In mathematics, a derivation of a commutative ring is called a locally nilpotent derivation (LND) if every element of is annihilated by some power of . One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.
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| - Locally nilpotent derivation (en)
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| - In mathematics, a derivation of a commutative ring is called a locally nilpotent derivation (LND) if every element of is annihilated by some power of . One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring. (en)
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| name
| - Bonnet's theorem (en)
- Principal ideal theorem (en)
- Freudenburg's theorem (en)
- Kaliman's theorem (en)
- Miyanishi's theorem (en)
- Rentschler's theorem (en)
- Zurkowski's theorem (en)
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| - In mathematics, a derivation of a commutative ring is called a locally nilpotent derivation (LND) if every element of is annihilated by some power of . One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring. Over a field of characteristic zero, to give a locally nilpotent derivation on the integral domain , finitely generated over the field, is equivalent to giving an action of the additive group to the affine variety . Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space. (en)
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| - Every fixed-point free action of on is conjugate to a translation. In other words, every such that the image of generates the unit ideal , admits a slice. This results answers one of the conjectures from Kraft's list.
Again, this result is not true for : e.g. consider the . The points and are in the same orbit of the corresponding -action if and only if ; hence the quotient is not even Hausdorff, let alone homeomorphic to . (en)
- Let . Then is faithfully flat over . Moreover, the ideal is principal in . (en)
- Assume that and is homogeneous relative to some positive grading of such that are homogeneous. Then for some homogeneous . Moreover, if are relatively prime, then are relatively prime as well. (en)
- The above necessary geometrical condition was later generalized by Freudenburg. To state his result, we need the following definition:
A corank of is a maximal number such that there exists a system of variables such that . Define as minus the corank of .
We have and if and only if in some coordinates, for some .
Theorem: If is triangulable, then any hypersurface contained in the fixed-point set of the corresponding -action is isomorphic to .
In particular, LND's of maximal rank cannot be triangulable. Such derivations do exist for : the first example is the -homogeneous derivation , and it can be easily generalized to any . (en)
- Every LND of can be conjugated to for some . This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions. (en)
- The kernel of every nontrivial LND of is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial -action on is isomorphic to .
In other words, for every there exist such that . In this case, is a Jacobian derivation: . (en)
- A quotient morphism of a -action is surjective. In other words, for every , the embedding induces a surjective morphism .
This is no longer true for , e.g. the image of a quotient map by a -action (en)
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