Spectra of rings differentially finitely generated over a subring
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1 Spectra of rings differentially finitely generated over a subring Dm. Trushin Department of Mechanics and Mathematics Moscow State University 15 April 2007 Dm. Trushin () -spectra of rings April 15, / 15
2 Preliminaries Ring = an associative, a commutative ring with an identity element (A, ) is a differential ring, where = { 1,..., n }, [ i, j ] = 0. Dm. Trushin () -spectra of rings April 15, / 15
3 Preliminaries Ring = an associative, a commutative ring with an identity element (A, ) is a differential ring, where = { 1,..., n }, [ i, j ] = 0. The main objects are the Ritt and Keigher algebras. Dm. Trushin () -spectra of rings April 15, / 15
4 Preliminaries Ring = an associative, a commutative ring with an identity element (A, ) is a differential ring, where = { 1,..., n }, [ i, j ] = 0. The main objects are the Ritt and Keigher algebras. = differential. B -FG over A = B is differentially finitely generated over A. Dm. Trushin () -spectra of rings April 15, / 15
5 Preliminaries Ring = an associative, a commutative ring with an identity element (A, ) is a differential ring, where = { 1,..., n }, [ i, j ] = 0. The main objects are the Ritt and Keigher algebras. = differential. B -FG over A = B is differentially finitely generated over A. Spec A (Max A) is the prime (maximal) spectrum of A. Spec A (Max A) is prime (maximal) -spectrum of A. Dm. Trushin () -spectra of rings April 15, / 15
6 Preliminaries Ring = an associative, a commutative ring with an identity element (A, ) is a differential ring, where = { 1,..., n }, [ i, j ] = 0. The main objects are the Ritt and Keigher algebras. = differential. B -FG over A = B is differentially finitely generated over A. Spec A (Max A) is the prime (maximal) spectrum of A. Spec A (Max A) is prime (maximal) -spectrum of A. The localization by {s n } n=0 is denoted by A s. Dm. Trushin () -spectra of rings April 15, / 15
7 Preliminaries Ring = an associative, a commutative ring with an identity element (A, ) is a differential ring, where = { 1,..., n }, [ i, j ] = 0. The main objects are the Ritt and Keigher algebras. = differential. B -FG over A = B is differentially finitely generated over A. Spec A (Max A) is the prime (maximal) spectrum of A. Spec A (Max A) is prime (maximal) -spectrum of A. The localization by {s n } n=0 is denoted by A s. Let f : A B be a homomorphism. A contraction of ideal is denoted by b c = f (b). An extension of ideal is denoted by a e. Dm. Trushin () -spectra of rings April 15, / 15
8 Relation between prime spectrum and prime -spectrum Let f : A B be a -homomorphism of Keigher algebras. Let f : Spec B Spec A; f : Spec B Spec A. Dm. Trushin () -spectra of rings April 15, / 15
9 Relation between prime spectrum and prime -spectrum Let f : A B be a -homomorphism of Keigher algebras. Let f : Spec B Spec A; f : Spec B Spec A. Lemma (fibre lemma) Let p Spec A. Then (f ) 1 (p) (f ) 1 (p). Dm. Trushin () -spectra of rings April 15, / 15
10 Relation between prime spectrum and prime -spectrum Let f : A B be a -homomorphism of Keigher algebras. Let f : Spec B Spec A; f : Spec B Spec A. Lemma (fibre lemma) Let p Spec A. Then (f ) 1 (p) (f ) 1 (p). Corollary (on -surjectivity) f is surjective = f is surjective. Dm. Trushin () -spectra of rings April 15, / 15
11 Technical definitions Let f : A B be a homomorphism. Definition f has the going-up property (GUP), if for any p 1... p m p m+1... p n q 1... q m p i Spec f (A) q i Spec B q i f (A) = p i there exists q 1... q m q m+1... q n q i f (A) = p i Dm. Trushin () -spectra of rings April 15, / 15
12 Technical definitions Let f : A B be a homomorphism. Definition f has the going-down property (GDP), if for any p 1... p m p m+1... p n q 1... q m p i Spec f (A) q i Spec B q i f (A) = p i there exists q 1... q m q m+1... q n q i f (A) = p i Dm. Trushin () -spectra of rings April 15, / 15
13 Pairs of properties Let f : A B be a -homomorphism of Keigher algebras. Dm. Trushin () -spectra of rings April 15, / 15
14 Pairs of properties Let f : A B be a -homomorphism of Keigher algebras. Consider pairs of properties A1 and A2 such that A1 characterizes f as a homomorphism, A2 characterizes f as a -homomorphism and A1 A2. Dm. Trushin () -spectra of rings April 15, / 15
15 Pairs of properties Let f : A B be a -homomorphism of Keigher algebras. Consider pairs of properties A1 and A2 such that A1 characterizes f as a homomorphism, A2 characterizes f as a -homomorphism and A1 A2. Lemma (f ) 1 (p) (f ) 1 (p), where p Spec A. f surjective = f surjective. f has GUP = f has GUP for -ideals. f has GDP = f has GDP for -ideals. Dm. Trushin () -spectra of rings April 15, / 15
16 Application of pairs of properties Let s generalize well-known going-up and going-down Cohen-Seidenberg theorems for Keigher algebras. Dm. Trushin () -spectra of rings April 15, / 15
17 Application of pairs of properties Let s generalize well-known going-up and going-down Cohen-Seidenberg theorems for Keigher algebras. Theorem ( Going-up theorem ) Let f : A B be a -homomorphism of Keigher algebras and B be integral over A. Then f has GUP for -ideals. Dm. Trushin () -spectra of rings April 15, / 15
18 Application of pairs of properties Let s generalize well-known going-up and going-down Cohen-Seidenberg theorems for Keigher algebras. Theorem ( Going-up theorem ) Let f : A B be a -homomorphism of Keigher algebras and B be integral over A. Then f has GUP for -ideals. Theorem ( Going-down theorem ) Let f : A B be a -homomorphism of Keigher algebras. Let also 1 B be integral domain; 2 f (A) be integrally closed; 3 B be integral over A. Then f has GDP for -ideals. Dm. Trushin () -spectra of rings April 15, / 15
19 In case of characteristic zero Lemma Let A B be Ritt algebras. Let 1 B = A{x 1,..., x n } be -FG over A 2 B be integral domain Then s B such that B s = A[x 1,..., x n ][y α ] α Λ, where {y α } are algebraic independent over A[x 1,..., x n ] and card Λ card N. Dm. Trushin () -spectra of rings April 15, / 15
20 In case of characteristic zero Lemma Let A B be Ritt algebras. Let 1 B = A{x 1,..., x n } be -FG over A 2 B be integral domain Then s B such that B s = A[x 1,..., x n ][y α ] α Λ, where {y α } are algebraic independent over A[x 1,..., x n ] and card Λ card N. Theorem (main theorem) Let A B be Ritt algebras. Let also 1 B = A{x 1,..., x n } be -FG over A; 2 B be integral domain. Then s A such that map Spec B s Spec A s is surjective. Dm. Trushin () -spectra of rings April 15, / 15
21 Corollary of main theorem Let B be -ring. Let also B be -FG over a -field K of characteristic zero; B be a simple -ring. Dm. Trushin () -spectra of rings April 15, / 15
22 Corollary of main theorem Let B be -ring. Let also B be -FG over a -field K of characteristic zero; B be a simple -ring. Corollary Let A B be -subalgebra over K. Then s A such that A s is a simple -algebra. Dm. Trushin () -spectra of rings April 15, / 15
23 Corollary of main theorem Let B be -ring. Let also B be -FG over a -field K of characteristic zero; B be a simple -ring. Corollary Let A B be -subalgebra over K. Then s A such that A s is a simple -algebra. Let s apply this corollary to the case A = K{y}, where y B. Dm. Trushin () -spectra of rings April 15, / 15
24 Corollary of main theorem Let B be -ring. Let also B be -FG over a -field K of characteristic zero; B be a simple -ring. Corollary Let A B be -subalgebra over K. Then s A such that A s is a simple -algebra. Let s apply this corollary to the case A = K{y}, where y B. Corollary Any y B is -algebraically depended over K. Dm. Trushin () -spectra of rings April 15, / 15
25 Corollary of main theorem Let B be -ring. Let also B be -FG over a -field K of characteristic zero; B be a simple -ring. Corollary Let A B be -subalgebra over K. Then s A such that A s is a simple -algebra. Let s apply this corollary to the case A = K{y}, where y B. Corollary Any y B is -algebraically depended over K. Let F be a field of fractions of B, and C K, C B, C F be constant rings, resp. Dm. Trushin () -spectra of rings April 15, / 15
26 Corollary of main theorem Let B be -ring. Let also B be -FG over a -field K of characteristic zero; B be a simple -ring. Corollary Let A B be -subalgebra over K. Then s A such that A s is a simple -algebra. Let s apply this corollary to the case A = K{y}, where y B. Corollary Any y B is -algebraically depended over K. Let F be a field of fractions of B, and C K, C B, C F be constant rings, resp. Corollary Then C B = C F, and C B /C K is algebraic field extension. Dm. Trushin () -spectra of rings April 15, / 15
27 In Noetherian case Let f : A B be a -homomorphism of Ritt algebras. And B is -FG over A. Suppose, additionally, that all spectra are Noetherian. Denote X = Spec A, Y = Spec B. Consider the properties of the map f : Y X. Dm. Trushin () -spectra of rings April 15, / 15
28 In Noetherian case Let f : A B be a -homomorphism of Ritt algebras. And B is -FG over A. Suppose, additionally, that all spectra are Noetherian. Denote X = Spec A, Y = Spec B. Consider the properties of the map f : Y X. Lemma Let E Y be constructible. Then f (E) is constructible. Dm. Trushin () -spectra of rings April 15, / 15
29 In Noetherian case Let f : A B be a -homomorphism of Ritt algebras. And B is -FG over A. Suppose, additionally, that all spectra are Noetherian. Denote X = Spec A, Y = Spec B. Consider the properties of the map f : Y X. Lemma Let E Y be constructible. Then f (E) is constructible. Lemma The map f has GUP for -deals = f is closed map. The map f has GDP for -deals = f is open map. Dm. Trushin () -spectra of rings April 15, / 15
30 Open maps Theorem Let A B be Ritt algebras. Let 1 B = A{x 1,..., x n } be -FG over A; 2 B be integral domain. Then s B such that the map Spec B s Spec A has GDP. Dm. Trushin () -spectra of rings April 15, / 15
31 Open maps Theorem Let A B be Ritt algebras. Let 1 B = A{x 1,..., x n } be -FG over A; 2 B be integral domain. Then s B such that the map Spec B s Spec A has GDP. Corollary Let A B be Ritt algebras. Let 1 B = A{x 1,..., x n } be -FG over A; 2 Spec A be Noetherian; 3 B be integral domain. Then s B such that the map Spec B s Spec A is open. Dm. Trushin () -spectra of rings April 15, / 15
32 Good very dense subset of prime -spectrum Let A be -FG over a field K with char K = 0. Dm. Trushin () -spectra of rings April 15, / 15
33 Good very dense subset of prime -spectrum Let A be -FG over a field K with char K = 0. Definition X := {p Spec A s A : (A/p) s is a simple -ring} Dm. Trushin () -spectra of rings April 15, / 15
34 Good very dense subset of prime -spectrum Let A be -FG over a field K with char K = 0. Definition X := {p Spec A s A : (A/p) s is a simple -ring} Lemma The set X is very dense subset of Spec A. Dm. Trushin () -spectra of rings April 15, / 15
35 Good very dense subset of prime -spectrum Let A be -FG over a field K with char K = 0. Definition Lemma X := {p Spec A s A : (A/p) s is a simple -ring} The set X is very dense subset of Spec A. Let f : A B be a -homomorphism of -FG algebras over the field K. Denote Y := {p Spec B s B : (B/p) s is a simple -ring}. Dm. Trushin () -spectra of rings April 15, / 15
36 Good very dense subset of prime -spectrum Let A be -FG over a field K with char K = 0. Definition Lemma X := {p Spec A s A : (A/p) s is a simple -ring} The set X is very dense subset of Spec A. Let f : A B be a -homomorphism of -FG algebras over the field K. Denote Y := {p Spec B s B : (B/p) s is a simple -ring}. Lemma The map f : Y X is well defined. Dm. Trushin () -spectra of rings April 15, / 15
37 Simple properties Lemma The set X is the smallest subset of Spec A such that B is -FG over field K, and f : A B is a -homomorphism, we have f (Max B) X. Dm. Trushin () -spectra of rings April 15, / 15
38 Simple properties Lemma The set X is the smallest subset of Spec A such that B is -FG over field K, and f : A B is a -homomorphism, we have f (Max B) X. Lemma Let f : A B be a -homomorphism. Then 1 f has GUP for -ideals = f is closed map from Y to X ; 2 f has GDP for -ideals = f is open map from Y to X ; 3 E is constructible = f (E) is constructible. Dm. Trushin () -spectra of rings April 15, / 15
39 Remark Consider any field F of characteristic zero with the property: For any algebra B such that } B is -FG over F = B = F. B is a simple -ring Dm. Trushin () -spectra of rings April 15, / 15
40 Remark Consider any field F of characteristic zero with the property: For any algebra B such that } B is -FG over F = B = F. B is a simple -ring Example Universal field extension U of any -field K (Kolchin). Dm. Trushin () -spectra of rings April 15, / 15
41 Remark Consider any field F of characteristic zero with the property: For any algebra B such that } B is -FG over F = B = F. B is a simple -ring Example Universal field extension U of any -field K (Kolchin). Let A = F{y 1,..., y n }/b. Dm. Trushin () -spectra of rings April 15, / 15
42 Remark Consider any field F of characteristic zero with the property: For any algebra B such that } B is -FG over F = B = F. B is a simple -ring Example Universal field extension U of any -field K (Kolchin). Let A = F{y 1,..., y n }/b. The following sets coincide: { x F n f b f (x) = 0 }; Max A = { p Spec A A/p is a simple -ring }; { p Spec A s A : (A/p) s is a simple -ring }. Dm. Trushin () -spectra of rings April 15, / 15
43 Reference M. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra. Addison-Wesley, Reading, Mass, I. Kaplansky. An introduction to Differential Algebra. Hermann, Paris, E. R. Kolchin. Differential Algebra and Algebraic Groups. Academic Press, New York, J. F. Ritt. Differential Algebra. volume 33 of American Mathematical Society Colloquium Publication. Lectures notes in computer science American Mathematical Society, New York, D. V. Trushin. The ideal of separants in the ring of differential polynomials. Fundamental and applied mathematics, vol. 13, num. 1, 2007, pp Dm. Trushin () -spectra of rings April 15, / 15
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