Locally nilpotent derivation

In mathematics, a derivation formula_1 of a commutative ring formula_2 is called a locally nilpotent derivation (LND) if every element of formula_2 is annihilated by some power of formula_1. 
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 formula_5 of characteristic zero, to give a locally nilpotent derivation on the integral domain formula_2, finitely generated over the field, is equivalent to giving an action of the additive group formula_7 to the affine variety formula_8. Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.
Definition.
Let formula_2 be a ring. Recall that a derivation of formula_2 is a map formula_11 satisfying the Leibniz rule formula_12 for any formula_13. If formula_2 is an algebra over a field formula_5, we additionally require formula_1 to be formula_5-linear, so formula_18.
A derivation formula_1 is called a "locally nilpotent derivation" (LND) if for every formula_20, there exists a positive integer formula_21 such that formula_22.
If formula_2 is graded, we say that a locally nilpotent derivation formula_1 is "homogeneous" (of degree formula_25) if formula_26 for every formula_27.
The set of locally nilpotent derivations of a ring formula_2 is denoted by formula_29. Note that this set has no obvious structure: it is neither closed under addition (e.g. if formula_30, formula_31 then formula_32 but formula_33, so formula_34) nor under multiplication by elements of formula_2 (e.g. formula_36, but formula_37). However, if formula_38 then formula_39 implies formula_40 and if formula_41, formula_42 then formula_43.
Relation to formula_44-actions.
Let formula_2 be an algebra over a field formula_5 of characteristic zero (e.g. formula_47). Then there is a one-to-one correspondence between the locally nilpotent formula_5-derivations on formula_2 and the actions of the additive group formula_44 of formula_5 on the affine variety formula_52, as follows. 
A formula_44-action on formula_52 corresponds to an formula_5-algebra homomorphism formula_56. Any such formula_57 determines a locally nilpotent derivation formula_1 of formula_2 by taking its derivative at zero, namely formula_60 where formula_61 denotes the evaluation at formula_62. 
Conversely, any locally nilpotent derivation formula_1 determines a homomorphism formula_56 by formula_65
It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if formula_66 and formula_41 then formula_68 and formula_69
The kernel algorithm.
The algebra formula_70 consists of the invariants of the corresponding formula_44-action. It is algebraically and factorially closed in formula_2. A special case of Hilbert's 14th problem asks whether formula_70 is finitely generated, or, if formula_74, whether the quotient formula_75 is affine. By Zariski's finiteness theorem, it is true if formula_76. On the other hand, this question is highly nontrivial even for formula_77, formula_78. For formula_79 the answer, in general, is negative. The case formula_80 is open.
However, in practice it often happens that formula_81 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 (by "linear" we mean homogeneous of degree zero with respect to the standard grading).
Assume formula_70 is finitely generated. If formula_83 is a finitely generated algebra over a field of characteristic zero, then formula_81 can be computed using van den Essen's algorithm, as follows. Choose a "local slice", i.e. an element formula_85 and put formula_86. Let formula_87 be the "Dixmier map" given by formula_88. Now for every formula_89, chose a minimal integer formula_90 such that formula_91, put formula_92, and define inductively formula_93 to be the subring of formula_2 generated by formula_95. By induction, one proves that formula_96 are finitely generated and if formula_97 then formula_98, so formula_99 for some formula_100. Finding the generators of each formula_93 and checking whether formula_97 is a standard computation using Gröbner bases.
Slice theorem.
Assume that formula_103 admits a "slice", i.e. formula_104 such that formula_105. The "slice theorem" asserts that formula_2 is a polynomial algebra formula_107 and formula_108.
For any local slice formula_109 we can apply the slice theorem to the localization formula_110, and thus obtain that formula_2 is "locally" a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient formula_112 is affine (e.g. when formula_76 by the Zariski theorem), then it has a Zariski-open subset formula_114 such that formula_115 is isomorphic over formula_114 to formula_117, where formula_44 acts by translation on the second factor.
However, in general it is not true that formula_119 is locally trivial. For example, let formula_120. Then formula_81 is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.
If formula_122 then formula_123 is a curve. To describe the formula_44-action, it is important to understand the geometry formula_125. Assume further that formula_47 and that formula_127 is smooth and contractible (in which case formula_128 is smooth and contractible as well) and choose formula_125 to be minimal (with respect to inclusion). Then Kaliman proved that each irreducible component of formula_125 is a "polynomial curve", i.e. its normalization is isomorphic to formula_131. The curve formula_125 for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in formula_133, so formula_125 may not be irreducible. However, it is conjectured that formula_125 is always contractible.
Examples.
The standard coordinate derivations formula_136 of a polynomial algebra formula_137 are locally nilpotent. The corresponding formula_138-actions are translations: formula_139, formula_140 for formula_141.
Let formula_142, formula_143, and let formula_1 be the Jacobian derivation formula_145. Then formula_146 and formula_147 (see below); that is, formula_1 annihilates no variable. The fixed point set of the corresponding formula_44-action equals formula_150.
Consider formula_151. The locally nilpotent derivation formula_152 of its coordinate ring corresponds to a natural action of formula_138 on formula_154 via right multiplication of upper triangular matrices. This action gives a nontrivial formula_138-bundle over formula_156. However, if formula_47 then this bundle is trivial in the smooth category
LND's of the polynomial algebra.
Let formula_5 be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case formula_47
) and let formula_160 be a polynomial algebra.
formula_161 (formula_44-actions on an affine plane).
Every LND of formula_163 can be conjugated to formula_164 for some formula_165. This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.
formula_166 (formula_44-actions on an affine 3-space).
The kernel of every nontrivial LND of formula_168 is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial formula_44-action on formula_170 is isomorphic to formula_171.
In other words, for every formula_172 there exist formula_173 such that formula_174 (but, in contrast to the case formula_161, formula_2 is not necessarily a polynomial ring over formula_70). In this case, formula_1 is a Jacobian derivation: formula_145.
Assume that formula_166 and formula_41 is homogeneous relative to some positive grading of formula_2 such that formula_183 are homogeneous. Then formula_184 for some homogeneous formula_185. Moreover, if formula_186 are relatively prime, then formula_187 are relatively prime as well.
A quotient morphism formula_188 of a formula_44-action is surjective. In other words, for every formula_172, the embedding formula_191 induces a surjective morphism formula_192.
This is no longer true for formula_193, e.g. the image of a quotient map formula_194 by a formula_44-action formula_196 (which corresponds to a LND given by formula_197 equals formula_198.
Every fixed-point free action of formula_44 on formula_170 is conjugate to a translation. In other words, every formula_201 such that the image of formula_1 generates the unit ideal (or, equivalently, formula_1 defines a nowhere vanishing vector field), admits a slice. This results answers one of the conjectures from Kraft's list.
Again, this result is not true for formula_193: e.g. consider the formula_205. The points formula_206 and formula_207 are in the same orbit of the corresponding formula_138-action if and only if formula_209; hence the (topological) quotient is not even Hausdorff, let alone homeomorphic to formula_210.
Let formula_103. Then formula_2 is faithfully flat over formula_81. Moreover, the ideal formula_214 is principal in formula_2.
Triangular derivations.
Let formula_216 be any system of variables of formula_2; that is, formula_218. A derivation of formula_2 is called "triangular" with respect to this system of variables, if formula_220 and formula_221 for formula_222. 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 formula_223 by Rentschler's theorem above, but it is not true for formula_224.
The derivation of formula_225 given by formula_226 is not triangulable. Indeed, the fixed-point set of the corresponding formula_44-action is a quadric cone formula_228, while by the result of Popov, a fixed point set of a triangulable formula_44-action is isomorphic to formula_230 for some affine variety formula_231; and thus cannot have an isolated singularity.
The above necessary geometrical condition was later generalized by Freudenburg. To state his result, we need the following definition:
A corank of formula_41 is a maximal number formula_233 such that there exists a system of variables formula_234 such that formula_235. Define formula_236 as formula_21 minus the corank of formula_1.
We have formula_239 and formula_240 if and only if in some coordinates, formula_241 for some formula_242.
Theorem: If formula_41 is triangulable, then any hypersurface contained in the fixed-point set of the corresponding formula_44-action is isomorphic to formula_245.
In particular, LND's of maximal rank formula_21 cannot be triangulable. Such derivations do exist for formula_224: the first example is the "(2,5)-homogeneous derivation" (see above), and it can be easily generalized to any formula_224.
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 formula_44-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 formula_210, it is not.