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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Zariski Main Theorem for Henselian Rigid Spaces

Fumiharu Kato

Kumamoto University

August 25, 2015

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Contents

Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ “Usual” rigid geometry:

formal scheme X

✤ rig

rigid space Xrig

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ “Usual” rigid geometry:

formal scheme X

✤ rig

rigid space Xrig

▶ Henselian rigid geometry:

henselian scheme X

✤ rig

rigid space Xrig

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

▶ Henselization A ! Ah is always flat. 4 / 30

Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

▶ Henselization A ! Ah is always flat. ▶ “Henselian” is preserved by inductive limits. 4 / 30

Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

▶ Henselization A ! Ah is always flat. ▶ “Henselian” is preserved by inductive limits. ▶ “Henselian” is “algebraic”; 4 / 30

Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

▶ Henselization A ! Ah is always flat. ▶ “Henselian” is preserved by inductive limits. ▶ “Henselian” is “algebraic”; ▶ henselization is like algebraic closure; 4 / 30

Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

▶ Henselization A ! Ah is always flat. ▶ “Henselian” is preserved by inductive limits. ▶ “Henselian” is “algebraic”; ▶ henselization is like algebraic closure; ▶ henselian top. finite type ext.

(= henselization of finite type ext.) can be nicely approximated by finite type extensions.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid geometry

▶ The generalities can be done almost parallel

to the usual case, sometimes with different proofs.

▶ Henselization A ! Ah is always flat. ▶ “Henselian” is preserved by inductive limits. ▶ “Henselian” is “algebraic”; ▶ henselization is like algebraic closure; ▶ henselian top. finite type ext.

(= henselization of finite type ext.) can be nicely approximated by finite type extensions.

▶ So one can say that hRG gives more

“pro-algebraic” hybrid between algebraic geometry and analytic geometry.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

▶ Joint-work with Shuji Saito (Tokyo Inst. of

Tech.)

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

▶ Joint-work with Shuji Saito (Tokyo Inst. of

Tech.)

▶ Our results illustrate the “algebraic nature”

• f hRG:

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

▶ Joint-work with Shuji Saito (Tokyo Inst. of

Tech.)

▶ Our results illustrate the “algebraic nature”

• f hRG:

▶ Henselian rigid ZMT for quasi-finite map

from affinoids to algebraic varieties,

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

▶ Joint-work with Shuji Saito (Tokyo Inst. of

Tech.)

▶ Our results illustrate the “algebraic nature”

• f hRG:

▶ Henselian rigid ZMT for quasi-finite map

from affinoids to algebraic varieties,

▶ “Scheme-theoretic closure” of rigid analytic

subspaces in algebraic varieties,

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

▶ Joint-work with Shuji Saito (Tokyo Inst. of

Tech.)

▶ Our results illustrate the “algebraic nature”

• f hRG:

▶ Henselian rigid ZMT for quasi-finite map

from affinoids to algebraic varieties,

▶ “Scheme-theoretic closure” of rigid analytic

subspaces in algebraic varieties,

▶ Application: Prolongation of flat families of

closed subspaces in a projective variety.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Results

▶ Joint-work with Shuji Saito (Tokyo Inst. of

Tech.)

▶ Our results illustrate the “algebraic nature”

• f hRG:

▶ Henselian rigid ZMT for quasi-finite map

from affinoids to algebraic varieties,

▶ “Scheme-theoretic closure” of rigid analytic

subspaces in algebraic varieties,

▶ Application: Prolongation of flat families of

closed subspaces in a projective variety.

▶ Further application to the construction of

“analytic Chow groups” (forthcoming work by

• M. Kerz & S. Saito).

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Contents

Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

I-adically henselian rings

▶ A: ring, I „ A: finitely generated ideal, ▶ S = Spec A, D = Spec A=I.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

I-adically henselian rings

▶ A: ring, I „ A: finitely generated ideal, ▶ S = Spec A, D = Spec A=I.

Definition

✓ ✏

▶ A is said to be I-adically henselian

def

, any étale morphism ffi: X ! S with ffi`1(D) ‰ = D admits a section. , I „ Jac(A), 8monic F (T ) 2 A[T ] with F (0) ” 0 mod I and F 0(0) 2 (A=I)ˆ; 9a 2 I such that F (a) = 0. ✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

I-adically henselian rings

▶ A: ring, I „ A: finitely generated ideal, ▶ S = Spec A, D = Spec A=I.

Definition

✓ ✏

▶ A is said to be I-adically henselian

def

, any étale morphism ffi: X ! S with ffi`1(D) ‰ = D admits a section. , I „ Jac(A), 8monic F (T ) 2 A[T ] with F (0) ” 0 mod I and F 0(0) 2 (A=I)ˆ; 9a 2 I such that F (a) = 0. ✒ ✑

Henselization

✓ ✏

▶ A ! Ah is flat, and is faithfully flat if

I „ Jac(A).

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian finite type algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g).

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian finite type algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g).

▶ We write

V fX1; : : : ; Xng

def

= (V [X1; : : : ; Xn])h:

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian finite type algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g).

▶ We write

V fX1; : : : ; Xng

def

= (V [X1; : : : ; Xn])h: Definition

✓ ✏

▶ A henselian finite type V -algebra is a

V -algebra isom. to V fX1; : : : ; Xng=a.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian finite type algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g).

▶ We write

V fX1; : : : ; Xng

def

= (V [X1; : : : ; Xn])h: Definition

✓ ✏

▶ A henselian finite type V -algebra is a

V -algebra isom. to V fX1; : : : ; Xng=a.

✒ ✑

▶ If A = V fX1; : : : ; Xng=a is V -flat, then a is

finitely generated.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian finite type algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g).

▶ We write

V fX1; : : : ; Xng

def

= (V [X1; : : : ; Xn])h: Definition

✓ ✏

▶ A henselian finite type V -algebra is a

V -algebra isom. to V fX1; : : : ; Xng=a.

✒ ✑

▶ If A = V fX1; : : : ; Xng=a is V -flat, then a is

finitely generated.

▶ If, moreover, V is of height 1, then A is what

we should call “henselian admissible” V -algebra.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian affinoid algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g),

▶ K = Frac(V ) = V [ 1 a] the fractional field.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian affinoid algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g),

▶ K = Frac(V ) = V [ 1 a] the fractional field.

Henselian Tate algebra

✓ ✏

KfX1; : : : ; Xng

def

= V fX1; : : : ; Xng ˙V K = V fX1; : : : ; Xng[ 1

a]:

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian affinoid algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g),

▶ K = Frac(V ) = V [ 1 a] the fractional field.

Henselian Tate algebra

✓ ✏

KfX1; : : : ; Xng

def

= V fX1; : : : ; Xng ˙V K = V fX1; : : : ; Xng[ 1

a]:

✒ ✑

▶ KfX1; : : : ; Xng is a Noetherian K-algebra.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian affinoid algebras

▶ V : a-adically separated a-adically henselian

valuation ring (a 2 mV n f0g),

▶ K = Frac(V ) = V [ 1 a] the fractional field.

Henselian Tate algebra

✓ ✏

KfX1; : : : ; Xng

def

= V fX1; : : : ; Xng ˙V K = V fX1; : : : ; Xng[ 1

a]:

✒ ✑

▶ KfX1; : : : ; Xng is a Noetherian K-algebra.

Henselian affinoid algebra

✓ ✏

A = KfX1; : : : ; Xng=a by an ideal a „ KfX1; : : : ; Xng.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Noether normalization

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Noether normalization

Theorem

✓ ✏

For any henselian affinoid algebra A over K, there exists an injective finite morphism KfX1; : : : ; Xdg , ` ! A:

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Noether normalization

Theorem

✓ ✏

For any henselian affinoid algebra A over K, there exists an injective finite morphism KfX1; : : : ; Xdg , ` ! A:

✒ ✑ Lemma ✓ ✏ Let A ! B be a finite type morphism of rings, and I „ A an ideal. If A=I ! B=IB is finite, then so is Ah ! Bh. ✒ ✑

▶ Classical (refined) ZMT: finiteness extends to an

étale neighborhood.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Noether normalization

Theorem

✓ ✏

For any henselian affinoid algebra A over K, there exists an injective finite morphism KfX1; : : : ; Xdg , ` ! A:

✒ ✑ Lemma ✓ ✏ Let A ! B be a finite type morphism of rings, and I „ A an ideal. If A=I ! B=IB is finite, then so is Ah ! Bh. ✒ ✑

▶ Classical (refined) ZMT: finiteness extends to an

étale neighborhood.

Theorem

✓ ✏

Henselian affinoid algebras are Jacobson.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Contents

Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian schemes: References

▶ Cox, D. A.: Algebraic tubular

• neighborhoods. I, II, Math. Scand. 42

(1978), no. 2, 211–228, 229–242.

▶ Greco, S.; Strano, R.: Quasicoherent sheaves

• ver affine Hensel schemes. Trans. Amer.
• Math. Soc. 268 (1981), no. 2, 445–465.

▶ Kurke, H.; Pfister, G.; Roczen, M.:

Henselsche Ringe und algebraische

• Geometrie. Mathematische Monographien,

Band II. VEB Deutscher Verlag der Wissenschaften, Berlin, 1975.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian spectrum

▶ A: I-adically henselian.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian spectrum

▶ A: I-adically henselian. ▶ The henselian spectrum

Sph A is defined similarly to the formal spectrum.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian spectrum

▶ A: I-adically henselian. ▶ The henselian spectrum

Sph A is defined similarly to the formal spectrum. Henselian spectrum

✓ ✏

Sph A = top. locally ringed space by

▶ the set of all open prime ideals of A ▶ with the subspace topology induced from

the Zariski topology of Spec A;

▶ for any f 2 A, D(f) = D(f) \ X and

OX(D(f)) = (Af)h, which gives a sheaf

• f top. rings on X.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian schemes

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian schemes

Definition

✓ ✏

▶ An affine henselian scheme is a top. loc.

ringed space isom. to (X = Sph A; OX) for an I-adically henselian ring A.

▶ A top. locally ringed space (X; OX) is

called a henselian scheme if it is covered by affine henselian schemes.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian schemes

Definition

✓ ✏

▶ An affine henselian scheme is a top. loc.

ringed space isom. to (X = Sph A; OX) for an I-adically henselian ring A.

▶ A top. locally ringed space (X; OX) is

called a henselian scheme if it is covered by affine henselian schemes.

✒ ✑

▶ A morphism between henselian schemes is a

morphism of top. locally ringed spaces.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian schemes

Definition

✓ ✏

▶ An affine henselian scheme is a top. loc.

ringed space isom. to (X = Sph A; OX) for an I-adically henselian ring A.

▶ A top. locally ringed space (X; OX) is

called a henselian scheme if it is covered by affine henselian schemes.

✒ ✑

▶ A morphism between henselian schemes is a

morphism of top. locally ringed spaces.

▶ A 7! Sph A gives duality between the cat. of

henselian rings with cont. homomorphisms and the cat. of affine henselian schemes.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian schemes

Definition

✓ ✏

▶ An affine henselian scheme is a top. loc.

ringed space isom. to (X = Sph A; OX) for an I-adically henselian ring A.

▶ A top. locally ringed space (X; OX) is

called a henselian scheme if it is covered by affine henselian schemes.

✒ ✑

▶ A morphism between henselian schemes is a

morphism of top. locally ringed spaces.

▶ A 7! Sph A gives duality between the cat. of

henselian rings with cont. homomorphisms and the cat. of affine henselian schemes.

▶ There is the notion of henselization XhjY of

schemes along closed subschemes.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid spaces

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid spaces

▶ CHs˜ = cat. of coherent (= q-cpt & q-sep)

henselian schemes with adic morphisms.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid spaces

▶ CHs˜ = cat. of coherent (= q-cpt & q-sep)

henselian schemes with adic morphisms. Coherent henselian rigid spaces

✓ ✏

CRh = CHs˜=fadm. blow-upsg:

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid spaces

▶ CHs˜ = cat. of coherent (= q-cpt & q-sep)

henselian schemes with adic morphisms. Coherent henselian rigid spaces

✓ ✏

CRh = CHs˜=fadm. blow-upsg:

✒ ✑

▶ Xrig = the rigid space associated to

X 2 CHs˜.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid spaces

▶ CHs˜ = cat. of coherent (= q-cpt & q-sep)

henselian schemes with adic morphisms. Coherent henselian rigid spaces

✓ ✏

CRh = CHs˜=fadm. blow-upsg:

✒ ✑

▶ Xrig = the rigid space associated to

X 2 CHs˜.

▶ General rigid space by “birational patching”

(similarly to [FK, Chap. II, §2.2.(c)]).

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Henselian rigid spaces

▶ CHs˜ = cat. of coherent (= q-cpt & q-sep)

henselian schemes with adic morphisms. Coherent henselian rigid spaces

✓ ✏

CRh = CHs˜=fadm. blow-upsg:

✒ ✑

▶ Xrig = the rigid space associated to

X 2 CHs˜.

▶ General rigid space by “birational patching”

(similarly to [FK, Chap. II, §2.2.(c)]).

▶ First appearance in literature:

▶ Fujiwara, K.: Theory of tubular

neighborhood in étale topology. Duke Math.

• J. 80 (1995), no. 1, 15–57.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Visualization and points

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Visualization and points

Zariski-Riemann space

✓ ✏

For a coherent henselian rigid space X = Xrig, hX i = lim `

X0!X

X0; where X0 runs over all adm. blow-ups of X.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Visualization and points

Zariski-Riemann space

✓ ✏

For a coherent henselian rigid space X = Xrig, hX i = lim `

X0!X

X0; where X0 runs over all adm. blow-ups of X.

✒ ✑

▶ hX i is sober and coherent.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Visualization and points

Zariski-Riemann space

✓ ✏

For a coherent henselian rigid space X = Xrig, hX i = lim `

X0!X

X0; where X0 runs over all adm. blow-ups of X.

✒ ✑

▶ hX i is sober and coherent. ▶ Points of hX i are in bijection with the equiv.

classes of rigid points, i.e., morphisms of the form (Sph V )rig ` ! X ; where V is an a-adically sep. and henselian valuation ring.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Contents

Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

GAGA

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

GAGA

Situation

▶ V : a-adically sep. and henselian valuation

ring, K = V [ 1

a]. ▶ A: henselian finite type V -algebra.

⇝ A = A[ 1

a]: henselian affinoid algebra over K.

▶ U = Spec A „ S = Spec A, D = Spec A=aA, ▶ S = (Spf A)rig: the affinoid associated to A.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

GAGA

Situation

▶ V : a-adically sep. and henselian valuation

ring, K = V [ 1

a]. ▶ A: henselian finite type V -algebra.

⇝ A = A[ 1

a]: henselian affinoid algebra over K.

▶ U = Spec A „ S = Spec A, D = Spec A=aA, ▶ S = (Spf A)rig: the affinoid associated to A.

The GAGA functor

ȷsep. of finite type

schemes over U

ff

` !

8 > < > :

locally of finite type henselian rigid spaces

• ver S

9 > = > ;

X 7` ! Xan:

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

GAGA

Situation

▶ V : a-adically sep. and henselian valuation

ring, K = V [ 1

a]. ▶ A: henselian finite type V -algebra.

⇝ A = A[ 1

a]: henselian affinoid algebra over K.

▶ U = Spec A „ S = Spec A, D = Spec A=aA, ▶ S = (Spf A)rig: the affinoid associated to A.

The GAGA functor

ȷsep. of finite type

schemes over U

ff

` !

8 > < > :

locally of finite type henselian rigid spaces

• ver S

9 > = > ;

X 7` ! Xan: GAGA theorems ( = GHGA theorems

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Affinoid valued points

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Affinoid valued points

▶ Any ¸: T = (Sph B)rig ! Xan from a finite

type affinoid canonically corresponds to a mor.

e

¸: s(T ) = Spec B[ 1

a] ! X of schemes.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Affinoid valued points

▶ Any ¸: T = (Sph B)rig ! Xan from a finite

type affinoid canonically corresponds to a mor.

e

¸: s(T ) = Spec B[ 1

a] ! X of schemes.

Theorem

✓ ✏

8 > < > :

morphism ¸: T ! Xan

• f rigid spaces

9 > = > ; ‰

` !

8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > :

pair (˛; h) consisting

• f

˛ : T ! S and h: s(T ) ! X such that the diagram s(T )

h

• s(˛)
• X

f

• U

commutes

9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ;

¸ 7` ! (fan ‹ ¸; e ¸)

✒ ✑

▶ Cf. [FK, Chap. II, Theorem 9.2.2].

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Contents

Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

▶ V : a-adically sep. and henselian valuation

ring of height 1.

▶ We consider locally of finite type henselian

rigid spaces over K = V [ 1

a].

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

▶ V : a-adically sep. and henselian valuation

ring of height 1.

▶ We consider locally of finite type henselian

rigid spaces over K = V [ 1

a].

Definition

✓ ✏

▶ A henselian rigid space Z is said to be

point-like if it is coherent and reduced, having a unique minimal point in hZ i.

▶ A classical point of a henselian rigid

space X is a point-like locally closed rigid subspace Z „ X .

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

▶ V : a-adically sep. and henselian valuation

ring of height 1.

▶ We consider locally of finite type henselian

rigid spaces over K = V [ 1

a].

Definition

✓ ✏

▶ A henselian rigid space Z is said to be

point-like if it is coherent and reduced, having a unique minimal point in hZ i.

▶ A classical point of a henselian rigid

space X is a point-like locally closed rigid subspace Z „ X .

✒ ✑

▶ Z „ X is in fact a closed subspace. ▶ Z = (Sph W )rig, where W is finite, flat, and

finitely presented over V .

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

▶ A: henselian admissible V -algebra, ▶ X = (Sph A)rig, ▶ s(X ) = Spec A[ 1 a] (Noetherian scheme)

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

▶ A: henselian admissible V -algebra, ▶ X = (Sph A)rig, ▶ s(X ) = Spec A[ 1 a] (Noetherian scheme)

Proposition

✓ ✏

For any classical point Z , ! X , the image

• f s(Z ) ! s(X ) is a closed point, and this

establishes a canonical bijection

ȷclassical points

• f X

ff ‰

` !

(closed points

• f s(X )

)

:

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Classical points

▶ A: henselian admissible V -algebra, ▶ X = (Sph A)rig, ▶ s(X ) = Spec A[ 1 a] (Noetherian scheme)

Proposition

✓ ✏

For any classical point Z , ! X , the image

• f s(Z ) ! s(X ) is a closed point, and this

establishes a canonical bijection

ȷclassical points

• f X

ff ‰

` !

(closed points

• f s(X )

)

:

✒ ✑

▶ X cl = the set of all classical points of X . ▶ X 7! X cl is functorial.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Quasi-finite morphisms

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Quasi-finite morphisms

▶ ’: X ! Y morphism between loc. of finite

type rigid spaces over K

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Quasi-finite morphisms

▶ ’: X ! Y morphism between loc. of finite

type rigid spaces over K Definition

✓ ✏

’ is said to be quasi-finite

def

, for any x 2 Y cl the fiber X ˆY x is of dimension 0, , for any x 2 Y cl the fiber X ˆY x consists

• f finitely many classical points.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Quasi-finite morphisms

▶ ’: X ! Y morphism between loc. of finite

type rigid spaces over K Definition

✓ ✏

’ is said to be quasi-finite

def

, for any x 2 Y cl the fiber X ˆY x is of dimension 0, , for any x 2 Y cl the fiber X ˆY x consists

• f finitely many classical points.

✒ ✑

▶ If f : X ! Y is a quasi-finite map between

• sep. finite type schemes over K, then

fan : Xan ! Y an is quasi-finite.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Contents

Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Statement

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Statement

Theorem

✓ ✏

Let X be a separated finite type scheme over K, U = (Sph A)rig a henselian affinoid space

• f finite type over K, and ’: U ! Xan a quasi-

finite K-morphism. Then there exists a finite morphism g : W ! X with the commutative diagram W an

gan

• U
• j
• ’

Xan;

where j is an open immersion.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Write A = lim

` ! A– such that

▶ each A– is finite type V -algebra; ▶ A– ! A— is étale with A–=aA– ‰

= A—=aA—;

▶ Ah

– = A for each –.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Write A = lim

` ! A– such that

▶ each A– is finite type V -algebra; ▶ A– ! A— is étale with A–=aA– ‰

= A—=aA—;

▶ Ah

– = A for each –.

▶ The map ’: U = (Sph A)rig ! Xan gives e

’: Spec A[ 1

a] ! X.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Write A = lim

` ! A– such that

▶ each A– is finite type V -algebra; ▶ A– ! A— is étale with A–=aA– ‰

= A—=aA—;

▶ Ah

– = A for each –.

▶ The map ’: U = (Sph A)rig ! Xan gives e

’: Spec A[ 1

a] ! X. ▶ Since X is of finite type over K, 9– such that

Spec A[ 1

a] e ’

• X

Spec A–[ 1

a]: e ’–

• 26 / 30

Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Write A = lim

` ! A– such that

▶ each A– is finite type V -algebra; ▶ A– ! A— is étale with A–=aA– ‰

= A—=aA—;

▶ Ah

– = A for each –.

▶ The map ’: U = (Sph A)rig ! Xan gives e

’: Spec A[ 1

a] ! X. ▶ Since X is of finite type over K, 9– such that

Spec A[ 1

a] e ’

• X

Spec A–[ 1

a]: e ’–

• ▶ Observe:

e

’– is quasi-finite (due to Chevalley’s Theorem).

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Hence by usual ZMT,

Spec A[ 1

a] e ’

• X

Spec A–[ 1

a] e ’–

• pen W:

g: finite

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Hence by usual ZMT,

Spec A[ 1

a] e ’

• X

Spec A–[ 1

a] e ’–

• pen W:

g: finite

• ▶ Take “an”:

Xan (Spec A–[ 1

a])an e ’an

–

• pen W an:

gan: finite

• 27 / 30

Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Hence by usual ZMT,

Spec A[ 1

a] e ’

• X

Spec A–[ 1

a] e ’–

• pen W:

g: finite

• ▶ Take “an”:

Xan (Spec A–[ 1

a])an e ’an

–

• pen W an:

gan: finite

• ▶ U = (Sph A)rig = (Sph Ah

–)rig is an affinoid

domain in (Spec A–[ 1

a])an.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Scheme-theoretic closure

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Scheme-theoretic closure

Corollary

✓ ✏

Let X be a separated finite type scheme over K, U a henselian rigid space of finite type over K, and ’: U , ! Xan an immersion. Then there exists a closed subscheme W „ X that is smallest among those containing the image

• f U as an open subspace.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Scheme-theoretic closure

Corollary

✓ ✏

Let X be a separated finite type scheme over K, U a henselian rigid space of finite type over K, and ’: U , ! Xan an immersion. Then there exists a closed subscheme W „ X that is smallest among those containing the image

• f U as an open subspace.

✒ ✑

Proof.

▶ Suffices to show 9W containing U () one

can take the intersection of all such W ’s).

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Scheme-theoretic closure

Corollary

✓ ✏

Let X be a separated finite type scheme over K, U a henselian rigid space of finite type over K, and ’: U , ! Xan an immersion. Then there exists a closed subscheme W „ X that is smallest among those containing the image

• f U as an open subspace.

✒ ✑

Proof.

▶ Suffices to show 9W containing U () one

can take the intersection of all such W ’s).

▶ We may assume U is an affinoid.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Scheme-theoretic closure

Corollary

✓ ✏

Let X be a separated finite type scheme over K, U a henselian rigid space of finite type over K, and ’: U , ! Xan an immersion. Then there exists a closed subscheme W „ X that is smallest among those containing the image

• f U as an open subspace.

✒ ✑

Proof.

▶ Suffices to show 9W containing U () one

can take the intersection of all such W ’s).

▶ We may assume U is an affinoid. ▶ Take W ! X as in the theorem.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Scheme-theoretic closure

Corollary

✓ ✏

Let X be a separated finite type scheme over K, U a henselian rigid space of finite type over K, and ’: U , ! Xan an immersion. Then there exists a closed subscheme W „ X that is smallest among those containing the image

• f U as an open subspace.

✒ ✑

Proof.

▶ Suffices to show 9W containing U () one

can take the intersection of all such W ’s).

▶ We may assume U is an affinoid. ▶ Take W ! X as in the theorem. ▶ Replace it by the scheme-theoretic image (in

the usual sense) in X.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Family of closed subspaces

▶ X: projective scheme over K, ▶ U (resp. U): finite type henselian rigid space

(resp. finite type scheme) over K.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Family of closed subspaces

▶ X: projective scheme over K, ▶ U (resp. U): finite type henselian rigid space

(resp. finite type scheme) over K.

▶ A flat family of closed subspaces in X over

U (resp. U) is a closed subspace Y „ Xan ˆK U (resp. Y „ X ˆK U) that is flat over U (resp. U).

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Family of closed subspaces

▶ X: projective scheme over K, ▶ U (resp. U): finite type henselian rigid space

(resp. finite type scheme) over K.

▶ A flat family of closed subspaces in X over

U (resp. U) is a closed subspace Y „ Xan ˆK U (resp. Y „ X ˆK U) that is flat over U (resp. U).

▶ N.B. In the scheme-situation, such families

are classified by the Hilbert scheme HilbX=K.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Family of closed subspaces

▶ X: projective scheme over K, ▶ U (resp. U): finite type henselian rigid space

(resp. finite type scheme) over K.

▶ A flat family of closed subspaces in X over

U (resp. U) is a closed subspace Y „ Xan ˆK U (resp. Y „ X ˆK U) that is flat over U (resp. U).

▶ N.B. In the scheme-situation, such families

are classified by the Hilbert scheme HilbX=K.

Proposition

✓ ✏

For any flat family Y „ Xan ˆK U over finite type henselian affinoid U , there exists an affine finite type scheme U such that (a) Uan contains U as an affinoid subdomain; (b) Y extends to a flat family Y „ X ˆK U

• ver U.

✒ ✑

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig. ▶ Take A = lim

` ! A– as before, U– = Spec A–[ 1

a],

and consider Xan ˆK Uan

–

= (X ˆK U–)an.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig. ▶ Take A = lim

` ! A– as before, U– = Spec A–[ 1

a],

and consider Xan ˆK Uan

–

= (X ˆK U–)an.

▶ Take the scheme-theoretic closure Y– of

Y „ Xan ˆK U „ Xan ˆK Uan

– .

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig. ▶ Take A = lim

` ! A– as before, U– = Spec A–[ 1

a],

and consider Xan ˆK Uan

–

= (X ˆK U–)an.

▶ Take the scheme-theoretic closure Y– of

Y „ Xan ˆK U „ Xan ˆK Uan

– . ▶ Since Y ! U is projective, it is “an” of

Y ! U = Spec A[ 1

a].

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig. ▶ Take A = lim

` ! A– as before, U– = Spec A–[ 1

a],

and consider Xan ˆK Uan

–

= (X ˆK U–)an.

▶ Take the scheme-theoretic closure Y– of

Y „ Xan ˆK U „ Xan ˆK Uan

– . ▶ Since Y ! U is projective, it is “an” of

Y ! U = Spec A[ 1

a]. ▶ Observe: Y ! U = Spec A[ 1 a] is the

projective limit of Y– ! U–.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig. ▶ Take A = lim

` ! A– as before, U– = Spec A–[ 1

a],

and consider Xan ˆK Uan

–

= (X ˆK U–)an.

▶ Take the scheme-theoretic closure Y– of

Y „ Xan ˆK U „ Xan ˆK Uan

– . ▶ Since Y ! U is projective, it is “an” of

Y ! U = Spec A[ 1

a]. ▶ Observe: Y ! U = Spec A[ 1 a] is the

projective limit of Y– ! U–.

▶ Y is U-flat, since Y is U -flat.

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Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results

Proof

▶ Set U = (Sph A)rig. ▶ Take A = lim

` ! A– as before, U– = Spec A–[ 1

a],

and consider Xan ˆK Uan

–

= (X ˆK U–)an.

▶ Take the scheme-theoretic closure Y– of

Y „ Xan ˆK U „ Xan ˆK Uan

– . ▶ Since Y ! U is projective, it is “an” of

Y ! U = Spec A[ 1

a]. ▶ Observe: Y ! U = Spec A[ 1 a] is the

projective limit of Y– ! U–.

▶ Y is U-flat, since Y is U -flat. ▶ By standard limit argument, there exists –

such that Y– is U–-flat, hence giving a desired extension.

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