Theorems A and B for Berkovich spaces over Z Jrme Poineau Universit - PowerPoint PPT Presentation

Theorems A and B for Berkovich spaces over Z Jérôme Poineau Université de Caen 08.25.2015 Jérôme Poineau (Caen) Theorems A and B 08.25.2015 1 / 14

Outline Definitions 1 Local properties 2 Coherent sheaves on disks 3 Affinoid spaces over Z 4 Jérôme Poineau (Caen) Theorems A and B 08.25.2015 2 / 14

The analytic space A n , an A Let ( A , �·� ) be a commutative Banach ring with unity. Let n be a non-negative integer. Jérôme Poineau (Caen) Theorems A and B 08.25.2015 3 / 14

The analytic space A n , an A Let ( A , �·� ) be a commutative Banach ring with unity. Let n be a non-negative integer. Definition (Berkovich) The analytic space A n , an is the set of multiplicative semi-norms on A A [ T 1 , . . . , T n ] bounded on A , Jérôme Poineau (Caen) Theorems A and B 08.25.2015 3 / 14

The analytic space A n , an A Let ( A , �·� ) be a commutative Banach ring with unity. Let n be a non-negative integer. Definition (Berkovich) The analytic space A n , an is the set of multiplicative semi-norms on A A [ T 1 , . . . , T n ] bounded on A , i.e. maps | . | : A [ T 1 , . . . , T n ] → R + such that 1 | 0 | = 0 and | 1 | = 1; 2 ∀ f , g ∈ A [ T 1 , . . . , T n ] , | f + g | ≤ | f | + | g | ; 3 ∀ f , g ∈ A [ T 1 , . . . , T n ] , | fg | = | f | | g | ; 4 ∀ f ∈ A , | f | ≤ � f � . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 3 / 14

The topology on A n , an A The topology on A n , an is the coarsest topology such that, for any f A in A [ T 1 , . . . , T n ] , the evaluation function A n , an → R + A | . | x �→ | f | x is continuous. Jérôme Poineau (Caen) Theorems A and B 08.25.2015 4 / 14

The topology on A n , an A The topology on A n , an is the coarsest topology such that, for any f A in A [ T 1 , . . . , T n ] , the evaluation function A n , an → R + A | . | x �→ | f | x is continuous. Theorem (Berkovich) The space A n , an is Hausdorff and locally compact. A Jérôme Poineau (Caen) Theorems A and B 08.25.2015 4 / 14

The structure sheaf on A n , an A Definition (Berkovich) For every open subset U of A n , an , O ( U ) is the set of maps A � f : U → H ( x ) x ∈ U such that 1 ∀ x ∈ U , f ( x ) ∈ H ( x ) ; 2 f is locally a uniform limit of rational functions without poles. Jérôme Poineau (Caen) Theorems A and B 08.25.2015 5 / 14

Outline Definitions 1 Local properties 2 Coherent sheaves on disks 3 Affinoid spaces over Z 4 Jérôme Poineau (Caen) Theorems A and B 08.25.2015 6 / 14

Local properties of A n , an Z Theorem (Lemanissier) The space A n , an is locally arcwise connected. Z Theorem (P.) For every x in A n , an , the local ring O x is henselian, noetherian, Z regular, excellent. The structure sheaf of A n , an is coherent. Z Jérôme Poineau (Caen) Theorems A and B 08.25.2015 7 / 14

Outline Definitions 1 Local properties 2 Coherent sheaves on disks 3 Affinoid spaces over Z 4 Jérôme Poineau (Caen) Theorems A and B 08.25.2015 8 / 14

Closed disks Let ( A , �·� ) be a Banach ring. Let r 1 , . . . , r n > 0. Set D A ( r 1 , . . . , r n ) = { x ∈ A n , an | ∀ i , | T i ( x ) | ≤ r i } . A Jérôme Poineau (Caen) Theorems A and B 08.25.2015 9 / 14

Closed disks Let ( A , �·� ) be a Banach ring. Let r 1 , . . . , r n > 0. Set D A ( r 1 , . . . , r n ) = { x ∈ A n , an | ∀ i , | T i ( x ) | ≤ r i } . A We have O ( D ) = lim O ( U ) − → U ⊃ D A � s − 1 1 T 1 , . . . , s − 1 = lim n T n � , − → s i > r i where b u T u | � b u � r u < ∞ A � s − 1 1 T 1 , . . . , s − 1 � � � � n T n � = . u ≥ 0 u ≥ 0 Jérôme Poineau (Caen) Theorems A and B 08.25.2015 9 / 14

Cousin-Runge systems I K − , K + compacts, L = K − ∩ K + Jérôme Poineau (Caen) Theorems A and B 08.25.2015 10 / 14

Cousin-Runge systems I K − , K + compacts, L = K − ∩ K + Cousin property There exists C > 0 such that, for every s ∈ O ( L ) , there exist s ± ∈ O ( K ± ) such that 1 s = s − + s + ; 2 � s ± � ≤ C � s � . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 10 / 14

Cousin-Runge systems I K − , K + compacts, L = K − ∩ K + Cousin property There exists C > 0 such that, for every s ∈ O ( L ) , there exist s ± ∈ O ( K ± ) such that 1 s = s − + s + ; 2 � s ± � ≤ C � s � . In fact, we need more: ∼ lim → C k − → O ( L ) − → B ± k → O ( K ± ) lim − ∀ k , B ± k → C k s ∈ C k , s ± ∈ B ± k Jérôme Poineau (Caen) Theorems A and B 08.25.2015 10 / 14

Cousin-Runge systems I K − , K + compacts, L = K − ∩ K + Cousin property There exists C > 0 such that, for every s ∈ O ( L ) , there exist s ± ∈ O ( K ± ) such that 1 s = s − + s + ; 2 � s ± � ≤ C � s � . = ⇒ multiplicative version with S ∈ GL n ( O ( L )) Jérôme Poineau (Caen) Theorems A and B 08.25.2015 10 / 14

Cousin-Runge systems I K − , K + compacts, L = K − ∩ K + Cousin property There exists C > 0 such that, for every s ∈ O ( L ) , there exist s ± ∈ O ( K ± ) such that 1 s = s − + s + ; 2 � s ± � ≤ C � s � . = ⇒ multiplicative version with S ∈ GL n ( O ( L )) Runge property For all finite families ( s i ) , ( t j ) in O ( L ) , there exist f in O ( K + ) invertible j ) in O ( K + ) that approximate ( f − 1 s i ) and families ( s ′ i ) in O ( K − ) and ( t ′ and ( f t j ) arbitrarily well. Jérôme Poineau (Caen) Theorems A and B 08.25.2015 10 / 14

Cousin-Runge systems II Let ( K − , K + ) be a Cousin-Runge system. Let F be a sheaf of finite type on M = K − ∪ K + . Proposition If F is generated by global sections on K ± , i.e. for every x in K ± , F x is generated by F ( K ± ) as an O x -module, then F is generated by global sections on M . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 11 / 14

Theorems A and B Let r 1 , . . . , r n > 0. Set D = D Z ( r 1 , . . . , r n ) . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 12 / 14

Theorems A and B Let r 1 , . . . , r n > 0. Set D = D Z ( r 1 , . . . , r n ) . Theorem (Theorem A) Every sheaf of finite type on D is generated by global sections. Jérôme Poineau (Caen) Theorems A and B 08.25.2015 12 / 14

Theorems A and B Let r 1 , . . . , r n > 0. Set D = D Z ( r 1 , . . . , r n ) . Theorem (Theorem A) Every sheaf of finite type on D is generated by global sections. Theorem (Theorem B) For every coherent sheaf F on D and every q ≥ 1 , we have H q ( D , F ) = 0 . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 12 / 14

Outline Definitions 1 Local properties 2 Coherent sheaves on disks 3 Affinoid spaces over Z 4 Jérôme Poineau (Caen) Theorems A and B 08.25.2015 13 / 14

Affinoid spaces over Z Definition (Affinoid space) An affinoid space X is of the form ( V ( I ) , O / I ) , where I is a coherent sheaf on some D = D Z ( r 1 , . . . , r n ) . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 14 / 14

Affinoid spaces over Z Definition (Affinoid space) An affinoid space X is of the form ( V ( I ) , O / I ) , where I is a coherent sheaf on some D = D Z ( r 1 , . . . , r n ) . Theorem Theorems A and B hold for affinoid spaces. Jérôme Poineau (Caen) Theorems A and B 08.25.2015 14 / 14

Affinoid spaces over Z Definition (Affinoid space) An affinoid space X is of the form ( V ( I ) , O / I ) , where I is a coherent sheaf on some D = D Z ( r 1 , . . . , r n ) . Theorem Theorems A and B hold for affinoid spaces. Theorem If I = ( f 1 , . . . , f m ) , then O ( X ) ≃ O ( D ) / ( f 1 , . . . , f m ) . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 14 / 14