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

1

Definitions

2

Local properties

3

Coherent sheaves on disks

4

Affinoid spaces over Z

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 2 / 14

The analytic space An,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 An,an

A

Let (A, ·) be a commutative Banach ring with unity. Let n be a non-negative integer.

Definition (Berkovich)

The analytic space An,an

A

is the set of multiplicative semi-norms on A[T1, . . . , Tn] bounded on A,

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 3 / 14

The analytic space An,an

A

Let (A, ·) be a commutative Banach ring with unity. Let n be a non-negative integer.

Definition (Berkovich)

The analytic space An,an

A

is the set of multiplicative semi-norms on A[T1, . . . , Tn] bounded on A, i.e. maps |.| : A[T1, . . . , Tn] → R+ such that

1 |0| = 0 and |1| = 1; 2 ∀f , g ∈ A[T1, . . . , Tn], |f + g| ≤ |f | + |g|; 3 ∀f , g ∈ A[T1, . . . , Tn], |fg| = |f | |g|; 4 ∀f ∈ A, |f | ≤ f . Jérôme Poineau (Caen) Theorems A and B 08.25.2015 3 / 14

The topology on An,an

A

The topology on An,an

A

is the coarsest topology such that, for any f in A[T1, . . . , Tn], the evaluation function An,an

A

→ R+ |.|x → |f |x is continuous.

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 4 / 14

The topology on An,an

A

The topology on An,an

A

is the coarsest topology such that, for any f in A[T1, . . . , Tn], the evaluation function An,an

A

→ R+ |.|x → |f |x is continuous.

Theorem (Berkovich)

The space An,an

A

is Hausdorff and locally compact.

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 4 / 14

The structure sheaf on An,an

A

Definition (Berkovich)

For every open subset U of An,an

A

, O(U) is the set of maps f : U →

• x∈U

H (x) 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

1

Definitions

2

Local properties

3

Coherent sheaves on disks

4

Affinoid spaces over Z

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 6 / 14

Local properties of An,an

Z

Theorem (Lemanissier)

The space An,an

Z

is locally arcwise connected.

Theorem (P.)

For every x in An,an

Z

, the local ring Ox is henselian, noetherian, regular, excellent. The structure sheaf of An,an

Z

is coherent.

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 7 / 14

Outline

1

Definitions

2

Local properties

3

Coherent sheaves on disks

4

Affinoid spaces over Z

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 8 / 14

Closed disks

Let (A, ·) be a Banach ring. Let r1, . . . , rn > 0. Set DA(r1, . . . , rn) = {x ∈ An,an

A

| ∀i, |Ti(x)| ≤ ri}.

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 9 / 14

Closed disks

Let (A, ·) be a Banach ring. Let r1, . . . , rn > 0. Set DA(r1, . . . , rn) = {x ∈ An,an

A

| ∀i, |Ti(x)| ≤ ri}. We have O(D) = lim − →

U⊃D

O(U) = lim − →

si>ri

As−1

1 T1, . . . , s−1 n Tn,

where As−1

1 T1, . . . , s−1 n Tn = u≥0

buT u |

• u≥0

buru < ∞

• .

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± ≤ Cs. 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± ≤ Cs.

In fact, we need more:

lim − → Ck

∼

− → O(L) lim − → B±

k → O(K ±)

∀k, B±

k → Ck

s ∈ Ck, 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± ≤ Cs.

= ⇒ multiplicative version with S ∈ GLn(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± ≤ Cs.

= ⇒ multiplicative version with S ∈ GLn(O(L))

Runge property

For all finite families (si), (tj) in O(L), there exist f in O(K +) invertible and families (s′

i) in O(K −) and (t′ j) in O(K +) that approximate (f −1si)

and (f tj) 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

• n M = K − ∪ K +.

Proposition

If F is generated by global sections on K ±, i.e. for every x in K ±, Fx is generated by F(K ±) as an Ox-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 r1, . . . , rn > 0. Set D = DZ(r1, . . . , rn).

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 12 / 14

Theorems A and B

Let r1, . . . , rn > 0. Set D = DZ(r1, . . . , rn).

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 r1, . . . , rn > 0. Set D = DZ(r1, . . . , rn).

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 Hq(D, F) = 0.

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 12 / 14

Outline

1

Definitions

2

Local properties

3

Coherent sheaves on disks

4

Affinoid spaces over Z

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 = DZ(r1, . . . , rn).

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 = DZ(r1, . . . , rn).

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 = DZ(r1, . . . , rn).

Theorem

Theorems A and B hold for affinoid spaces.

Theorem

If I = (f1, . . . , fm), then O(X) ≃ O(D)/(f1, . . . , fm).

Jérôme Poineau (Caen) Theorems A and B 08.25.2015 14 / 14