Imaginaries in valued fields with operators Silvain Rideau UC - PowerPoint PPT Presentation

Imaginaries in valued fields with operators Silvain Rideau UC Berkeley Mai 25 2016 1 / 8

Valued fields the field of p -adic numbers. Example 2 / 8 ▸ Let k be a field. On k ( X ) , v X ( X n P / Q ) = n ∈ Z when X ∧ P = X ∧ Q = 1. Its competion is k (( X )) = { ∑ i > i 0 c i X i ∶ c i ∈ k } . ▸ On Q , v p ( p n a / b ) = n ∈ Z when p ∧ a = p ∧ b = 1. Its compeltion is Q p ▸ Let k be a field and Γ be an ordered Abelian group: c γ t γ ∶ { γ ∶ c γ ≠ 0 } well-ordered } k (( t Γ )) = { ∑ γ ∈ Γ and v ( ∑ γ c γ t γ ) = min { γ ∶ c γ ≠ 0 } . ▸ Let k be a perfect characteristic p > 0 field. c p − i c p − i i p i ∶ c i ∈ k } and v ( ∑ W ( k ) = { ∑ i p i ) = min { i ∶ c i ≠ 0 } . i > i 0 i > i 0

Operators 3 / 8 verifies: ▸ Contractive derivations: an additive morphism ∂ ∶ K → K that ▸ the Leibniz rule ∂ ( xy ) = ∂ ( x ) y + x ∂ ( y ) ▸ v ( ∂ ( x )) ≥ v ( x ) . ▸ (Iterative) Hasse derivations: a collection ( ∂ n ) n ≥ 0 of additive morphisms K → K that verify ▸ D 0 ( x ) = x ; ▸ The generalised Leibniz rule: ∂ n ( xy ) = ∑ ∂ i ( x ) ∂ j ( y ) ; i + j = n ▸ D n ( D m ( x )) = ( m + n n ) ∂ m + n ( x ) . ▸ Automorphisms (of the valued field).

Examples imperfection degree e with e commuting Hasse derivations. Let K be difference field. 4 / 8 ▸ Scanlon, 2000: Model completion of valued fields with a contractive derivation. Let k ,∂ be differentially closed: k (( t Q )) and ∂ ( ∑ c γ t γ ) = ∑ ∂ ( c γ ) t γ . γ γ ▸ Hils-Kamensky-R., 2015: Strict separably closed valued fields of finite a separably closed such that K = K p ( b 1 ... b e ) . Take ( ∂ i , j ) 1 ≤ i ≤ e , j ≥ 0 such that ∂ i , 1 ( b i ) = 1, ∂ i , 0 ( b l ) = b l and ∂ i , j ( b l ) = 0 otherwise. ▸ Bélair-Macinyre-Scanlon, 2007: ( W ( k ) , W ( σ )) where k is a ▸ k ⊧ ACF p with the Frobenius automorphism. ▸ k ⊧ ACFA p . ▸ Durhan-Onay, 2015: k (( t Γ )) where k ⊧ ACFA 0 , Γ an ordered Abelian group with an automorphism and σ ( ∑ γ c γ t γ ) = ∑ γ σ ( c γ ) t σ ( γ ) . ▸ Γ is divisible with an ω -increasing automorphism. ▸ Γ a Z -group with the identity.

Imaginaries Example Definition Theorem (Poizat, 1983) Algebraically closed fields and characteristic zero differentially closed fields eliminate imaginaries in the (differential) ring language. 5 / 8 An imaginary is an equivalent class of an ∅ -definable equivalence relation. ▸ Let ( X y ) y ∈ Y be an ∅ -definable family of sets. ▸ Define y 1 ≡ y 2 whenever X y 1 = X y 2 . ▸ The set Y / ≡ is a moduli space for the family ( X y ) y ∈ Y . A theory T eliminates imaginaries if for all ∅ -definable equivalence relation E ⊆ D 2 , there exists an ∅ -definable function f defined on D such that for all x , y ∈ D : ⇒ f ( x ) = f ( y ) . xEy ⇐

Imaginaries in valued fields Theorem (Haskell-Hrushovski-Macpherson, 2006) 6 / 8 Let ( K , v ) be a valued field, we define: ▸ S n ∶ = GL n ( K )/ GL n ( O ) . ▸ It is the moduli space of rank n free O -submodules of K n . ▸ T n ∶ = GL n ( K )/ GL n , n ( O ) ▸ GL n , n ( O ) consists of the matrices M ∈ GL n ( O ) whose reduct modulo M has only zeroes on the last column but for a 1 in the last entry. ▸ It is the moduli space of ⋃ s ∈ S n s / M s = { a + M s ∶ s ∈ S n and a ∈ s } . Let L G ∶ = { K , ( S n ) n ∈ N > 0 , ( T n ) n ∈ N > 0 ; L div ,σ n ∶ K n 2 → S n ,τ n ∶ K n 2 → T n } . The L G -theory of algebraically closed valued fields eliminates imaginaries.

Imaginaries and definable/invariant types Proposition (Hrushovski, 2014) 3. Finite sets have canonical parameters. Then T eliminates imaginaries. Remark It suffjces to prove hypothesis 1 in dimension 1. 7 / 8 Let T be a theory such that, for all A = acl eq ( A ) ⊆ M eq ⊧ T eq : 1. Any L eq ( A ) -definable set is consistent with an L eq ( A ) -definable type. 2. Any L eq ( A ) -definable type p is L ( A ∩ M ) -definable.

Imaginaries and definable/invariant types Proposition type. 3. Finite sets have canonical parameters. Then T eliminates imaginaries. Remark If T is NIP, it suffjces to prove hypothesis 1 in dimension 1. 7 / 8 Let T be a theory such that, for all A = acl eq ( A ) ⊆ M eq ⊧ T eq : 1. Any L eq ( A ) -definable set is consistent with an Aut ( M eq / A ) -invariant 2. Any Aut ( M eq / A ) -invariant type p is Aut ( M eq / A ∩ M ) -invariant.

Results Theorem (R., 2014) The model completion of valued fields with a contractive derivation eliminates imaginaries in the geometric language (with a new symbol for the derivation). Theorem (Hils-Kamensky-R., 2015) Strict separably closed valued field of imperfection degree e with e commuting Hasse derivations eliminate imaginaries in the geometric language (with new symbols for the Hasse derivations). Conjecture All the other examples eliminate imaginaries in the geometric language (with a new symbol for the automorphism). 8 / 8