Valued hyperfields, truncated DVRs, and valued fields Junguk Lee - PowerPoint PPT Presentation

Preliminaries Main theorems Valued hyperfields, truncated DVRs, and valued fields Junguk Lee Institute of mathematics, Wroclaw University Logic Colloquium 2018 Udine, Italy 23-28, July, 2018 1 / 11

Preliminaries Main theorems [1] P. Deligne. Les corps locaux de caract´ eristique p , limits de corps locaux de caract´ eristique 0. J.-N. Bernstein, P. Deligne, D. Kazhdan, M.-F. Vigneras, Representations des groupes reductifs sur un corps local, Travaux en cours, Hermann, Paris, 119-157, (1984). [2] M. Krasner. Approximation des corps valu´ es complets de caract´ eristique p � = 0 par ceux caract´ eristique 0. 1957 Colloque d’alg` ebre sup´ erieure, tenu ´ a Bruxelles du 19 au 22 d´ ecembre. [3] J. Lee and W. Lee. On the structure of certain valued fields, preprint. [4] J. Tolliver. An equivalence between two approaches to limits of local fields, J. of Number Theory, (2016), 473-492. 2 / 11

Preliminaries Main theorems ( K , ν, k , Γ , R , m ) ( K , ν ) is a henselian valued field of mixed characteristic (0 , p ). k is perfect ( K , ν ) is finitely ramified. We have e ν ( p ) is finite, where for each x ∈ R we define e ν ( x ) := |{ γ ∈ Γ | 0 < γ < ν ( x ) }| . Note e ν ( p ) is the ramification index. If there is no confusion, we denote e ν by e . π ∈ R is a uniformizer so that Γ is generated by ν ( π ). For each n > 0, R n := R / m n , called the n-th residue ring . If ( K , ν ) is a discrete complete valued fields, we may assume that ν is normalized, that is, ν ( p ) = 1. And we still write ν for the unique extension of ν in K alg 3 / 11

Preliminaries Main theorems Main Goal : Let K 1 and K 2 be complete discrete valued fields of mixed characteristic with perfect residue fields. There is an positive integer N depending only on the ramification indices such that if the N -th valued hyper valued fields of valued fields are isomorphic over p , i.e., H N ( K 1 ) ∼ = H N ( { p } ) H N ( K 2 ) , then K 1 and K 2 are isomorphic, i.e., K 1 ∼ = K 2 . 4 / 11

Preliminaries Main theorems For each n > 0, we define the n-th valued hyperfield of K H n ( K ) := ( K / (1 + m n ); + H , − H , · H ; 0 H , 1 H ; ν H ) , where for α := [ a ] , β := [ b ] ∈ K / (1 + m n ), 0 H := [0] and 1 H := [1]; α + H β := { [ x + y ] | x ∈ a (1 + m n ) , y ∈ b (1 + m n ) } ; β := ( − H α ) is the unique element such that 0 H ∈ α + β ; α · H β := [ ab ]; and ν H ( α ) := ν ( a ), If there is no confusion, we omit H . 5 / 11

Preliminaries Main theorems Definition Let K 1 and K 2 be valued fields. Fix n > 0. 1 A map f from H n ( K 1 ) to H n ( K 2 ) is called a homomorphism if for α, β ∈ H n ( K 1 ), f (0) = 0 and f (1) = 1; f ( αβ ) = f ( α ) f ( β ); f ( α + β ) ⊂ f ( α ) + f ( β ); and ν 1 ( α ) < ν 1 ( β ) ⇔ ν 2 ( f ( α )) < ν 2 ( f ( β )). We denote Hom( H n ( K 1 ) , H n ( K 2 )) for the set of homomorphisms from H n ( K 1 ) to H n ( K 2 ). 2 A homomorphism f from H n ( K 1 ) to H n ( K 2 ) is called over p if f ([ p ]) = [ p ]. We denote Hom Z ( H n ( K 1 ) , H n ( K 2 )) for the set of homomorphisms from H n ( K 1 ) to H n ( K 2 ), which is over p . 6 / 11

Preliminaries Main theorems Let ( K 1 , ν 1 , k 1 , Γ 1 ) and ( K 2 , ν 2 , k 2 , Γ 2 ) be finitely ramified henselian valued fields of mixed characteristic with perfect residue fields. Let e 1 and e 2 be the ramification indices of K 1 and K 2 . Set � 1 if p � | e 1 (, tamely ramified) N = . e 2 ( p )(1 + e 2 1 ( p )) + 1 if p | e 1 (, wildly ramified) Here, e 2 1 ( p ) = e 1 ( e 1 ( p )) is well-defined because K 1 is of characteristic 0 and N is a subset of K 1 . 7 / 11

Preliminaries Main theorems Main Theorem 1 Theorem Suppose ( K 1 , ν 1 ) and ( K 2 , ν 2 ) be complete discrete valued fields. Then there is a unique lifting map L : Hom Z ( H N ( K 1 ) , H N ( K 2 )) → Hom( K 1 , K 2 ) such that for any homomorphism φ : H N ( K 1 ) → H N ( K 2 ) , set g = L( φ ) , and we have There exists a representaive β of φ ([ π 1 ]) which satisfies � � ν 2 ( g ( π 1 ) − β > M ( K 1 ) , where M ( K 1 ) := max { ν 1 ( π 1 − σ ( π 1 )) | σ ∈ Hom L 1 ( K 1 , K alg 1 ) , σ ( π 1 ) � = π 1 } and L 1 ( ⊂ K 1 ) is the quotient field of the Witt ring W ( k 1 ) . φ red , 1 ◦ H 1 = H 1 ◦ g, where φ red , 1 : H 1 ( K 1 ) → H 1 ( K 2 ) is the natural reduction map of φ and H 1 is the natural projection map to the first valued hyperfield. 8 / 11

Preliminaries Main theorems Main Theorem 1 g K 1 K 2 H 1 H 1 φ red , 1 H 1 ( K 1 ) H 1 ( K 2 ) Example √ 3) and K 2 = Q 3 ( √− 3). Since e 1 = e 2 = 2, N = 1. Set Let K 1 = Q 3 ( √ 3 and π 2 = √− 3. Note that the sets of Teichm¨ π 1 = uller representatives of K 1 and K 2 are ( S :=) S 1 = S 2 = {− 1 , 0 , 1 } . Consider a √ 3] �→ [ √− 3]. homomorphism f ∈ Hom( H 1 ( K 1 ) , H 1 ( K 2 )) by sending [ Note that for each a ∈ S , f ([ a ]) = [ a ]. We have that √ √ f : H 1 ( K 1 ) ∼ = H 1 ( K 2 ) , [ 3] �→ [ − 3] which is not over p . But K 1 and K 2 are never isomorphic. 9 / 11

Preliminaries Main theorems Main Theorem 2 Theorem Let K 1 and K 2 be finitely ramified henselian valued fields of mixed characteristic (0 , p ) with perfect residue fields. The followings are equivalent: 1 K 1 ≡ K 2 . 2 H N ( K 1 ) ≡ H N ( { p } ) H N ( K 1 ) . Furthermore, if we take N ′ = e 2 ( p )(1 + e 2 1 ( p )) + 1 , 3 (S. Basarab, L. and W.Lee)R 1 , N ′ ≡ R 2 , N ′ and Γ 1 ≡ Γ 2 . Example √ 3) and K 2 = Q 3 ( √− 3). Then e 1 = e 2 = 2, and N = 1 Let K 1 = Q 3 ( and N ′ = 3. We have that K 1 �≡ K 2 by the sentence ∃ X ( X 2 + 3 =)0, and so H 1 ( K 1 ) �≡ H 1 ( { 3 } ) H 1 ( K 2 ) and R 1 , 3 �≡ R 2 , 3 . But R 1 , 1 = R 2 , 1 = F 3 . 10 / 11

Preliminaries Main theorems Grazie! 11 / 11