Surjectivity of quotient maps for algebraic (C,+)-actions and polynomial maps with contractible fibers

Bonnet, Philippe

doi:10.1007/s00031-002-0001-6

Cited by 7 publications

(23 citation statements)

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“…In the case k = C, assertion (1) was proved by Miyanishi [20] and (2) and (3) by Bonnet [3]. See [9] for the generalizations to all fields of characteristic zero.…”

Section: If a ∈ Klnd(b) Thenmentioning

confidence: 89%

“…This proves ( * * ). Now let k be any field of characteristic zero and consider B = k[X, Y, Z] = k [3] , A ∈ KLND(B) and Q : Spec B → Spec A. We leave it to the reader to show that there exist a subfield k 0 of k finitely generated over Q and an element…”

Section: ( * * ) If K /K Is An Extension Of Fields Of Characteristic mentioning

confidence: 99%

“…Indeed, consider B = k [3] , A ∈ KLND(B) and Q : [3] and A = k ⊗ k A; note that A ∈ KLND(B ) and let Q : Spec B → Spec A be the morphism determined by the inclusion A → B . Then in the diagram…”

Section: ( * * ) If K /K Is An Extension Of Fields Of Characteristic mentioning

confidence: 99%

“…Until the end of this section, let k be an algebraically closed field of characteristic zero and let B = k [3] . We propose the idea that the following elements of B are interesting:…”

Section: Definitionmentioning

confidence: 99%

“…We shall need the following facts: Proof of 2.12. Let k be a field of characteristic zero and B = k [3] . We say that "2.12 is valid over k" if for every A ∈ KLND(B) the set X A is constructible in Spec A, or equivalently, for every…”

Section: Lemma Ifmentioning

confidence: 99%

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On polynomials in three variables annihilated by two locally nilpotent derivations

Daigle

2007

Journal of Algebra

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Let B be a polynomial ring in three variables over an algebraically closed field k of characteristic zero. We are interested in irreducible polynomials f ∈ B satisfying the following condition: there exist nonzero locally nilpotent derivations D 1 , D 2 : B → B such that ker(D 1 ) = ker(D 2 ) and D 1 (f ) = 0 = D 2 (f ). The main result asserts that a nonconstant polynomial f ∈ B satisfies the above requirement if and only if its "generic fiber" k(f ) ⊗ k[f ] B is isomorphic, as an algebra over the field K = k(f ), to K[X, Y, Z]/ (XY − ϕ(Z)) for some nonconstant ϕ(Z) ∈ K [Z].

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“…In the case k = C, assertion (1) was proved by Miyanishi [20] and (2) and (3) by Bonnet [3]. See [9] for the generalizations to all fields of characteristic zero.…”

Section: If a ∈ Klnd(b) Thenmentioning

confidence: 89%

Section: ( * * ) If K /K Is An Extension Of Fields Of Characteristic mentioning

confidence: 99%

Section: ( * * ) If K /K Is An Extension Of Fields Of Characteristic mentioning

confidence: 99%

“…Until the end of this section, let k be an algebraically closed field of characteristic zero and let B = k [3] . We propose the idea that the following elements of B are interesting:…”

Section: Definitionmentioning

confidence: 99%

Section: Lemma Ifmentioning

confidence: 99%

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On polynomials in three variables annihilated by two locally nilpotent derivations

Daigle

2007

Journal of Algebra

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Triangulable locally nilpotent derivations in dimension three

Barkatou

Houari

Kahoui

2008

Journal of Pure and Applied Algebra

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Pure Subrings of Commutative Rings

Chakraborty¹,

Gurjar²,

Miyanishi³

2016

Nagoya Math. J.

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We study subalgebras A of affine or local algebras B such that A → B is a pure extension from algebraic and geometric viewpoints. §0. IntroductionLet ϕ : A → B be a homomorphism of commutative rings. We say that ϕ is a pure homomorphism, or pure embedding, or pure extension if the canonical homomorphism ϕ M : M ⊗ A A → M ⊗ A B is injective, for any A-module M . This condition implies that ϕ is injective. Therefore, identifying A with ϕ(A), we often say that A is a pure subring of B, or that B is a pure extension of A. The notion of purity was first raised by Warfield [15]. The associated morphism a ϕ : Spec B → Spec A is also called a pure morphism of affine schemes.A morphism of schemes p : Y → X is called a pure morphism if p is an affine morphism, and there exists an affine open covering {U i } i∈I of X, such that p| p −1 (U i ) : p −1 (U i ) → U i is a pure morphism for every i ∈ I. The formal properties of pure subalgebras and pure morphisms are summarized in Section 1.In this paper, we are interested in pure subalgebras of an affine or local domain. The underlying field k, unless otherwise specified, is always assumed to be algebraically closed and of characteristic 0. Sometimes we also assume k to be the field of complex numbers C to apply results of local analytic algebras. Let A be a pure k-subalgebra of an affine domain B. Then, by [5], A itself is affine. Furthermore, if B is normal then A is also normal, and the associated morphism a ϕ : Spec B → Spec A is surjective by Lemma 1.1 below.

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Surjectivity of quotient maps for algebraic (C,+)-actions and polynomial maps with contractible fibers

Abstract: Abstract. In this paper we establish two results concerning algebraic (C, +)-actions on C n .

Cited by 7 publications

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On polynomials in three variables annihilated by two locally nilpotent derivations

On polynomials in three variables annihilated by two locally nilpotent derivations

Triangulable locally nilpotent derivations in dimension three

Pure Subrings of Commutative Rings

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