differential schemes and differential algebraic varieties Dmitry - PowerPoint PPT Presentation

differential schemes and differential algebraic varieties Dmitry Trushin Department of Mechanics and Mathematics Moscow State University October 2010 Dmitry Trushin () Differential schemes October, 2010 1 / 21

Contents The differential spectrum of the ring of global sections 1 Differential integral dependence 2 Differential catenarity 3 Dmitry Trushin () Differential schemes October, 2010 2 / 21

Contents The differential spectrum of the ring of global sections 1 Differential integral dependence 2 Differential catenarity 3 Dmitry Trushin () Differential schemes October, 2010 3 / 21

Differential spectrum Ring = commutative, associative, and with an identity Dmitry Trushin () Differential schemes October, 2010 4 / 21

Differential spectrum Ring = commutative, associative, and with an identity ∆-ring = ring + ∆ = { δ 1 , . . . , δ m } δ i δ j = δ j δ i Dmitry Trushin () Differential schemes October, 2010 4 / 21

Differential spectrum Ring = commutative, associative, and with an identity ∆-ring = ring + ∆ = { δ 1 , . . . , δ m } δ i δ j = δ j δ i X = Spec ∆ R Dmitry Trushin () Differential schemes October, 2010 4 / 21

Differential spectrum Ring = commutative, associative, and with an identity ∆-ring = ring + ∆ = { δ 1 , . . . , δ m } δ i δ j = δ j δ i X = Spec ∆ R Construction O R ( U ) = regular functions in U X = Spec ∆ � � � R = O R ( X ) R ι ∗ : � ι : R → � R X → X Dmitry Trushin () Differential schemes October, 2010 4 / 21

Differential spectrum Ring = commutative, associative, and with an identity ∆-ring = ring + ∆ = { δ 1 , . . . , δ m } δ i δ j = δ j δ i X = Spec ∆ R Construction O R ( U ) = regular functions in U X = Spec ∆ � � � R = O R ( X ) R ι ∗ : � ι : R → � R X → X Conjecture ι ∗ : � X → X is a homeomorphism Dmitry Trushin () Differential schemes October, 2010 4 / 21

Auxiliary sheaf Construction O ′ R ( U ) = regular functions in U R ′ = O ′ X ′ = Spec ∆ � � � R ′ R ( X ) X ′ → X ι r : R → � r : � R ′ ι ∗ Dmitry Trushin () Differential schemes October, 2010 5 / 21

Auxiliary sheaf Construction O ′ R ( U ) = regular functions in U R ′ = O ′ X ′ = Spec ∆ � � � R ′ R ( X ) X ′ → X ι r : R → � r : � R ′ ι ∗ Theorem r : Spec ∆ D → Spec ∆ R is a homeomorphism. ι r : R → D ⊆ � R ′ . Then ι ∗ Dmitry Trushin () Differential schemes October, 2010 5 / 21

Auxiliary sheaf Construction O ′ R ( U ) = regular functions in U R ′ = O ′ X ′ = Spec ∆ � � � R ′ R ( X ) X ′ → X ι r : R → � r : � R ′ ι ∗ Theorem r : Spec ∆ D → Spec ∆ R is a homeomorphism. ι r : R → D ⊆ � R ′ . Then ι ∗ Corollary X ′ → X is a homeomorphism. r : � The mapping ι ∗ Dmitry Trushin () Differential schemes October, 2010 5 / 21

Keigher rings Definition (Keigher ring) I ⊆ R is a ∆-ideal ⇒ r ( I ) is a ∆-ideal. Dmitry Trushin () Differential schemes October, 2010 6 / 21

Keigher rings Definition (Keigher ring) I ⊆ R is a ∆-ideal ⇒ r ( I ) is a ∆-ideal. R is a Ritt algebra ( Q ⊆ R ) ⇒ R is a Keigher ring. Dmitry Trushin () Differential schemes October, 2010 6 / 21

Keigher rings Definition (Keigher ring) I ⊆ R is a ∆-ideal ⇒ r ( I ) is a ∆-ideal. R is a Ritt algebra ( Q ⊆ R ) ⇒ R is a Keigher ring. Theorem R is a Keigher ring, ι : R → D ⊆ � R. Then ι ∗ : Spec ∆ D → Spec ∆ R is a homeomorphism. Dmitry Trushin () Differential schemes October, 2010 6 / 21

Keigher rings Definition (Keigher ring) I ⊆ R is a ∆-ideal ⇒ r ( I ) is a ∆-ideal. R is a Ritt algebra ( Q ⊆ R ) ⇒ R is a Keigher ring. Theorem R is a Keigher ring, ι : R → D ⊆ � R. Then ι ∗ : Spec ∆ D → Spec ∆ R is a homeomorphism. Corollary R is a Keigher ring. Then ι ∗ : � X → X is a homeomorphism. Dmitry Trushin () Differential schemes October, 2010 6 / 21

Iterative derivations Definition Let δ = { δ k } k � 0 , δ k : R → R : 3) δ k ( ab ) = � 1) δ 0 ( x ) = x µ + ν = k δ µ ( a ) δ ν ( b ) � k + m � 4) δ k δ m = 2) δ k ( a + b ) = δ k ( a ) + δ k ( b ) δ k + m k Dmitry Trushin () Differential schemes October, 2010 7 / 21

Iterative derivations Definition Let δ = { δ k } k � 0 , δ k : R → R : 3) δ k ( ab ) = � 1) δ 0 ( x ) = x µ + ν = k δ µ ( a ) δ ν ( b ) � k + m � 4) δ k δ m = 2) δ k ( a + b ) = δ k ( a ) + δ k ( b ) δ k + m k Construction O R ( U ) = regular functions in U X = Spec ∆ � � � R = O R ( X ) R ι ∗ : � ι : R → � R X → X Dmitry Trushin () Differential schemes October, 2010 7 / 21

Iterative derivations Definition Let δ = { δ k } k � 0 , δ k : R → R : 3) δ k ( ab ) = � 1) δ 0 ( x ) = x µ + ν = k δ µ ( a ) δ ν ( b ) � k + m � 4) δ k δ m = 2) δ k ( a + b ) = δ k ( a ) + δ k ( b ) δ k + m k Construction O R ( U ) = regular functions in U X = Spec ∆ � � � R = O R ( X ) R ι ∗ : � ι : R → � R X → X Theorem The mapping ι ∗ : � X → X is a homeomorphism. Dmitry Trushin () Differential schemes October, 2010 7 / 21

The structure sheaves � � ι R R ι ∗ � � X � X � � p � p Dmitry Trushin () Differential schemes October, 2010 8 / 21

The structure sheaves � � ι R R ι ∗ � � X � X � � p � p Question Does O R coincide with O � R ? Dmitry Trushin () Differential schemes October, 2010 8 / 21

The structure sheaves � � ι R R ι ∗ � � X � X � � p � p Question Does O R coincide with O � R ? O � p may contain more nilpotent elements than O R , p R , � Dmitry Trushin () Differential schemes October, 2010 8 / 21

The structure sheaves � � ι R R ι ∗ � � X � X � � p � p Question Does O R coincide with O � R ? O � p may contain more nilpotent elements than O R , p R , � Fact If R is reduced. Then O R = O � R . Dmitry Trushin () Differential schemes October, 2010 8 / 21

Contents The differential spectrum of the ring of global sections 1 Differential integral dependence 2 Differential catenarity 3 Dmitry Trushin () Differential schemes October, 2010 9 / 21

Differential integral dependence What do we want? Dmitry Trushin () Differential schemes October, 2010 10 / 21

Differential integral dependence What do we want? Properties of the integral dependence in commutative case: (Integral dependence = ID) 1 Noether’s normalization ⇒ ID appears often 2 ID has simple geometric behavior 3 ID describes universally closed morphisms of affine schemes (complete affine varieties) Dmitry Trushin () Differential schemes October, 2010 10 / 21

Differential integral dependence What do we want? Properties of the integral dependence in commutative case: (Integral dependence = ID) 1 Noether’s normalization ⇒ ID appears often 2 ID has simple geometric behavior 3 ID describes universally closed morphisms of affine schemes (complete affine varieties) We are seeking universally closed morphisms of affine ∆-schemes Dmitry Trushin () Differential schemes October, 2010 10 / 21

Differential integral dependence What do we want? Properties of the integral dependence in commutative case: (Integral dependence = ID) 1 Noether’s normalization ⇒ ID appears often 2 ID has simple geometric behavior 3 ID describes universally closed morphisms of affine schemes (complete affine varieties) We are seeking universally closed morphisms of affine ∆-schemes From now all differential rings are Ritt algebras ( Q ⊆ R ) Dmitry Trushin () Differential schemes October, 2010 10 / 21

Valuation rings and integral dependence Let K be a field and A , B ⊆ K be local rings with maximal ideals m , n Dmitry Trushin () Differential schemes October, 2010 11 / 21

Valuation rings and integral dependence Let K be a field and A , B ⊆ K be local rings with maximal ideals m , n Definition B dominates A : A � B iff A ⊆ B and n ∩ A = m Dmitry Trushin () Differential schemes October, 2010 11 / 21

Valuation rings and integral dependence Let K be a field and A , B ⊆ K be local rings with maximal ideals m , n Definition B dominates A : A � B iff A ⊆ B and n ∩ A = m Fact (Valuation ring) Let A ⊆ K. The following condition are equivalent: ∀ x � = 0 either x ∈ A or x − 1 ∈ A (or both) A is a maximal element with respect to � Dmitry Trushin () Differential schemes October, 2010 11 / 21

Valuation rings and integral dependence Let K be a field and A , B ⊆ K be local rings with maximal ideals m , n Definition B dominates A : A � B iff A ⊆ B and n ∩ A = m Fact (Valuation ring) Let A ⊆ K. The following condition are equivalent: ∀ x � = 0 either x ∈ A or x − 1 ∈ A (or both) A is a maximal element with respect to � Let A ⊆ B be integral domains and K = Q( B ). Dmitry Trushin () Differential schemes October, 2010 11 / 21