Algebraic Theory of Locally Nilpotent Derivations by Gene Freudenburg

By Gene Freudenburg

This booklet explores the speculation and alertness of in the community nilpotent derivations, that is an issue of growing to be curiosity and value not just between these in commutative algebra and algebraic geometry, but in addition in fields reminiscent of Lie algebras and differential equations. the writer offers a unified remedy of the topic, starting with sixteen First ideas on which the complete thought is predicated. those are used to set up classical effects, similar to Rentschler's Theorem for the airplane, correct as much as the newest effects, equivalent to Makar-Limanov's Theorem for in the neighborhood nilpotent derivations of polynomial jewelry. issues of distinctive curiosity comprise: growth within the size 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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Proof. That (1) implies (2) follows immediately from part (d) of Princ. 11, together with Princ. 9. , bn ] d for some bi ∈ B. Since dt (bi ) ∈ K[t] for each i, there exists s ∈ S so that d d s dt (B) ⊂ B. Since s ∈ K, s dt is locally nilpotent. If D denotes the restriction d of s dt to B, it follows that D is also locally nilpotent, and ker D = B ∩ d ) = B ∩ K = A. ⊓ ⊔ ker (s dt Principle 14. (Compare to Princ. II of [97]) Suppose B is the graded ring B = ⊕i∈Z Bi , and let D ∈ LND(B) be given. Suppose that, for integers m ≤ n, D admits a decomposition D = m≤i≤n Di , where each Di ∈ Derk (B) is homogeneous of degree i relative to this grading, and where Dm = 0 and Dn = 0.

Conversely, suppose P (t) ∈ ker ( dt ). If deg P ≥ d 1, then since this kernel is algebraically closed, it would follow that t ∈ ker ( dt ), d a contradiction. Therefore, ker ( dt ) = A. For (c), let D ∈ LNDA (B) be given, D = 0. By Prop. 8(c), for any d d p(t) ∈ A[t], D(p(t)) = p′ (t)Dt. Consequently, D = Dt dt . Since both D and dt are locally nilpotent, Princ. 7 implies that Dt ∈ A. Therefore, LNDA (A[t]) ⊆ d A · dt . The reverse inclusion is implied by Princ. 7. ⊓ ⊔ Principle 9. Let S ⊂ B − {0} be a multiplicatively closed set, and let D ∈ Derk (B) be given.

Let A = ker D. If r is a local slice of D and f = Dr, then B ⊂ Bf = Af [r]. We therefore have frac(B) ⊂ frac(A)(r) ⊂ frac(B), which implies frac(B) = frac(A)(r). Now suppose δg = 0 for g ∈ frac(B), and write g = P (r) for the rational function P having coefficients in frac(A). Then 0 = P ′ (r)δr, and since δr = 0, P ′ (r) = 0. It follows that g = P (r) ∈ frac(A), which shows ker δ ⊂ frac(ker D). The reverse containment is obvious. 24. k S = 1 and LND(S) = {0}, then S = K [1] , where K is a field algebraic over k.

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