck's Galois theory § Internal covers and This holds, for example, in ACF. the Tannakian formalism § 0 Homology and More generally, let GalLpT {Aq be the connected cohomology component of the identity. This is isomorphic to eq AutpM{ acl pAqq and we always have a short exact sequence 0 eq 1 Ñ GalLpT {Aq Ñ GalLpT {Aq Ñ Gpacl pAq{Aq Ñ 1: 0 and if T is stable, GalLpT {Aq “ 1. Model-theoretic Galois theory A hyperimaginary Galois theory Jesse Han Theorem. (Lascar, Pillay) Taking fixpoints and taking Introduction stabilizers yields a Galois correspondence between the closed Poizat's imaginary Galois theory subgroups of GalLpT {Aq and definably-closed sets of The Lascar group hyperimaginaries over A. Grothendieck's Galois theory Proof sketch. Internal covers and § Show that G Gal A is closed if and only if the Tannakian Ď LpM{ q formalism G “ Stabpeq for some bounded hyperimaginary e.
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