Toward an imaginary Ax-Kochen-Ershov principle Work in progress with Martin Hils Silvain Rideau CNRS, IMJ-PRG, Université Paris Diderot March 9 2018 1 / 16
A crash course on imaginaries = T eq . = T eq , there Definition denotes the collection of A -induced imaginary sorts of D . 2 / 16 For all L -theory T , we define: ▶ L eq = L ∪ ∪ X ⊆ Y × Z ∅ -definable { E X , f X : Y → E X } . ▶ T eq = T ∪ ∪ X { f X induces an bijection E X ≃ Y /( X y 1 = X y 2 ) } . = T has a unique L eq -enrichment M eq | ▶ Every M | ▶ If D is a collection of stably embedded A -definable set, D eq ▶ Any M -definable set X has a smallest definably closed set of definition ⌜ X ⌝ in M eq . Let T be a theory and D a collection of ∅ -interpretable sets. ▶ T eliminates imaginaries up to D if, for all e ∈ M eq | exists d ∈ D ( dcl ( e )) such that e ∈ dcl ( d ) . ▶ T weakly eliminates imaginaries up to D if, for all e ∈ M eq | = T eq , there exists d ∈ D ( acl ( e )) such that e ∈ dcl ( d ) .
Theorem (Haskell-Hrushovski-Macpherson, 2006) Imaginaries in valued fields Unreasonable Hope (Imaginary AKE, first attempt) 3 / 16 In Hen 0 , 0 , certain quotients cannot be eliminated: ▶ Γ = K × / O × . ▶ k = O / m . ▶ S n = GL n ( K )/ GL n ( O ) , the moduli space of lattices in K n . ▶ For all s ∈ S n , V s = O s / m s , a dimension n k -vector space. ▶ T n = ∪ s ∈ S n V s . ACVF eliminates imaginaries up to G = K ∪ ∪ n ( S n ∪ T n ) . ▶ k eq and Γ eq . Hen 0 , 0 weakly eliminates imaginaries up to G ∪ k eq ∪ Γ eq .
Some more imaginaries A New Hope (Imaginary AKE, second attempt) = Hen eq 4 / 16 Certain quotients cannot be eliminated in G ∪ k eq ∪ Γ eq : ▶ K / K n and, more generally, ( K / K n ) eq . ▶ Solved by considering RV eq , where RV = K × /1 + m = T 1 . ▶ K / I for some I ⊆ O definable ideal which is not a multiple of O or m , and higher dimensional equivalent. ▶ Prevented by requiring the value group to be definably complete, e.g ordered groups elementarily equivalent to Z or Q . ▶ R b = { b ′ ⊆ b maximal open subball } and, more generally, R eq b , if R b ( dcl ( b )) = ∅ . ▶ Solved by considering V eq s for some s ∈ S n ( dcl ( b )) . For all M | = Hen 0 , 0 and A = acl ( A ) ⊆ G ( M ) , let St A = ∪ s ∈ S n ( A ) V s and D A = A ∪ RV ∪ St A . Let e ∈ M eq | 0 , 0 and A = G ( acl ( e )) . Assume Γ( M ) is divisible or a Z -group. Then e is weakly coded in D eq A .
A local look at imaginaries Proposition Then T weakly eliminates imaginaries up to D . Sometimes, it is easier to look for a definable p . One can then proceed in two steps: 5 / 16 Let D be a collection of ∅ -interpretable sets in T . Assume: ▶ For every definable X , there exists a D ( acl ( ⌜ X ⌝ )) -invariant type p ( x ) such that p ( x ) ⊢ x ∈ X . ▶ For every definable X , find a acl ( ⌜ X ⌝ ) -definable type p ( x ) such that p ( x ) ⊢ x ∈ X . ▶ For any A = acl ( A ) ⊆ M eq show that any A -definable type p is D ( A ) -definable.
(Almost) Theorem = T . There exists a quantifier free account of elimination of imaginaries in ACVF. arguments. 6 / 16 Density of quantifier free definable types Hen 0 , 0 Let T ⊇ Hen 0 , 0 be a complete theory in an RV-enrichment of L div . Assume k and Γ are stably embedded and algebraically bounded. Assume also that Γ is definably complete. ▶ For all A ⊆ M eq | = T eq and quantifier free A -definable L div -type p , then p is G ( dcl ( A )) -definable. ▶ Let X be definable in M | acl ( ⌜ X ⌝ ) -definable L div -type p consistent with X . ▶ The first statement is essentially proved by Johnson in his ▶ The proof of the second statement is a mix of existing
Completing quantifier free types An alternative formulation of field quantifier elimination Then 7 / 16 Let M ≼ C | = T and a ∈ K be a tuple. Assume rv ( M ( a ))) ⊆ dcl 0 ( M ρ ( a )) , where ρ ( a ) ∈ RV ( dcl 0 ( Ma )) . tp 0 ( a / M ) ∪ tp ( ρ ( a )/ rv ( M )) ⊢ tp ( a / M ) . ▶ If a is generic in some ball b over M and c ∈ b ( M ) , then rv ( M ( a )) ⊆ dcl 0 ( rv ( M ) rv ( a − c )) . ▶ Moreover, if b is open, ρ ( a ) = rv ( a − c ) does not depend on the choice of c ∈ b ( M ) . ▶ So [ ρ ] q , the germ of ρ over the b -definable type q = tp 0 ( a / M ) , is in dcl ( b ) . ▶ It follows that tp ( a / M ) is b RV ( M ) -invariant.
Proposition 8 / 16 Computing rv ( M ( a )) Assume tp 0 ( a / M ) is N -definable for some N ≼ M , then there exists ρ ( a ) ∈ dcl 0 ( Na ) such that rv ( M ( ac )) ⊆ dcl 0 ( rv ( M ) ρ ( a )) . Let c ∈ K be such that p = tp 0 ( ac / M ) is A -definable for some A ⊆ M eq and q = tp 0 ( a / M ) . Assume one of the following holds: ▶ c is generic in an open ball or a strict intersection of balls over M ( a ) ; ▶ c is generic in a closed ball b over M ( a ) and there exists g ( a ) ∈ R b ( dcl 0 ( Ma )) with [ q ] g ∈ dcl ( A ) ; ▶ c ∈ M ( a ) alg . Then there exists ρ ( a ) ∈ RV ( dcl 0 ( Ma )) with [ ρ ] p ∈ dcl ( A ) and rv ( M ( ac )) ⊆ dcl 0 ( rv ( M ( a )) ρ ( ac )) .
Finding invariant types Corollary which is internal to RV. 9 / 16 Assume tp 0 ( a / M ) is N -definable for some N ≼ M , then tp ( a / M ) is N RV ( M ) -invariant. Assume k and Γ are stably embedded and algebraically bounded. Assume also that Γ is definably complete. ▶ Pick any e ∈ M eq and let A = G ( acl ( e )) . Let f be ∅ -definable and a ∈ K n such that e = f ( a ) . ▶ We find a quantifier free A -definable L div -type p consistent with f − 1 ( e ) . So we may assume tp 0 ( a / M ) is A -definable. ▶ So tp ( a / M ) — and hence tp ( e / M ) — is N RV ( M ) -invariant, for any A ⊆ N ≼ M . ▶ Since RV is stably embedded, e ∈ dcl ( N RV ( M )) . ▶ It follows that there exists some G ( acl ( e )) -definable set E
constants are added to the residue field. (Almost) Theorem 10 / 16 Imaginaries in Hen 0 , 0 , take one Assume k and Γ are stably embedded and algebraically bounded, Γ is definably complete and for all A ⊆ M eq and any A -definable ball b , either b isolates a complete type or R b ( dcl ( A )) ̸ = ∅ . ▶ for all A ⊆ M eq , there exists N ⊇ G ( A ) such that tp ( N / G ( A )) ⊢ tp ( N / A ) ; ▶ T weakly eliminates imaginaries up to G ∪ RV eq . ▶ Let k be a characteristic zero bounded PAC field, then k (( t )) and k (( t Q )) eliminate imaginaries up to G , provided certain ▶ The above result still holds if one adds angular components; i.e. a section of 1 → k × → RV → Γ → 0 . ▶ With some tweaking, similar results should hold for k elementarily equivalent to a finite extension of Q p .
(Almost) Theorem Theorem 11 / 16 Imaginaries in Hen 0 , 0 , take two Assume that for all A ⊆ G ( M ) and ϵ ∈ St A ( dcl 0 ( C )) , there is η ∈ St A ( C ) with ϵ ∈ dcl 0 ( A η ) and η is definable over A ϵ in ( C alg , C ) . ▶ If tp 0 ( a / M ) is stably dominated over A and c is generic, over M ( a ) , in a closed ball b ∈ dcl 0 ( Aa ) , then rv ( M ( ac )) ⊆ dcl 0 ( rv ( M ( a )) St A ( M ) ac ) . ▶ For all A ⊆ G ( M ) , there exists N ⊇ G ( A ) such that tp ( N / M ) is AD A ( M ) -invariant. If tp 0 ( a / M ) is A -definable then tp ( a / M ) is AD A ( M ) -invariant. Assume that k is stably embedded and algebraically bounded and Γ is a pure ordered group which is either divisible or a Z -group. Then any e ∈ M eq is weakly coded in D eq A , where A = G ( acl ( e )) .
Valued fields with operators Corollary Theorem (R.,R.-Simon) 12 / 16 Let δ = { δ i : K → K | i ∈ I } , L δ = L ∪ δ . Let T δ ⊇ T ⊇ ACVF 0 , 0 and M ≼ C | = T δ . Assume that for all tuples a ∈ K , tp ( δ ( a )/ M ) ⊢ tp δ ( a / M ) . If tp 0 ( δ ( a )/ M ) is A -definable, for some A ⊆ G ( M ) , then tp δ ( a / M ) is AD A ( M ) -invariant. Assume that k , Γ are stably embedded and k eq , Γ eq eliminate ∃ ∞ . ▶ For any L δ ( M ) -definable X , there exists a ∈ X such that tp 0 ( a / M ) is L δ ( acl L δ ( ⌜ X ⌝ )) -definable. ▶ Assume, moreover that any externally L -definable subset of Γ n ( M ) which is L δ ( M ) -definable is L ( M ) -definable. Then, for every A = dcl δ ( A ) ⊆ M eq , any L δ ( A ) -definable quantifier free L div -type is L ( G ( A )) -definable.
13 / 16 By results of Hrushovski, Durhan and Pal: particular, it is o -minimal. The asymptotic theory of ( F p ( t ) alg , Φ p ) Let VFA 0 be the theory of equicharacteristic zero existentially closed σ -Henselian fields with an ω -increasing automorphism: ▶ σ ( O ) = O ; ▶ if x ∈ m , for all n ∈ Z > 0 , v ( σ ( c )) > v ( c ) . We work in L RV σ with sorts K and RV, the ring language on both K and RV, and maps rv : K → RV, σ : K → K and σ RV : RV → RV. ▶ For all ( k , σ k ) | = ACFA 0 and (Γ , σ Γ ) | = ω DOAG, γ a γ t γ ) = ∑ γ σ k ( a γ ) t σ ( γ ) . ( k ((Γ)) , σ ) | = VFA 0 where σ ( ∑ ▶ For every non-principal ultrafilter U on the set of primes, ∏ p → U ( F p ( t ) alg , Φ p ) | = VFA 0 . ▶ VFA 0 eliminates field quantifiers. ▶ k is stably embedded and a pure model of ACFA 0 . ▶ Γ is stably embedded and a pure model of ω DOAG. In
Recommend
More recommend
