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

Imaginaries in valued fields with operators Silvain Rideau UC Berkeley April 15 2016 1 / 22

Valued fields Example (Hahn series field, Witt vectors) It is the unique complete, rank 1, mixed characteristic valued field 2 / 22 Let ( K , v ) be a valued field: ▸ Γ = v ( K ) its value group; ▸ O = { x ∈ K ∶ v ( x ) ≥ 0 } its valuation ring; ▸ M = { x ∈ K ∶ v ( x ) > 0 } its maximal ideal; ▸ k = O / M its residue field. ▸ Let k be a field and Γ be an ordered Abelian group: c γ t γ ∶ well-ordered support } . k (( t Γ )) = { ∑ γ ∈ Γ ▸ Let k be a perfect characteristic p > 0 field. a p − i W ( k ) = { ∑ i p i } . i > i 0 whose residue field is k .

Operators On a field K we consider: rule: 3 / 22 ▸ Automorphisms (of the field). ▸ Derivations: an additive morphism ∂ ∶ K → K that verifies the Leibniz ∂ ( xy ) = ∂ ( x ) y + x ∂ ( y ) . ▸ (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 )

Operators Example (Automorphisms) Example (Derivations) Example (Hasse Derivations) 4 / 22 ▸ ( F p alg , F p ) . ▸ Ultraproducts of the above. ▸ Meromorphic functions on some open subset of C . ▸ Germs at +∞ of infinitely differentiable real functions. ▸ For ( k ,∂ ) a differential field, k (( t Γ )) with ∂ ( ∑ γ c γ t γ ) = ∑ γ ∂ ( c γ ) t γ . ▸ Let K be a characteristic p > 0 field and ( b i ) i ∈ I a p -basis of K . There exists a a Hasse derivation ∂ i on K such that ∂ i , 1 ( b i ) = 1 and ∂ i , n ( b j ) = 0 otherwise.

Valued fields with operators We want to consider fields with both structures. Durhan-van den Dries, Hrushovski, Pal, Durhan-Onay). We will only consider existentially closed fields with operators. 5 / 22 ▸ You can either not assume any interaction: ▸ Separably closed valued fields (Delon,Hong,Hils-Kamensky-R.); ▸ Differentially closed valued fields (Michaux,Guzy-Point); ▸ Or force some level of interaction: ▸ Contractive derivations: v ( D ( x )) ≥ v ( x ) (Scanlon, R.); ▸ Valued field automorphism: σ ( O ) = O (Bélair-Macintyre-Scanlon, ▸ A priori, this rules out transseries. ▸ Actually, we will only need a certain form of quantifier elimination.

Contractive derivations Theorem (Scanlon, 2000) The theory of equicharacteristic zero valued fields with a contractive eliminates quantifiers. Example 6 / 22 In L ∂, div ∶ = { K ; 0 , 1 , + , − , ⋅ ,∂, div } : derivation has a model completion VDF EC which is complete and The theory VDF EC contains: ▸ The field is ∂ -Henselian; ▸ v ( C K ) = v ( K ) where C K = { x ∈ K ∶ ∂ ( x ) = 0 } ; ▸ The residue field is differentially closed; ▸ The value group is divisible. If ( k ,∂ ) is differentially closed and Γ is divisible, then k (( t Γ )) ⊧ VDF EC .

Separably closed fields Theorem complete and eliminates quantifiers. alg . 7 / 22 ▸ Let e be a positive integer. ▸ Let K be a characteristic p > 0 field with e commuting Hasse derivations ∂ i : ∂ i , n ○ ∂ j , m = ∂ j , m ○ ∂ i , n . K ∶ = { x ∈ K ∶ ∀ i , ∂ i , 1 ( x ) = 0 } = K p . ▸ The field K is strict if C 1 In L e , div ∶ = { K ; 0 , 1 , + , − , ⋅ , ( ∂ i , n ) 0 ≤ i < e , 0 ≤ n , div } : The theory SCVH p , e of characteristic p > 0 strict separably closed valued fields with e commuting Hasse derivations such that [ K ∶ K p ] = p e is Let K ⊧ SCVH p , e : ▸ v ( K ) is divisible and k ( K ) is algebraically closed; ▸ K is dense in K

Imaginaries Example is interpretable but a priori not definable. Definition 8 / 22 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 . ▸ The imaginary ⌜ X y ⌝ ∶ = y / ≡ is the canonical parameter of X y . ▸ Let G be a definable group and H � G be a subgroup. The group G / H 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 ⇐

Shelah’s eq construction Definition and Remark we denote M eq . 9 / 22 Let T be a theory. For all ∅ -definable equivalence relation E ⊆ ∏ i S i , let S E be a new sort and f E ∶ ∏ S i → S E be a new function symbol. Let L eq ∶ = L ∪ { S E , f E ∶ E is an ∅ -definable equivalence relation } T eq ∶ = T ∪ { f E is onto and ∀ x , y ( f E ( x ) = f E ( y ) ↔ xEy )} . ▸ Let M ⊧ T , then M can naturally be enriched into a model of T eq that ▸ The theory T eq eliminates imaginaries.

Imaginaries in fields Theorem (Poizat, 1983) imaginaries. One cannot hope for such a theorem to hold for algebraically closed continuum. 10 / 22 The theory of algebraically closed fields in L rg ∶ = { K ; 0 , 1 , + , − , ⋅ } and the theory of differentially closed fields in L ∂ ∶ = L rg ∪ { ∂ } both eliminate valued fields in L div ∶ = L rg ∪ { div } . Indeed, ▸ K = C (( t Q )) ⊧ ACVF; ▸ Q = K ⋆ / O ⋆ is both interpretable and countable; ▸ All definable set X ⊆ K n are either finite or have cardinality

Imaginaries in valued fields Theorem (Haskell-Hrushovski-Macpherson, 2006) 11 / 22 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 types Proposition (Hrushovski, 2014) 3. Finite sets have canonical parameters. Then T eliminates imaginaries. Remark It suffjces to prove hypothesis 1 in dimension 1. 12 / 22 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.

An aside: the invariant extension property Definition Proposition The following are equivalent: (i) The theory T has the invariant extension property. Remark If T is NIP then the above are also equivalent to: (iii) 13 / 22 We say that T has the invariant extension property if for all M ⊧ T and A = acl eq ( A ) ⊆ M eq , every type over A has a global A -invariant extension. (ii) For all A = acl eq ( A ) ⊆ M eq ⊧ T eq , any L eq ( A ) -definable set is consistent with an L eq ( A ) -definable type. ▸ Forking equals dividing ▸ Lascar strong type, Kim-Pillay strong type and strong type coincide.

Differentially closed and separably closed fields Let T be either the theory of characteristic zero differentially closed fields 14 / 22 or the theory of strict characteristic p > 0 separably closed fields with e commuting Hasse derivations such that [ K ∶ K p ] = p e . ▸ Hypothesis 1 is true by stability ▸ Hypothesis 3 is true because it is true in algebraically closed fields. ▸ As for Hypothesis 2: ▸ Let A = acl eq ( A ) ⊆ M eq ⊧ T eq and p ( x ) be an L ( A ) -definable type. ▸ Let ∂ ω ( x ) denote either ( ∂ n ( x )) n ∈ Z ≥ 0 or ( ∂ 0 , i 0 ○ . . . ∂ e − 1 , i e − 1 ( x )) i j ∈ Z ≥ 0 . ▸ Let a ⊧ p and q = tp L rg ( ∂ ω ( a )/ M ) . ▸ By elimination of imaginaries in ACF, q is L rg ( A ∩ M ) -definable. ▸ So p is L ( A ∩ M ) -definable.

Prolongations Hypothesis 1 and 2 (almost) reduce to questions about ACVF. 15 / 22 Let L be either L ∂, div or L e , div and T denote either VDF EC or SCVH p , e . ▸ Let M ⊧ T . For all p ∈ S L x ( M ) , we define: ω ( p ) ∶ = { φ ( x ω ; m ) ∶ φ ( ∂ ω ( x ) ; m ) ∈ p } ∈ S L div x ω ( M ) . ∇ ▸ By quantifier elimination, the map ∇ ω is injective. ▸ Let A = acl eq ( A ) ⊆ M eq ⊧ T eq , ⇒ ∇ ω ( p ) is consistent with ∂ ω ( X ) ; p is consistent with X ⇐ p is L eq ( A ) -definable ⇐ ⇒ ∇ ω ( p ) is L eq ( A ) -definable . ▸ The defining scheme of p consists of L ( M ) -formulas and not L div ( M ) formulas.

Proving Hypothesis 1 It is proved by a technical construction. eliminate imaginaries, 16 / 22 ▸ Given an enrichment T of ACVF in a language L , such that k and Γ ▸ A set A = acl eq ( A ) ⊆ M eq ⊧ T eq , ▸ An L eq ( A ) -definable set X , ▸ A finite set ∆ of L div -formulas, ▸ We find an L eq ( A ) -definable ∆ -type p consistent with X .