Zariski Main Theorem for Henselian affinoid algebras Henselian - PowerPoint PPT Presentation

I -adically henselian rings Zariski Main Theorem for Henselian Rigid Spaces ▶ A : ring, I „ A : finitely generated ideal, Fumiharu Kato ▶ S = Spec A , D = Spec A=I . Introduction ✓ ✏ Definition Henselian affinoid algebras ▶ A is said to be I -adically henselian Henselian rigid spaces def , any étale morphism ffi : X ! S with Henselian rigid ffi ` 1 ( D ) ‰ = D admits a section. GAGA , I „ Jac( A ) , 8 monic F ( T ) 2 A [ T ] with Classical points Main results F (0) ” 0 mod I and F 0 (0) 2 ( A=I ) ˆ ; 9 a 2 I such that F ( a ) = 0 . ✒ ✑ ✓ ✏ Henselization ▶ A ! A h is flat, and is faithfully flat if I „ Jac(A) . ✒ ✑ 7 / 30

Henselian finite type algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ). Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 8 / 30

Henselian finite type algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ). Introduction ▶ We write Henselian affinoid algebras def = ( V [ X 1 ; : : : ; X n ]) h : V f X 1 ; : : : ; X n g Henselian rigid spaces Henselian rigid GAGA Classical points Main results 8 / 30

Henselian finite type algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ). Introduction ▶ We write Henselian affinoid algebras def = ( V [ X 1 ; : : : ; X n ]) h : V f X 1 ; : : : ; X n g Henselian rigid spaces ✓ ✏ Henselian rigid Definition GAGA Classical points ▶ A henselian finite type V -algebra is a Main results V -algebra isom. to V f X 1 ; : : : ; X n g = a . ✒ ✑ 8 / 30

Henselian finite type algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ). Introduction ▶ We write Henselian affinoid algebras def = ( V [ X 1 ; : : : ; X n ]) h : V f X 1 ; : : : ; X n g Henselian rigid spaces ✓ ✏ Henselian rigid Definition GAGA Classical points ▶ A henselian finite type V -algebra is a Main results V -algebra isom. to V f X 1 ; : : : ; X n g = a . ✒ ✑ ▶ If A = V f X 1 ; : : : ; X n g = a is V -flat, then a is finitely generated. 8 / 30

Henselian finite type algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ). Introduction ▶ We write Henselian affinoid algebras def = ( V [ X 1 ; : : : ; X n ]) h : V f X 1 ; : : : ; X n g Henselian rigid spaces ✓ ✏ Henselian rigid Definition GAGA Classical points ▶ A henselian finite type V -algebra is a Main results V -algebra isom. to V f X 1 ; : : : ; X n g = a . ✒ ✑ ▶ If A = V f X 1 ; : : : ; X n g = 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. 8 / 30

Henselian affinoid algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ), Introduction ▶ K = Frac( V ) = V [ 1 a ] the fractional field. Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 9 / 30

Henselian affinoid algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ), Introduction ▶ K = Frac( V ) = V [ 1 a ] the fractional field. Henselian affinoid algebras ✓ ✏ Henselian Tate algebra Henselian rigid spaces def K f X 1 ; : : : ; X n g = V f X 1 ; : : : ; X n g ˙ V K Henselian rigid GAGA = V f X 1 ; : : : ; X n g [ 1 a ] : Classical points ✒ ✑ Main results 9 / 30

Henselian affinoid algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ), Introduction ▶ K = Frac( V ) = V [ 1 a ] the fractional field. Henselian affinoid algebras ✓ ✏ Henselian Tate algebra Henselian rigid spaces def K f X 1 ; : : : ; X n g = V f X 1 ; : : : ; X n g ˙ V K Henselian rigid GAGA = V f X 1 ; : : : ; X n g [ 1 a ] : Classical points ✒ ✑ Main results ▶ K f X 1 ; : : : ; X n g is a Noetherian K -algebra. 9 / 30

Henselian affinoid algebras Zariski Main Theorem for Henselian Rigid Spaces ▶ V : a -adically separated a -adically henselian Fumiharu Kato valuation ring ( a 2 m V n f 0 g ), Introduction ▶ K = Frac( V ) = V [ 1 a ] the fractional field. Henselian affinoid algebras ✓ ✏ Henselian Tate algebra Henselian rigid spaces def K f X 1 ; : : : ; X n g = V f X 1 ; : : : ; X n g ˙ V K Henselian rigid GAGA = V f X 1 ; : : : ; X n g [ 1 a ] : Classical points ✒ ✑ Main results ▶ K f X 1 ; : : : ; X n g is a Noetherian K -algebra. ✓ ✏ Henselian affinoid algebra A = K f X 1 ; : : : ; X n g = a by an ideal a „ K f X 1 ; : : : ; X n g . ✒ ✑ 9 / 30

Noether normalization Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 10 / 30

Noether normalization Zariski Main Theorem for ✓ ✏ Henselian Rigid Theorem Spaces Fumiharu Kato For any henselian affinoid algebra A over K , there exists an injective finite morphism Introduction Henselian affinoid algebras K f X 1 ; : : : ; X d g , ` ! A : Henselian rigid ✒ ✑ spaces Henselian rigid GAGA Classical points Main results 10 / 30

Noether normalization Zariski Main Theorem for ✓ ✏ Henselian Rigid Theorem Spaces Fumiharu Kato For any henselian affinoid algebra A over K , there exists an injective finite morphism Introduction Henselian affinoid algebras K f X 1 ; : : : ; X d g , ` ! A : Henselian rigid ✒ ✑ spaces ✓ ✏ Lemma Henselian rigid GAGA Let A ! B be a finite type morphism of rings, and Classical points I „ A an ideal. If A=I ! B=IB is finite, then so is Main results A h ! B h . ✒ ✑ ▶ Classical (refined) ZMT: finiteness extends to an étale neighborhood. 10 / 30

Noether normalization Zariski Main Theorem for ✓ ✏ Henselian Rigid Theorem Spaces Fumiharu Kato For any henselian affinoid algebra A over K , there exists an injective finite morphism Introduction Henselian affinoid algebras K f X 1 ; : : : ; X d g , ` ! A : Henselian rigid ✒ ✑ spaces ✓ ✏ Lemma Henselian rigid GAGA Let A ! B be a finite type morphism of rings, and Classical points I „ A an ideal. If A=I ! B=IB is finite, then so is Main results A h ! B h . ✒ ✑ ▶ Classical (refined) ZMT: finiteness extends to an étale neighborhood. ✓ ✏ Theorem Henselian affinoid algebras are Jacobson. ✒ ✑ 10 / 30

Contents Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Introduction Henselian affinoid algebras Henselian affinoid algebras Henselian rigid spaces Henselian rigid spaces Henselian rigid GAGA Classical points Henselian rigid GAGA Main results Classical points Main results 11 / 30

Henselian schemes: References Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato ▶ Cox, D. A.: Algebraic tubular Introduction neighborhoods. I, II , Math. Scand. 42 Henselian affinoid algebras (1978), no. 2, 211–228, 229–242. Henselian rigid ▶ Greco, S.; Strano, R.: Quasicoherent sheaves spaces Henselian rigid over affine Hensel schemes . Trans. Amer. GAGA Math. Soc. 268 (1981), no. 2, 445–465. Classical points ▶ Kurke, H.; Pfister, G.; Roczen, M.: Main results Henselsche Ringe und algebraische Geometrie . Mathematische Monographien, Band II. VEB Deutscher Verlag der Wissenschaften, Berlin, 1975. 12 / 30

Henselian spectrum Zariski Main Theorem for Henselian Rigid Spaces ▶ A : I -adically henselian. Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 13 / 30

Henselian spectrum Zariski Main Theorem for Henselian Rigid Spaces ▶ A : I -adically henselian. Fumiharu Kato ▶ The henselian spectrum Introduction Henselian Sph A affinoid algebras Henselian rigid is defined similarly to the formal spectrum. spaces Henselian rigid GAGA Classical points Main results 13 / 30

Henselian spectrum Zariski Main Theorem for Henselian Rigid Spaces ▶ A : I -adically henselian. Fumiharu Kato ▶ The henselian spectrum Introduction Henselian Sph A affinoid algebras Henselian rigid is defined similarly to the formal spectrum. spaces Henselian rigid ✓ ✏ Henselian spectrum GAGA Classical points Sph A = top. locally ringed space by Main results ▶ 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 O X ( D ( f )) = ( A f ) h , which gives a sheaf of top. rings on X . ✒ ✑ 13 / 30

Henselian schemes Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 14 / 30

Henselian schemes Zariski Main Theorem for ✓ ✏ Henselian Rigid Definition Spaces Fumiharu Kato ▶ An affine henselian scheme is a top. loc. ringed space isom. to ( X = Sph A; O X ) Introduction for an I -adically henselian ring A . Henselian affinoid algebras ▶ A top. locally ringed space ( X; O X ) is Henselian rigid spaces called a henselian scheme if it is covered Henselian rigid by affine henselian schemes. GAGA ✒ ✑ Classical points Main results 14 / 30

Henselian schemes Zariski Main Theorem for ✓ ✏ Henselian Rigid Definition Spaces Fumiharu Kato ▶ An affine henselian scheme is a top. loc. ringed space isom. to ( X = Sph A; O X ) Introduction for an I -adically henselian ring A . Henselian affinoid algebras ▶ A top. locally ringed space ( X; O X ) is Henselian rigid spaces called a henselian scheme if it is covered Henselian rigid by affine henselian schemes. GAGA ✒ ✑ Classical points Main results ▶ A morphism between henselian schemes is a morphism of top. locally ringed spaces. 14 / 30

Henselian schemes Zariski Main Theorem for ✓ ✏ Henselian Rigid Definition Spaces Fumiharu Kato ▶ An affine henselian scheme is a top. loc. ringed space isom. to ( X = Sph A; O X ) Introduction for an I -adically henselian ring A . Henselian affinoid algebras ▶ A top. locally ringed space ( X; O X ) is Henselian rigid spaces called a henselian scheme if it is covered Henselian rigid by affine henselian schemes. GAGA ✒ ✑ Classical points Main results ▶ 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. 14 / 30

Henselian schemes Zariski Main Theorem for ✓ ✏ Henselian Rigid Definition Spaces Fumiharu Kato ▶ An affine henselian scheme is a top. loc. ringed space isom. to ( X = Sph A; O X ) Introduction for an I -adically henselian ring A . Henselian affinoid algebras ▶ A top. locally ringed space ( X; O X ) is Henselian rigid spaces called a henselian scheme if it is covered Henselian rigid by affine henselian schemes. GAGA ✒ ✑ Classical points Main results ▶ 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 X h j Y of schemes along closed subschemes. 14 / 30

Henselian rigid spaces Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 15 / 30

Henselian rigid spaces Zariski Main Theorem for Henselian Rigid Spaces ▶ CHs ˜ = cat. of coherent ( = q-cpt & q-sep) Fumiharu Kato henselian schemes with adic morphisms. Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 15 / 30

Henselian rigid spaces Zariski Main Theorem for Henselian Rigid Spaces ▶ CHs ˜ = cat. of coherent ( = q-cpt & q-sep) Fumiharu Kato henselian schemes with adic morphisms. Introduction ✓ ✏ Coherent henselian rigid spaces Henselian affinoid algebras CRh = CHs ˜ = f adm. blow-ups g : Henselian rigid ✒ ✑ spaces Henselian rigid GAGA Classical points Main results 15 / 30

Henselian rigid spaces Zariski Main Theorem for Henselian Rigid Spaces ▶ CHs ˜ = cat. of coherent ( = q-cpt & q-sep) Fumiharu Kato henselian schemes with adic morphisms. Introduction ✓ ✏ Coherent henselian rigid spaces Henselian affinoid algebras CRh = CHs ˜ = f adm. blow-ups g : Henselian rigid ✒ ✑ spaces Henselian rigid ▶ X rig = the rigid space associated to GAGA Classical points X 2 CHs ˜ . Main results 15 / 30

Henselian rigid spaces Zariski Main Theorem for Henselian Rigid Spaces ▶ CHs ˜ = cat. of coherent ( = q-cpt & q-sep) Fumiharu Kato henselian schemes with adic morphisms. Introduction ✓ ✏ Coherent henselian rigid spaces Henselian affinoid algebras CRh = CHs ˜ = f adm. blow-ups g : Henselian rigid ✒ ✑ spaces Henselian rigid ▶ X rig = the rigid space associated to GAGA Classical points X 2 CHs ˜ . Main results ▶ General rigid space by “birational patching” (similarly to [FK, Chap. II , §2.2.(c)]). 15 / 30

Henselian rigid spaces Zariski Main Theorem for Henselian Rigid Spaces ▶ CHs ˜ = cat. of coherent ( = q-cpt & q-sep) Fumiharu Kato henselian schemes with adic morphisms. Introduction ✓ ✏ Coherent henselian rigid spaces Henselian affinoid algebras CRh = CHs ˜ = f adm. blow-ups g : Henselian rigid ✒ ✑ spaces Henselian rigid ▶ X rig = the rigid space associated to GAGA Classical points X 2 CHs ˜ . Main results ▶ 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. 15 / 30

Visualization and points Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 16 / 30

Visualization and points Zariski Main Theorem for Henselian Rigid ✓ ✏ Spaces Zariski-Riemann space Fumiharu Kato For a coherent henselian rigid space X = X rig , Introduction X 0 ; Henselian h X i = lim ` affinoid algebras X 0 ! X Henselian rigid spaces where X 0 runs over all adm. blow-ups of X . Henselian rigid ✒ ✑ GAGA Classical points Main results 16 / 30

Visualization and points Zariski Main Theorem for Henselian Rigid ✓ ✏ Spaces Zariski-Riemann space Fumiharu Kato For a coherent henselian rigid space X = X rig , Introduction X 0 ; Henselian h X i = lim ` affinoid algebras X 0 ! X Henselian rigid spaces where X 0 runs over all adm. blow-ups of X . Henselian rigid ✒ ✑ GAGA Classical points ▶ h X i is sober and coherent. Main results 16 / 30

Visualization and points Zariski Main Theorem for Henselian Rigid ✓ ✏ Spaces Zariski-Riemann space Fumiharu Kato For a coherent henselian rigid space X = X rig , Introduction X 0 ; Henselian h X i = lim ` affinoid algebras X 0 ! X Henselian rigid spaces where X 0 runs over all adm. blow-ups of X . Henselian rigid ✒ ✑ GAGA Classical points ▶ h X i is sober and coherent. Main results ▶ Points of h X 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. 16 / 30

Contents Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Introduction Henselian affinoid algebras Henselian affinoid algebras Henselian rigid spaces Henselian rigid spaces Henselian rigid GAGA Classical points Henselian rigid GAGA Main results Classical points Main results 17 / 30

GAGA Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 18 / 30

GAGA Zariski Main Theorem for Henselian Rigid Spaces Situation Fumiharu Kato ▶ V : a -adically sep. and henselian valuation ring, K = V [ 1 Introduction a ] . Henselian ▶ A : henselian finite type V -algebra. affinoid algebras ⇝ A = A [ 1 Henselian rigid a ] : henselian affinoid algebra over K . spaces ▶ U = Spec A „ S = Spec A , D = Spec A=aA , Henselian rigid GAGA ▶ S = (Spf A ) rig : the affinoid associated to A . Classical points Main results 18 / 30

GAGA Zariski Main Theorem for Henselian Rigid Spaces Situation Fumiharu Kato ▶ V : a -adically sep. and henselian valuation ring, K = V [ 1 Introduction a ] . Henselian ▶ A : henselian finite type V -algebra. affinoid algebras ⇝ A = A [ 1 Henselian rigid a ] : henselian affinoid algebra over K . spaces ▶ U = Spec A „ S = Spec A , D = Spec A=aA , Henselian rigid GAGA ▶ S = (Spf A ) rig : the affinoid associated to A . Classical points The GAGA functor Main results 8 9 locally of finite type > > ȷ sep. of finite type ff < = ` ! henselian rigid spaces > > schemes over U : ; over S ! X an : X 7` 18 / 30

GAGA Zariski Main Theorem for Henselian Rigid Spaces Situation Fumiharu Kato ▶ V : a -adically sep. and henselian valuation ring, K = V [ 1 Introduction a ] . Henselian ▶ A : henselian finite type V -algebra. affinoid algebras ⇝ A = A [ 1 Henselian rigid a ] : henselian affinoid algebra over K . spaces ▶ U = Spec A „ S = Spec A , D = Spec A=aA , Henselian rigid GAGA ▶ S = (Spf A ) rig : the affinoid associated to A . Classical points The GAGA functor Main results 8 9 locally of finite type > > ȷ sep. of finite type ff < = ` ! henselian rigid spaces > > schemes over U : ; over S ! X an : X 7` GAGA theorems ( = GHGA theorems 18 / 30

Affinoid valued points Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 19 / 30

Affinoid valued points Zariski Main Theorem for Henselian Rigid Spaces ▶ Any ¸ : T = (Sph B ) rig ! X an from a finite Fumiharu Kato type affinoid canonically corresponds to a ¸ : s ( T ) = Spec B [ 1 Introduction mor. a ] ! X of schemes. e Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 19 / 30

� � � Affinoid valued points Zariski Main Theorem for Henselian Rigid Spaces ▶ Any ¸ : T = (Sph B ) rig ! X an from a finite Fumiharu Kato type affinoid canonically corresponds to a ¸ : s ( T ) = Spec B [ 1 Introduction mor. a ] ! X of schemes. e Henselian ✓ ✏ Theorem affinoid algebras 8 9 Henselian rigid pair ( ˛; h ) consisting > > spaces > > > > > > > > > of ˛ : T ! S and > Henselian rigid > > > > > > GAGA > > 8 9 > > h : s ( T ) ! X such > > morphism > > > > > > Classical points < = < = ‰ that the diagram ! X an ¸ : T ` ! Main results > > > > : ; > h > > > > s ( T ) X commutes > of rigid spaces > > > > > > > > > > > > f > > > > > > s ( ˛ ) > > : ; U ! ( f an ‹ ¸; e ¸ 7` ¸ ) ✒ ✑ ▶ Cf. [FK, Chap. II , Theorem 9.2.2]. 19 / 30

Contents Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Introduction Henselian affinoid algebras Henselian affinoid algebras Henselian rigid spaces Henselian rigid spaces Henselian rigid GAGA Classical points Henselian rigid GAGA Main results Classical points Main results 20 / 30

Classical points Zariski Main Theorem for Henselian Rigid ▶ V : a -adically sep. and henselian valuation Spaces ring of height 1 . Fumiharu Kato ▶ We consider locally of finite type henselian Introduction rigid spaces over K = V [ 1 a ] . Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 21 / 30

Classical points Zariski Main Theorem for Henselian Rigid ▶ V : a -adically sep. and henselian valuation Spaces ring of height 1 . Fumiharu Kato ▶ We consider locally of finite type henselian Introduction rigid spaces over K = V [ 1 a ] . Henselian affinoid algebras ✓ ✏ Definition Henselian rigid spaces ▶ A henselian rigid space Z is said to be Henselian rigid GAGA point-like if it is coherent and reduced, Classical points having a unique minimal point in h Z i . Main results ▶ A classical point of a henselian rigid space X is a point-like locally closed rigid subspace Z „ X . ✒ ✑ 21 / 30

Classical points Zariski Main Theorem for Henselian Rigid ▶ V : a -adically sep. and henselian valuation Spaces ring of height 1 . Fumiharu Kato ▶ We consider locally of finite type henselian Introduction rigid spaces over K = V [ 1 a ] . Henselian affinoid algebras ✓ ✏ Definition Henselian rigid spaces ▶ A henselian rigid space Z is said to be Henselian rigid GAGA point-like if it is coherent and reduced, Classical points having a unique minimal point in h Z i . Main results ▶ 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 . 21 / 30

Classical points Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 22 / 30

Classical points Zariski Main Theorem for Henselian Rigid Spaces ▶ A : henselian admissible V -algebra, Fumiharu Kato ▶ X = (Sph A ) rig , Introduction ▶ s ( X ) = Spec A [ 1 a ] (Noetherian scheme) Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 22 / 30

Classical points Zariski Main Theorem for Henselian Rigid Spaces ▶ A : henselian admissible V -algebra, Fumiharu Kato ▶ X = (Sph A ) rig , Introduction ▶ s ( X ) = Spec A [ 1 a ] (Noetherian scheme) Henselian affinoid algebras ✓ ✏ Henselian rigid Proposition spaces Henselian rigid For any classical point Z , ! X , the image GAGA of s ( Z ) ! s ( X ) is a closed point, and this Classical points establishes a canonical bijection Main results ( closed points ) ȷ classical points ff ‰ ` ! : of s ( X ) of X ✒ ✑ 22 / 30

Classical points Zariski Main Theorem for Henselian Rigid Spaces ▶ A : henselian admissible V -algebra, Fumiharu Kato ▶ X = (Sph A ) rig , Introduction ▶ s ( X ) = Spec A [ 1 a ] (Noetherian scheme) Henselian affinoid algebras ✓ ✏ Henselian rigid Proposition spaces Henselian rigid For any classical point Z , ! X , the image GAGA of s ( Z ) ! s ( X ) is a closed point, and this Classical points establishes a canonical bijection Main results ( closed points ) ȷ classical points ff ‰ ` ! : of s ( X ) of X ✒ ✑ ▶ X cl = the set of all classical points of X . ▶ X 7! X cl is functorial. 22 / 30

Quasi-finite morphisms Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 23 / 30

Quasi-finite morphisms Zariski Main Theorem for Henselian Rigid Spaces ▶ ’ : X ! Y morphism between loc. of finite Fumiharu Kato type rigid spaces over K Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 23 / 30

Quasi-finite morphisms Zariski Main Theorem for Henselian Rigid Spaces ▶ ’ : X ! Y morphism between loc. of finite Fumiharu Kato type rigid spaces over K Introduction ✓ ✏ Henselian Definition affinoid algebras Henselian rigid ’ is said to be quasi-finite spaces , for any x 2 Y cl the fiber X ˆ Y x is of def Henselian rigid GAGA dimension 0 , Classical points , for any x 2 Y cl the fiber X ˆ Y x consists Main results of finitely many classical points. ✒ ✑ 23 / 30

Quasi-finite morphisms Zariski Main Theorem for Henselian Rigid Spaces ▶ ’ : X ! Y morphism between loc. of finite Fumiharu Kato type rigid spaces over K Introduction ✓ ✏ Henselian Definition affinoid algebras Henselian rigid ’ is said to be quasi-finite spaces , for any x 2 Y cl the fiber X ˆ Y x is of def Henselian rigid GAGA dimension 0 , Classical points , for any x 2 Y cl the fiber X ˆ Y x consists Main results of finitely many classical points. ✒ ✑ ▶ If f : X ! Y is a quasi-finite map between sep. finite type schemes over K , then f an : X an ! Y an is quasi-finite. 23 / 30

Contents Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Introduction Henselian affinoid algebras Henselian affinoid algebras Henselian rigid spaces Henselian rigid spaces Henselian rigid GAGA Classical points Henselian rigid GAGA Main results Classical points Main results 24 / 30

Statement Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 25 / 30

�� � � Statement Zariski Main Theorem for Henselian Rigid Spaces ✓ ✏ Theorem Fumiharu Kato Let X be a separated finite type scheme over Introduction K , U = (Sph A ) rig a henselian affinoid space Henselian of finite type over K , and ’ : U ! X an a quasi- affinoid algebras Henselian rigid finite K -morphism. Then there exists a finite spaces morphism g : W ! X with the commutative Henselian rigid GAGA diagram Classical points Main results W an g an j � X an ; U ’ where j is an open immersion . ✒ ✑ 25 / 30

Proof Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 26 / 30

Proof Zariski Main Theorem for Henselian Rigid Spaces ▶ Write A = lim ! A – such that ` Fumiharu Kato ▶ each A – is finite type V -algebra; ▶ A – ! A — is étale with A – =aA – ‰ = A — =aA — ; Introduction ▶ A h – = A for each – . Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 26 / 30

Proof Zariski Main Theorem for Henselian Rigid Spaces ▶ Write A = lim ! A – such that ` Fumiharu Kato ▶ each A – is finite type V -algebra; ▶ A – ! A — is étale with A – =aA – ‰ = A — =aA — ; Introduction ▶ A h – = A for each – . Henselian affinoid algebras ▶ The map ’ : U = (Sph A ) rig ! X an gives Henselian rigid ’ : Spec A [ 1 spaces a ] ! X . e Henselian rigid GAGA Classical points Main results 26 / 30

� � � Proof Zariski Main Theorem for Henselian Rigid Spaces ▶ Write A = lim ! A – such that ` Fumiharu Kato ▶ each A – is finite type V -algebra; ▶ A – ! A — is étale with A – =aA – ‰ = A — =aA — ; Introduction ▶ A h – = A for each – . Henselian affinoid algebras ▶ The map ’ : U = (Sph A ) rig ! X an gives Henselian rigid ’ : Spec A [ 1 spaces a ] ! X . e Henselian rigid ▶ Since X is of finite type over K , 9 – such that GAGA Classical points ’ e Main results Spec A [ 1 a ] X ’ – e Spec A – [ 1 a ] : 26 / 30

� � � Proof Zariski Main Theorem for Henselian Rigid Spaces ▶ Write A = lim ! A – such that ` Fumiharu Kato ▶ each A – is finite type V -algebra; ▶ A – ! A — is étale with A – =aA – ‰ = A — =aA — ; Introduction ▶ A h – = A for each – . Henselian affinoid algebras ▶ The map ’ : U = (Sph A ) rig ! X an gives Henselian rigid ’ : Spec A [ 1 spaces a ] ! X . e Henselian rigid ▶ Since X is of finite type over K , 9 – such that GAGA Classical points ’ e Main results Spec A [ 1 a ] X ’ – e Spec A – [ 1 a ] : ▶ Observe: ’ – is quasi-finite e (due to Chevalley’s Theorem). 26 / 30

Proof Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 27 / 30

� � � � Proof Zariski Main Theorem for Henselian Rigid ▶ Hence by usual ZMT, Spaces Fumiharu Kato ’ e Spec A [ 1 a ] X Introduction Henselian affinoid algebras g : finite ’ – e Henselian rigid open � W: Spec A – [ 1 spaces a ] � � Henselian rigid GAGA Classical points Main results 27 / 30

� � � � � � Proof Zariski Main Theorem for Henselian Rigid ▶ Hence by usual ZMT, Spaces Fumiharu Kato ’ e Spec A [ 1 a ] X Introduction Henselian affinoid algebras g : finite ’ – e Henselian rigid open � W: Spec A – [ 1 spaces a ] � � Henselian rigid GAGA ▶ Take “an”: Classical points Main results X an ’ an e – g an : finite (Spec A – [ 1 open � W an : a ]) an � � 27 / 30

� � � � � � Proof Zariski Main Theorem for Henselian Rigid ▶ Hence by usual ZMT, Spaces Fumiharu Kato ’ e Spec A [ 1 a ] X Introduction Henselian affinoid algebras g : finite ’ – e Henselian rigid open � W: Spec A – [ 1 spaces a ] � � Henselian rigid GAGA ▶ Take “an”: Classical points Main results X an ’ an e – g an : finite (Spec A – [ 1 open � W an : a ]) an � � ▶ U = (Sph A ) rig = (Sph A h – ) rig is an affinoid domain in (Spec A – [ 1 a ]) an . 27 / 30

Scheme-theoretic closure Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 28 / 30

Scheme-theoretic closure Zariski Main Theorem for Henselian Rigid ✓ ✏ Corollary Spaces Fumiharu Kato Let X be a separated finite type scheme over K , U a henselian rigid space of finite type over Introduction ! X an an immersion. Henselian K , and ’ : U , Then affinoid algebras there exists a closed subscheme W „ X that Henselian rigid spaces is smallest among those containing the image Henselian rigid of U as an open subspace. GAGA ✒ ✑ Classical points Main results 28 / 30

Scheme-theoretic closure Zariski Main Theorem for Henselian Rigid ✓ ✏ Corollary Spaces Fumiharu Kato Let X be a separated finite type scheme over K , U a henselian rigid space of finite type over Introduction ! X an an immersion. Henselian K , and ’ : U , Then affinoid algebras there exists a closed subscheme W „ X that Henselian rigid spaces is smallest among those containing the image Henselian rigid of U as an open subspace. GAGA ✒ ✑ Classical points Proof. Main results ▶ Suffices to show 9 W containing U ( ) one can take the intersection of all such W ’s). 28 / 30

Scheme-theoretic closure Zariski Main Theorem for Henselian Rigid ✓ ✏ Corollary Spaces Fumiharu Kato Let X be a separated finite type scheme over K , U a henselian rigid space of finite type over Introduction ! X an an immersion. Henselian K , and ’ : U , Then affinoid algebras there exists a closed subscheme W „ X that Henselian rigid spaces is smallest among those containing the image Henselian rigid of U as an open subspace. GAGA ✒ ✑ Classical points Proof. Main results ▶ Suffices to show 9 W containing U ( ) one can take the intersection of all such W ’s). ▶ We may assume U is an affinoid. 28 / 30

Scheme-theoretic closure Zariski Main Theorem for Henselian Rigid ✓ ✏ Corollary Spaces Fumiharu Kato Let X be a separated finite type scheme over K , U a henselian rigid space of finite type over Introduction ! X an an immersion. Henselian K , and ’ : U , Then affinoid algebras there exists a closed subscheme W „ X that Henselian rigid spaces is smallest among those containing the image Henselian rigid of U as an open subspace. GAGA ✒ ✑ Classical points Proof. Main results ▶ Suffices to show 9 W 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. 28 / 30

Scheme-theoretic closure Zariski Main Theorem for Henselian Rigid ✓ ✏ Corollary Spaces Fumiharu Kato Let X be a separated finite type scheme over K , U a henselian rigid space of finite type over Introduction ! X an an immersion. Henselian K , and ’ : U , Then affinoid algebras there exists a closed subscheme W „ X that Henselian rigid spaces is smallest among those containing the image Henselian rigid of U as an open subspace. GAGA ✒ ✑ Classical points Proof. Main results ▶ Suffices to show 9 W 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 . 28 / 30

Family of closed subspaces Zariski Main Theorem for Henselian Rigid ▶ X : projective scheme over K , Spaces ▶ U (resp. U ): finite type henselian rigid space Fumiharu Kato (resp. finite type scheme) over K . Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 29 / 30

Family of closed subspaces Zariski Main Theorem for Henselian Rigid ▶ X : projective scheme over K , Spaces ▶ U (resp. U ): finite type henselian rigid space Fumiharu Kato (resp. finite type scheme) over K . Introduction ▶ A flat family of closed subspaces in X over Henselian U (resp. U ) is a closed subspace affinoid algebras Y „ X an ˆ K U (resp. Y „ X ˆ K U ) that is Henselian rigid flat over U (resp. U ). spaces Henselian rigid GAGA Classical points Main results 29 / 30

Family of closed subspaces Zariski Main Theorem for Henselian Rigid ▶ X : projective scheme over K , Spaces ▶ U (resp. U ): finite type henselian rigid space Fumiharu Kato (resp. finite type scheme) over K . Introduction ▶ A flat family of closed subspaces in X over Henselian U (resp. U ) is a closed subspace affinoid algebras Y „ X an ˆ K U (resp. Y „ X ˆ K U ) that is Henselian rigid flat over U (resp. U ). spaces ▶ N.B. In the scheme-situation, such families Henselian rigid GAGA are classified by the Hilbert scheme Hilb X=K . Classical points Main results 29 / 30

Family of closed subspaces Zariski Main Theorem for Henselian Rigid ▶ X : projective scheme over K , Spaces ▶ U (resp. U ): finite type henselian rigid space Fumiharu Kato (resp. finite type scheme) over K . Introduction ▶ A flat family of closed subspaces in X over Henselian U (resp. U ) is a closed subspace affinoid algebras Y „ X an ˆ K U (resp. Y „ X ˆ K U ) that is Henselian rigid flat over U (resp. U ). spaces ▶ N.B. In the scheme-situation, such families Henselian rigid GAGA are classified by the Hilbert scheme Hilb X=K . Classical points ✓ ✏ Proposition Main results For any flat family Y „ X an ˆ K U over finite type henselian affinoid U , there exists an affine finite type scheme U such that (a) U an contains U as an affinoid subdomain ; (b) Y extends to a flat family Y „ X ˆ K U over U . ✒ ✑ 29 / 30

Proof Zariski Main Theorem for Henselian Rigid Spaces Fumiharu Kato Introduction Henselian affinoid algebras Henselian rigid spaces Henselian rigid GAGA Classical points Main results 30 / 30