Locally nilpotent derivation


In mathematics, a derivation of a commutative ring is called a locally nilpotent derivation 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.

Definition

Let be a ring. Recall that a derivation of is a map satisfying the Leibniz rule for any. If is an algebra over a field, we additionally require to be -linear, so.
A derivation is called a locally nilpotent derivation if for every, there exists a positive integer such that.
If is graded, we say that a locally nilpotent derivation is homogeneous if for every.
The set of locally nilpotent derivations of a ring is denoted by. Note that this set has no obvious structure: it is neither closed under addition nor under multiplication by elements of . However, if then implies and if, then.

Relation to \mathbb{G}_{a}-actions

Let be an algebra over a field of characteristic zero. Then there is a one-to-one corresponence between the locally nilpotent -derviations on and the actions of the additive group of on the affine variety, as follows.
A -action on corresponds to an -algebra homomorphism. Any such determines a locally nilpotent derivation of by taking its derivative at zero, namely where denotes the evaluation at.
Conversely, any locally nilpotent derivation determines a homomorphism by
It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if and then and

The kernel algorithm

The algebra consists of the invariants of the corresponding -action. It is algebraically and factorially closed in. A special case of Hilbert's 14th problem asks whether is finitely generated, or, if, whether the quotient is affine. By Zariski's finiteness theorem, it is true if. On the other hand, this question is highly nontrivial even for,. For the answer, in general, is negative. The case is open.
However, in practice it often happens that is known to be finitely generated: notably, by the Maurer-Weitzenböck theorem, it is the case for linear LND's of the polynomial algebra over a field of characteristic zero.
Assume is finitely generated. If is a finitely generated algebra over a field of characteristic zero, then can be computed using van den Essen's algorithm, as follows. Choose a local slice, i.e. an element and put. Let be the Dixmier map given by. Now for every, chose a minimal integer such that, put, and define inductively to be the subring of generated by. By induction, one proves that are finitely generated and if then, so for some. Finding the generators of each and checking whether is a standard computation using Gröbner bases.

Slice theorem

Assume that admits a slice, i.e. such that. The slice theorem asserts that is a polynomial algebra and.
For any local slice we can apply the slice theorem to the localization, and thus obtain that is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient is affine, then it has a Zariski-open subset such that is isomorphic over to, where acts by translation on the second factor.
However, in general it is not true that is locally trivial. For example, let. Then is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.
If then is a curve. To describe the -action, it is important to understand the geometry. Assume further that and that is smooth and contractible and choose to be minimal. Then Kaliman proved that each irreducible component of is a polynomial curve, i.e. its normalization is isomorphic to. The curve for the action given by Freudenburg's -derivation is a union of two lines in, so may not be irreducible. However, it is conjectured that is always contractible.

Examples

;Example 1
The standard coordinate derivations of a polynomial algebra are locally nilpotent. The corresponding -actions are translations:, for.
;Example 2
Let,, and let be the Jacobian derivation. Then and ; that is, annihilates no variable. The fixed point set of the corresponding -action equals.
;Example 3
Consider. The locally nilpotent derivation of its coordinate ring corresponds to a natural action of on via right multiplication of upper triangular matrices. This action gives a nontrivial -bundle over. However, if then this bundle is trivial in the smooth category

LND's of the polynomial algebra

Let be a field of characteristic zero and let be a polynomial algebra.

n=2 (\mathbb{G}_{a}-actions on an affine plane)

;Rentschler's theorem
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.

n=3 (\mathbb{G}_{a}-actions on an affine 3-space)

;Miyanishi's theorem
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:.
;Zurkowski's theorem
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.
;Bonnet's theorem
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 , 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.
;Principal ideal theorem
Let. Then is faithfully flat over. Moreover, the ideal is principal in.

Triangular derivations

Let be any system of variables of ; that is,. A derivation of is called triangular with respect to this system of variables, if and for. A derivation is called triangulable if it is conjugate to a triangular one, or, equivalently, if it is triangular with respect to some system of variables. Every triangular derivation is locally nilpotent. The converse is true for by Rentschler's theorem above, but it is not true for.
;Bass's example
The derivation of given by is not triangulable. Indeed, the fixed-point set of the corresponding -action is a quadric cone, while by the result of Popov, a fixed point set of a triangulable -action is isomorphic to for some affine variety ; and thus cannot have an isolated singularity.
;Freudenburg's theorem
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.

Makar-Limanov invariant

The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all -actions, is called "Makar-Limanov invariant" and is an important algebraic invariant of an affine variety. For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to, it is not.