Extensions of valuations to the Henselization and completion Steven - PowerPoint PPT Presentation

Extensions of valuations to the Henselization and completion Steven Dale Cutkosky Steven Dale Cutkosky

K a field with a valuation ν Φ ν value group V ν valuation ring with maximal ideal m ν ( R , m R ) local domain with QF K Steven Dale Cutkosky

Semigroup: S R ( ν ) = { ν ( f ) | f ∈ R \ { 0 }} Associated graded ring of R along ν : � � P γ ( R ) / P + P γ ( R ) / P + gr ν ( R ) = γ ( R ) = γ ( R ) γ ∈ Φ ν γ ∈ S R ( ν ) P γ = { f ∈ R | ν ( f ) ≥ γ } , P + γ ( R ) = { f ∈ R | ν ( f ) > γ } Steven Dale Cutkosky

Question 1 Suppose R is a Noetherian (excellent) local domain which is dominated by a valuation ν . Does there exist a regular local ring R ′ of the quotient field K of R such that ν dominates R ′ and R ′ dominates R , a prime ideal P of the m R -adic completion � R ′ such that P ∩ R ′ = (0) and an extension ˆ ν of ν to the QF of � R ′ / P which dominates � R ′ / P such that gr ν ( R ′ ) ∼ ν ( � R ′ / P )? = gr ˆ Steven Dale Cutkosky

V ν → V ˆ ν ↑ ↑ � R ′ → R ′ / P ↑ R Steven Dale Cutkosky

If ν has rank 1, then setting P ( ˆ R ) ∞ = { f ∈ ˆ R | ν ( f ) = ∞} we have gr ν ( R ) ∼ ν ( ˆ R / P ( ˆ = gr ˆ R ) ∞ ) . so if ν has rank 1, then Question 1 has a positive answer for local domains R and rank 1 valuations ν which admit local uniformization. If R is essentially of finite type over a field of char. 0, then we can even take P so that � R ′ / P is a regular local ring. Steven Dale Cutkosky

Question 2 Suppose R is a Noetherian (excellent) local domain which is dominated by a valuation ν . Does there exist a regular local ring R ′ of the quotient field K of R such that ν dominates R ′ and R ′ dominates R , and an extension ν h of ν to the QF of the Henselization ( R ′ ) h of R ′ which dominates ( R ′ ) h such that gr ν ( R ′ ) ∼ ν (( R ′ ) h )? = gr ˆ Steven Dale Cutkosky

V ν → V ν h ↑ ↑ R ′ ( R ′ ) h → ↑ R Steven Dale Cutkosky

If Question 1 is true then so is Question 2. A start on answering Question 2 is the following proposition. Proposition 3 [C] Suppose R and S are normal local rings such that R is excellent, S lies over R and S is unramified over R , ˜ ν is a valuation of the QF L of S which dominates S and ν is the restriction of ˜ ν to the QF K of R . Suppose L is finite over K . Then there exists a normal local ring R ′ of K which is dominated by ν and dominates R ′ such that if R ′′ is a normal local ring of K which is dominated by ν and dominates R ′ , S ′′ is the normal local ν and lies over R ′′ , then R ′′ → S ′′ ring of L which is dominated by ˜ is unramified and ν ( S ′′ ) ∼ = gr ν ( R ′′ ) ⊗ R ′′ / m R ′′ S ′′ / m S ′′ . gr ˆ Steven Dale Cutkosky

→ V ν V ˜ ν ↑ ↑ R ′′ S ′′ → ↑ R ′ ↑ R → S Steven Dale Cutkosky

Questions 1 and 2 have a negative answer in general (even in equicharacteristic 0). Steven Dale Cutkosky

Theorem 4 Suppose k is an algebraically closed field. Then there exists a 3 dimensional regular local ring T 0 , which is a localization of a finite type k -algebra, with residue field k , and a valuation ϕ of the quotient field K of T 0 which dominates T 0 and whose residue field is k , such that if T is a regular local ring of K which is dominated by ϕ and dominates T 0 , T h is the Henselization of T and ϕ h is an extension of ϕ to the quotient field of T h which dominates T h , then S T h ( ϕ h ) � = S T ( ϕ ) under the natural inclusion S T ( ϕ ) ⊂ S T h ( ϕ h ). Steven Dale Cutkosky

Theorem 4 gives a counterexample to Questions 1 and 2. T ⊂ T h ⊂ ˆ T / P if P ∩ T = (0) Steven Dale Cutkosky

Outline of proof of Theorem 4 R 0 = k [ x , y , t ] ( x , y ) ∼ = k ( t )[ x , y ] ( x , y ) Define a valuation ν dominating R 0 by constructing a generating sequence P 0 = x , P 1 = y , P 2 , . . . Let p 1 , p 2 , . . . be the sequence of prime numbers, excluding the characteristic of k . Define a 1 = p 1 + 1 and inductively define a i by a i +1 = p i p i +1 a i + 1 . Steven Dale Cutkosky

Define p 2 − (1 + t ) x p i a i P i +1 = P i i for i ≥ 1. Set ν ( x ) = 1, ν ( P i ) = a i p i for i ≥ 1. 1 Φ ν = ∪ i ≥ 1 Z p 1 p 2 · · · p i Steven Dale Cutkosky

Let k be an algebraic closure of k ( t ), and α i ∈ k be a root of f i ( u ) = u p i − (1 + t ) ∈ k [ u ] for i ≥ 1. f i ( u ) is the minimal polynomial of α i over k ( α 1 , . . . , α i − 1 ). 1 pi | i ≥ 1 } ] V ν / m ν = k ( { α i | i ≥ 1 } ) = k [ { (1 + t ) � � P p i i α i = x a i Steven Dale Cutkosky

Suppose A is a regular local ring of the QF K of R 0 which is dominated by ν and dominates R 0 . Then there exists a generating sequence of ν in A Q 0 = u , Q 1 = w , Q 2 = w pc − (1 + t ) τ p z pe , . . . where p = p 1+ l for some l and τ is a unit in A . Let λ be a p -th root of 1 + t in an algebraic closure of K , L = K ( λ ) and ν be an extension of ν to L . Let ε ∈ k be a primitive p -th root of unity. Steven Dale Cutkosky

Let B = A [ λ ], C = B m ν ∩ B . A → C is unramifed, so C is a regular local ring with regular parameters z , w . Proposition 5 S C ( ν ) � = S A ( ν ). Steven Dale Cutkosky

proof: S A ( ν ) = S ( { ν ( Q i ) | i ≥ 0 } ). Q 0 = z , Q 1 = w , Q 2 = w pc − (1 + t ) τ p z pe , . . . � w c � γ 1 = ∈ V ν / m ν ⊂ V ν / m ν z e Let 0 � = β = [ λτ ] ∈ V ν / m ν ., h j = w c − ε j λτ z e ∈ C . Steven Dale Cutkosky

If ε j β � = γ , then ν ( h j ) = e ν ( z ), p � ν ( h j ) = ν ( Q 2 ) > pe ν ( z ) j =1 implies there exists a unique value of j such that ε j β = γ 1 and ν ( h j ) > e ν ( z ). If ν ( h j ) ∈ S A ( ν ), then ν ( h j ) ∈ S ( ν ( z ) , ν ( w )) since ν ( h j ) = ν ( Q 2 ) − ( p − 1) e ν ( z ) < ν ( Q 2 ) . Thus ν ( Q 2 ) ∈ G ( ν ( z ) , ν ( w )), a contradiction. Steven Dale Cutkosky

Let µ be a valuation of 1 pi | i ≥ 1 } ] V ν / m ν = k ( t )[ { 1 + t ) which is an extension of the ( t )-adic valuation on k [ t ] ( t ) . The value group of µ is Z . Let ϕ be the composite valuation of ν and µ on K , so that V ϕ = π − 1 ( V µ ), where π : V ν → V ν / m ν . V ϕ / m ϕ = V µ / m µ = k . Let T 0 = k [ t , x , y ] ( t , x , y ) which is dominated by ϕ . Proposition 5 implies Steven Dale Cutkosky

Proposition 6 Suppose T is a regular local ring of K which dominates T 0 and is dominated by ϕ . Then there exists a finite separable extension field L of K such that T is unramified in L . Further, if ϕ is an extension of ϕ to L and if U is the normal local ring of L which lies over T and is dominated by ϕ , then 1) U is a regular local ring 2) T → U is unramified with no residue field extension 3) S U ( ϕ ) � = S T ( ϕ ) Steven Dale Cutkosky

Proof of Theorem 4: Construction of T h (after Nagata). Let N be a separable closure of K . N is an (infinite) Galois extension of K with Galois group G ( N / K ). Let E be a local ring of the integral closure of T in N . G s ( E / T ) = { σ ∈ G ( N / K ) | σ ( E ) = E } . T h = E G s ( E / T ) with QF M = N G s ( E / T ) . Steven Dale Cutkosky

Let K → L be the field extension of Proposition 6. Choose an embedding K → L → N . Let U be the local ring of the integral closure of T in L which is dominated by E . U is unramifed over T with no residue field extension, so L ⊂ M and U is dominated by T h . Let ϕ = ϕ h | L . Then ϕ dominates U and T h dominates U , so S U ( ϕ ) ⊂ S T h ( ϕ h ). But S U ( ϕ ) � = S T ( ϕ ) by Proposition 6, so S T h ( ϕ h ) � = S T ( ϕ ). Steven Dale Cutkosky