Local Global Principles for Torsors over Arithmetic Curves Julia Hartmann RWTH Aachen University jt. work with D. Harbater, University of Pennsylvania and D. Krashen, University of Georgia
Patching Patching in algebraic terms: F ≤ F 1 , F 2 ≤ F 0 fields. Base change functor Θ : Vect( F ) − → Vect( F 1 ) × Vect( F 0 ) Vect( F 2 ) . RAGE 2011 Julia Hartmann
Patching Patching in algebraic terms: F ≤ F 1 , F 2 ≤ F 0 fields. Base change functor Θ : Vect( F ) − → Vect( F 1 ) × Vect( F 0 ) Vect( F 2 ) . patching problems RAGE 2011 Julia Hartmann
Patching Patching in algebraic terms: F ≤ F 1 , F 2 ≤ F 0 fields. Base change functor Θ : Vect( F ) − → Vect( F 1 ) × Vect( F 0 ) Vect( F 2 ) . solutions patching problems RAGE 2011 Julia Hartmann
Patching Patching in algebraic terms: F ≤ F 1 , F 2 ≤ F 0 fields. Base change functor Θ : Vect( F ) − → Vect( F 1 ) × Vect( F 0 ) Vect( F 2 ) . Question: When do patching problems have solutions? RAGE 2011 Julia Hartmann
Existence of Solutions Let F ≤ F 1 , F 2 ≤ F 0 , and consider a patching problem ( V 1 , V 2 , φ ) RAGE 2011 Julia Hartmann
Existence of Solutions Let F ≤ F 1 , F 2 ≤ F 0 , and consider a patching problem ( V 1 , V 2 , φ ) , i.e., V i is an F i -vector space and φ : V 1 ⊗ F 1 F 0 → V 2 ⊗ F 2 F 0 is an isomorphism RAGE 2011 Julia Hartmann
Existence of Solutions Let F ≤ F 1 , F 2 ≤ F 0 , and consider a patching problem ( V 1 , V 2 , φ ) , i.e., V i is an F i -vector space and φ : V 1 ⊗ F 1 F 0 → V 2 ⊗ F 2 F 0 is an isomorphism Proposition: Suppose that F 1 ∩ F 2 = F . Then the following are equivalent: • The patching problem has a solution. RAGE 2011 Julia Hartmann
Existence of Solutions Let F ≤ F 1 , F 2 ≤ F 0 , and consider a patching problem ( V 1 , V 2 , φ ) , i.e., V i is an F i -vector space and φ : V 1 ⊗ F 1 F 0 → V 2 ⊗ F 2 F 0 is an isomorphism Proposition: Suppose that F 1 ∩ F 2 = F . Then the following are equivalent: • The patching problem has a solution. • V 1 ∩ V 2 has the maximum dimension dim F i ( V i ) . RAGE 2011 Julia Hartmann
Existence of Solutions Let F ≤ F 1 , F 2 ≤ F 0 , and consider a patching problem ( V 1 , V 2 , φ ) , i.e., V i is an F i -vector space and φ : V 1 ⊗ F 1 F 0 → V 2 ⊗ F 2 F 0 is an isomorphism Proposition: Suppose that F 1 ∩ F 2 = F . Then the following are equivalent: • The patching problem has a solution. • V 1 ∩ V 2 has the maximum dimension dim F i ( V i ) . Notice: A solution (if it exists) is automatically unique. RAGE 2011 Julia Hartmann
A Criterion for Patching Let F ≤ F 1 , F 2 ≤ F 0 . Proposition: The base change Θ is an equivalence of categories if and only if RAGE 2011 Julia Hartmann
A Criterion for Patching Let F ≤ F 1 , F 2 ≤ F 0 . Proposition: The base change Θ is an equivalence of categories if and only if (1) F = F 1 ∩ F 2 ≤ F 0 ( Intersection ) RAGE 2011 Julia Hartmann
A Criterion for Patching Let F ≤ F 1 , F 2 ≤ F 0 . Proposition: The base change Θ is an equivalence of categories if and only if (1) F = F 1 ∩ F 2 ≤ F 0 ( Intersection ) (2) n ∈ N , g ∈ GL n ( F 0 ) ⇒ g i ∈ GL n ( F i ) s.t. g = g 1 g 2 ( Factorization ) RAGE 2011 Julia Hartmann
A Criterion for Patching Let F ≤ F 1 , F 2 ≤ F 0 . Proposition: The base change Θ is an equivalence of categories if and only if (1) F = F 1 ∩ F 2 ≤ F 0 ( Intersection ) (2) n ∈ N , g ∈ GL n ( F 0 ) ⇒ g i ∈ GL n ( F i ) s.t. g = g 1 g 2 ( Factorization ) From now on, we will assume that our fields satisfy these two conditions. RAGE 2011 Julia Hartmann
More equivalences of categories Suppose Θ is an equivalence of categories. Then so are the base change functors for the following categories: • associative finite dimensional F -algebras • commutative finite dimensional F -algebras • separable commutative F -algebras • G -Galois F -algebras • central simple F -algebras • differential modules, Frobenius modules • . . . RAGE 2011 Julia Hartmann
Patching of Division Algebras Corollary: If Θ is an equivalence of categories, then the natural map Br( F ) → Br( F 1 ) × Br( F 0 ) Br( F 2 ) is a group isomorphism. RAGE 2011 Julia Hartmann
Patching of Division Algebras Corollary: If Θ is an equivalence of categories, then the natural map Br( F ) → Br( F 1 ) × Br( F 0 ) Br( F 2 ) is a group isomorphism. New aspect: An object is globally trivial if and only if it is locally trivial. So patching leads to local-global principles. RAGE 2011 Julia Hartmann
Patching Torsors Let F ≤ F 1 , F 2 ≤ F 0 be as above. Let G be a linear algebraic group defined over F . RAGE 2011 Julia Hartmann
Patching Torsors Let F ≤ F 1 , F 2 ≤ F 0 be as above. Let G be a linear algebraic group defined over F . A patching problem of G -torsors is a triple ( T 1 , T 2 , φ ) where each T i is a G F i -torsor and φ : T 1 × F 1 F 0 → T 2 × F 2 F 0 is an isomorphism of G F 0 -torsors. RAGE 2011 Julia Hartmann
Patching Torsors Let F ≤ F 1 , F 2 ≤ F 0 be as above. Let G be a linear algebraic group defined over F . A patching problem of G -torsors is a triple ( T 1 , T 2 , φ ) where each T i is a G F i -torsor and φ : T 1 × F 1 F 0 → T 2 × F 2 F 0 is an isomorphism of G F 0 -torsors. Notice: The coordinate ring F [ G ] is not a finite dimensional F -algebra (unless G is finite), so this is different from the situation considered before. RAGE 2011 Julia Hartmann
Patching Torsors Let F ≤ F 1 , F 2 ≤ F 0 be as above. Let G be a linear algebraic group defined over F . A patching problem of G -torsors is a triple ( T 1 , T 2 , φ ) where each T i is a G F i -torsor and φ : T 1 × F 1 F 0 → T 2 × F 2 F 0 is an isomorphism of G F 0 -torsors. Theorem: Every G -torsor patching problem has a unique solution. RAGE 2011 Julia Hartmann
Galois Cohomology Let F be any field, and let G be a linear algebraic group over F . RAGE 2011 Julia Hartmann
Galois Cohomology Let F be any field, and let G be a linear algebraic group over F . Recall that there is a 1 − 1 correspondence { G − torsors over F } / ∼ H 1 ( F, G ) := H 1 (Gal( F sep /F ) , G ( F sep )) ↔ = RAGE 2011 Julia Hartmann
Galois Cohomology Let F be any field, and let G be a linear algebraic group over F . Recall that there is a 1 − 1 correspondence { G − torsors over F } / ∼ H 1 ( F, G ) := H 1 (Gal( F sep /F ) , G ( F sep )) ↔ = Notice that both are sets with a distinguished point. RAGE 2011 Julia Hartmann
� An exact sequence Let F, F i be as before. We have seen: There are exact sequences of pointed sets id · () − 1 1 � G ( F ) � G ( F 1 ) × G ( F 2 ) � G ( F 0 ) and H 1 ( F, G ) � H 1 ( F 1 , G ) × H 1 ( F 2 , G ) � H 1 ( F 0 , G ) RAGE 2011 Julia Hartmann
� � An exact sequence Let F, F i be as before. Theorem: There is an exact sequence of pointed sets 1 � G ( F ) � G ( F 1 ) × G ( F 2 ) � G ( F 0 ) �� �� H 1 ( F, G ) � H 1 ( F 1 , G ) × H 1 ( F 2 , G ) � H 1 ( F 0 , G ) RAGE 2011 Julia Hartmann
� � An exact sequence Let F, F i be as before. Theorem: There is an exact sequence of pointed sets � H 0 ( F, G ) � H 0 ( F 1 , G ) × H 0 ( F 2 , G ) � H 0 ( F 0 , G ) 1 �� �� H 1 ( F, G ) � H 1 ( F 1 , G ) × H 1 ( F 2 , G ) � H 1 ( F 0 , G ) RAGE 2011 Julia Hartmann
Local-Global-Principles and Factorization Corollary: There is a local-global principle for G -torsors over F if and only if factori- zation holds in G . RAGE 2011 Julia Hartmann
Local-Global-Principles and Factorization Corollary: There is a local-global principle for G -torsors over F if and only if factori- zation holds in G . So far, this is all abstract. RAGE 2011 Julia Hartmann
Our Setup - Simplest Case T = k [[ t ]] a complete discrete valuation ring with residue field k X = P 1 T with closed fibre X = P 1 k F = k (( t ))( x ) , F 1 = frac( k [ x − 1 ][[ t ]]) , F 2 = k (( x, t )) , F 0 = k (( x ))(( t )) Then F ≤ F 1 , F 2 ≤ F 0 satisfy the intersection and factorization condition. RAGE 2011 Julia Hartmann
Our Setup T a complete discrete valuation ring X a regular connected projective T -curve with closed fibre X and function field F F 1 : related to the set P of closed points of X at which distinct irreducible components meet F 2 : related to the set of components of the complement of P in X F 0 : related to the set of branches at points of P . RAGE 2011 Julia Hartmann
Our Setup T a complete discrete valuation ring X a regular connected projective T -curve with closed fibre X and function field F F 1 : related to the set P of closed points of X at which distinct irreducible components meet F 2 : related to the set of components of the complement of P in X F 0 : related to the set of branches at points of P . These are in general not fields, but finite direct products of fields. For ex- ample, F 1 is the product over the fraction fields F P of the complete local rings at points P ∈ P . RAGE 2011 Julia Hartmann
Recommend
More recommend
