Gene Freudenburg's Algebraic Theory of Locally Nilpotent Derivations PDF

By Gene Freudenburg

ISBN-10: 3540295216

ISBN-13: 9783540295211

This booklet explores the idea and alertness of in the community nilpotent derivations, that is a subject matter of growing to be curiosity and value not just between these in commutative algebra and algebraic geometry, but additionally in fields comparable to Lie algebras and differential equations. the writer offers a unified remedy of the topic, starting with sixteen First ideas on which the whole idea relies. those are used to set up classical effects, comparable to Rentschler's Theorem for the aircraft, correct as much as the latest effects, comparable to Makar-Limanov's Theorem for in the neighborhood nilpotent derivations of polynomial jewelry. themes of specified curiosity contain: growth within the measurement 3 case, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation challenge and the Embedding challenge. The reader also will discover a wealth of pertinent examples and open difficulties and an up to date source for study.

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Example text

The defect of D relative to deg is then defined to be def(B), and is denoted by def(D). The reason for defining the defect of a derivation is that, if B = ∪i∈Z Bi is the filtration of B induced by the degree function deg, then D (nonzero) respects this filtration if and only if def(D) is finite. The defect has the following basic properties. 13. Let a, b ∈ B, and let S be a non-empty subset of B. (a) def(S) = −∞ if and only if S ⊂ ker D. (b) def(D) = −∞ if and only if D = 0. 4 The Defect of a Derivation (c) (d) (e) (f ) (g) (h) 41 D is homogeneous relative to deg if and only if def is constant on B − 0.

Suppose D ∈ LND(B) is irreducible, and set A = ker D. By Prop. 4, D has a minimal local slice y. Suppose Dy ∈ B ∗ . Then there exists irreducible x ∈ B dividing Dy. Since A is factorially closed, x ∈ A. ¯ = D (mod x) on B ¯ = B (mod x). Since D is irreducible, D ¯ = 0. In Let D ¯ = 1. By Cor. 24, it follows that B ¯ = k [1] and ker D ¯ = k. k B ¯ y = 0, we have that y ∈ xB + k. Write y = xz + λ for some z ∈ B Since D¯ and λ ∈ k. Then y − λ = xz is irreducible, by Prop. 5. But this implies z ∈ B ∗ ⊂ A, and thus y = xz + λ ∈ A, a contradiction.

Suppose D ∈ LND(B) is irreducible, and set A = ker D. By Prop. 4, D has a minimal local slice y. Suppose Dy ∈ B ∗ . Then there exists irreducible x ∈ B dividing Dy. Since A is factorially closed, x ∈ A. ¯ = D (mod x) on B ¯ = B (mod x). Since D is irreducible, D ¯ = 0. In Let D ¯ = 1. By Cor. 24, it follows that B ¯ = k [1] and ker D ¯ = k. k B ¯ y = 0, we have that y ∈ xB + k. Write y = xz + λ for some z ∈ B Since D¯ and λ ∈ k. Then y − λ = xz is irreducible, by Prop. 5. But this implies z ∈ B ∗ ⊂ A, and thus y = xz + λ ∈ A, a contradiction.

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Algebraic Theory of Locally Nilpotent Derivations by Gene Freudenburg


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