Locally nilpotent derivation explained

\partial of a commutative ring A is called a locally nilpotent derivation (LND) if every element of A is annihilated by some power of \partial .

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.^[1]

Over a field

k of characteristic zero, to give a locally nilpotent derivation on the integral domain A , finitely generated over the field, is equivalent to giving an action of the additive group (k, + ) to the affine variety X = \operatorname{Spec}(A) . Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.^[2]

Definition

Let

A be a ring. Recall that a derivation of A is a map \partial\colon A\to A satisfying the Leibniz rule \partial (ab) = (\partial a)b + a(\partial b) for any a,b\in A . If A is an algebra over a field k , we additionally require \partial to be k -linear, so k\subseteq \ker \partial .

A derivation

\partial is called a locally nilpotent derivation (LND) if for every a \in A , there exists a positive integer n such that \partial ^{n} (a) =0 .

If

A is graded, we say that a locally nilpotent derivation \partial is homogeneous (of degree d ) if \deg \partial a = \deg a + d for every a\in A .

The set of locally nilpotent derivations of a ring

A is denoted by \operatorname{LND}(A) . Note that this set has no obvious structure: it is neither closed under addition (e.g. if \partial _{1} = y\tfrac{\partial}{\partial x} , \partial _{2} = x\tfrac{\partial}{\partial y} then \partial _{1} ,\partial _{2} \in \operatorname{LND}(k[x,y]) but (\partial _{1} + \partial _{2} ) ^{2} (x) = x , so \partial _{1} + \partial _{2} \not\in \operatorname{LND}(k[x,y]) ) nor under multiplication by elements of A (e.g. \tfrac{\partial}{\partial x}\in \operatorname{LND}(k[x]) , but x\tfrac{\partial}{\partial x}\not\in\operatorname{LND}(k[x]) ). However, if [\partial _{1} ,\partial _{2} ] =0 then \partial _{1} ,\partial _{2} \in \operatorname{LND}(A) implies \partial _{1} + \partial _{2} \in \operatorname{LND}(A) and if \partial\in \operatorname{LND}(A) , h\in\ker\partial then h\partial\in \operatorname{LND}(A) .

Relation to -actions

Let

A be an algebra over a field k of characteristic zero (e.g. k = C ). Then there is a one-to-one correspondence between the locally nilpotent k -derivations on A and the actions of the additive group G _{a} of k on the affine variety \operatorname{Spec} A , as follows.^[3] A G _{a} -action on \operatorname{Spec} A corresponds to a k -algebra homomorphism \rho\colon A\to A[t] . Any such \rho determines a locally nilpotent derivation \partial of A by taking its derivative at zero, namely \partial = \epsilon \circ \tfrac{d}{dt}\circ \rho, where \epsilon denotes the evaluation at t =0 . Conversely, any locally nilpotent derivation \partial determines a homomorphism \rho\colon A\to A[t] by \rho = \exp (t\partial) = \sum ^{inf ty} _{n =0} (t ^{n})/(n!) \partial ^{n} .

It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if

\alpha\in \operatorname{Aut} A and \partial\in \operatorname{LND}(A) then \alpha\circ\partial \circ \alpha ^{-1} \in \operatorname{LND}(A) and \exp(t ⋅ \alpha\circ\partial \circ \alpha ^{-1} ) = \alpha \circ \exp(t\partial)\circ \alpha ^{-1}

The kernel algorithm

The algebra

\ker \partial consists of the invariants of the corresponding G _{a} -action. It is algebraically and factorially closed in A . A special case of Hilbert's 14th problem asks whether \ker \partial is finitely generated, or, if A = k[X] , whether the quotient X// G _{a} is affine. By Zariski's finiteness theorem,^[4] it is true if \dim X\leq 3 . On the other hand, this question is highly nontrivial even for X = C ^{n} , n\geq 4 . For n\geq 5 the answer, in general, is negative.^[5] The case n =4 is open.

However, in practice it often happens that

\ker\partial is known to be finitely generated: notably, by the Maurer - Weitzenböck theorem,^[6] 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

\ker \partial is finitely generated. If A = k[g _{1,...,} g _{n]} is a finitely generated algebra over a field of characteristic zero, then \ker\partial can be computed using van den Essen's algorithm,^[7] as follows. Choose a local slice, i.e. an element r\in \ker \partial ^{2} \setminus \ker \partial and put f = \partial r\in \ker\partial . Let \pi _{r} \colon A\to (\ker \partial) _{f} be the Dixmier map given by \pi _{r} (a) = \sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(n!) \partial ^{n} (a) (r ^{n})/(f ^{n}) . Now for every i =1 , ... , n , chose a minimal integer m _{i} such that h _{i} \colon = f ^{m _{i}} \pi _{r} (g _{i} )\in \ker\partial , put B _{0} = k[h _{1} , ... , h _{n} ,f]\subseteq \ker \partial , and define inductively B _{i} to be the subring of A generated by \{h\in A: fh\in B _{i-1} \} . By induction, one proves that B _{0} \subset B _{1} \subset ... \subset\ker \partial are finitely generated and if B _{i} = B _{i+1} then B _{i} = \ker \partial , so B _{N} = \ker \partial for some N . Finding the generators of each B _{i} and checking whether B _{i} = B _{i+1} is a standard computation using Gröbner bases.

Slice theorem

Assume that

\partial\in\operatorname{LND}(A) admits a slice, i.e. s\in A such that \partial s =1 . The slice theorem asserts that A is a polynomial algebra (\ker\partial) [s] and \partial = \tfrac{d}{ds} .

For any local slice

r\in\ker\partial \setminus \ker\partial ^{2} we can apply the slice theorem to the localization A _{\partial} , and thus obtain that A is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient \pi\colonX\to X// G _{a} is affine (e.g. when \dim X\leq 3 by the Zariski theorem), then it has a Zariski-open subset U such that \pi ^{-1} (U) is isomorphic over U to U x A ^{1} , where G _{a} acts by translation on the second factor.

However, in general it is not true that

X\to X// G _{a} is locally trivial. For example,^[8] let \partial = u\tfrac{\partial}{\partial x} + v\tfrac{\partial}{\partial y} + ( 1+ uy ^{2)\tfrac{\partial}{\partial} z}\in \operatorname{LND}( C [x,y,z,u,v]) . Then \ker\partial is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.

If

\dim X =3 then \Gamma = X\setminus U is a curve. To describe the G _{a} -action, it is important to understand the geometry \Gamma . Assume further that k = C and that X is smooth and contractible (in which case S is smooth and contractible as well^[9]) and choose \Gamma to be minimal (with respect to inclusion). Then Kaliman proved that each irreducible component of \Gamma is a polynomial curve, i.e. its normalization is isomorphic to C ^{1} . The curve \Gamma for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in C ^{2} , so \Gamma may not be irreducible. However, it is conjectured that \Gamma is always contractible.^[10]

Examples

Example 1

The standard coordinate derivations

\tfrac{\partial}{\partial x _{i}} of a polynomial algebra k[x _{1,...,} x _{n]} are locally nilpotent. The corresponding G _{a} -actions are translations: t ⋅ x _{i} = x _{i} + t , t ⋅ x _{j} = x _{j} for j ≠ i .

Example 2 (Freudenburg's (2,5)-homogeneous derivation^[11])

Let

f _{1=x} _{1x} _{3-x} ^{2} ^{2} , f _{2=x} _{3f} ^{2 x} ^{2 f} ^{5} ^{1+ x} , and let \partial be the Jacobian derivation $\backslash partial(f\_)=\backslash det\; \backslash left[\backslash tfrac\{\backslash partial\; f\_\{i\}\}\{\backslash partial\; x\_\{j\}\}\backslash right]\_$. Then \partial\in \operatorname{LND}(k[x _{1,x} _{2,x} _{3])} and \operatorname{rank}\partial =3 (see below); that is, \partial annihilates no variable. The fixed point set of the corresponding G _{a} -action equals \{x _{1=x} _{2=0\}} .

Example 3

Consider

Sl _{2(k)=\{ad-bc=1\}\subseteq} k ^{4} . The locally nilpotent derivation a\tfrac{\partial}{\partial b} + c\tfrac{\partial}{\partial d} of its coordinate ring corresponds to a natural action of G _{a} on Sl _{2(k)} via right multiplication of upper triangular matrices. This action gives a nontrivial G _{a} -bundle over A ^{2} \setminus \{( 0 , 0 )\} . However, if k = C then this bundle is trivial in the smooth category^[12]

LND's of the polynomial algebra

Let

k be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case k = C ^[13]) and let A = k[x _{1,...,} x _{n]} be a polynomial algebra.

Triangular derivations

Let

f _{1,...,f} _{n} be any system of variables of A ; that is, A = k[f _{1,...,} f _{n]} . A derivation of A is called triangular with respect to this system of variables, if \partial f _{1\in} k and \partial f _{i} \in k[f _{1,...,f} _{i-1} ] for i =2 , ... ,n . 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 \leq 2 by Rentschler's theorem above, but it is not true for n\geq 3 .

Bass's exampleThe derivation of

k[x _{1,x} _{2,x} _{3]} given by x _{1\tfrac{\partial}{\partial} x _{2}+2x} _{2x} _{1\tfrac{\partial}{\partial} x _{3}} is not triangulable.^[14] Indeed, the fixed-point set of the corresponding G _{a} -action is a quadric cone x _{2x} _{3=x} ^{2} ^{2} , while by the result of Popov,^[15] a fixed point set of a triangulable G _{a} -action is isomorphic to Z x A ^{1} for some affine variety Z ; and thus cannot have an isolated singularity.

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

G _{a} -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 C ^{3} , it is not.^[16]

Further reading

• A Nowicki, the fourteenth problem of hilbert for polynomial derivations

Notes and References

1. Web site: Daigle . Daniel . Hilbert's Fourteenth Problem and Locally Nilpotent Derivations . . 11 September 2018.
2. Arzhantsev. I.. Flenner. H.. Kaliman. S.. Kutzschebauch. F.. Zaidenberg. M.. Flexible varieties and automorphism groups. Duke Math. J.. 2013. 162. 4. 767–823. 10.1215/00127094-2080132. 1011.5375. 53412676 .
3. Book: Freudenburg. G.. Algebraic theory of locally nilpotent derivations. 2006. Springer-Verlag. Berlin. 978-3-540-29521-1. 10.1.1.470.10.
4. Zariski. O.. Interprétations algébrico-géométriques du quatorzième problème de Hilbert. Bull. Sci. Math. (2). 1954. 78. 155–168.
5. Derksen. H. G. J.. The kernel of a derivation. J. Pure Appl. Algebra. 1993. 84. 1. 13–16. 10.1016/0022-4049(93)90159-Q. free.
6. Seshadri. C.S.. On a theorem of Weitzenböck in invariant theory. J. Math. Kyoto Univ.. 1962. 1. 3. 403–409. 10.1215/kjm/1250525012. free.
7. Book: van den Essen. A.. Polynomial automorphisms and the Jacobian conjecture. 2000. Birkhäuser Verlag. Basel. 978-3-7643-6350-5. 10.1007/978-3-0348-8440-2. 252433637 .
8. Deveney. J.. Finston. D.. A proper G _{a} -action on C ^{5} which is not locally trivial. Proc. Amer. Math. Soc.. 1995. 123. 3. 651–655. 10.1090/S0002-9939-1995-1273487-0 . free. 2160782.
9. Kaliman. S. Saveliev. N.. C _{+} -Actions on contractible threefolds. Michigan Math. J.. 2004. 52. 3. 619–625. 10.1307/mmj/1100623416. math/0209306. 15020160.
10. Book: Kaliman. S.. Actions of C ^{*} and C _{+} on affine algebraic varieties. 2009. Algebraic geometry-Seattle 2005. Part 2. 80. 629–654. 10.1090/pspum/080.2/2483949. http://www.math.miami.edu/~kaliman/library/seattle.paper.pdf. Proceedings of Symposia in Pure Mathematics. 9780821847039.
11. Freudenburg. G.. Actions of G _{a} on A ^{3} defined by homogeneous derivations. Journal of Pure and Applied Algebra. 1998. 126. 1. 169–181. 10.1016/S0022-4049(96)00143-0. free.
12. Dubouloz. A.. Finston. D.. On exotic affine 3-spheres. J. Algebraic Geom.. 2014. 23. 3. 445–469. 10.1090/S1056-3911-2014-00612-3. 1106.2900. 119651964 .
13. Daigle. D.. Kaliman. S.. A note on locally nilpotent derivations and variables of k[X,Y,Z ]. Canad. Math. Bull.. 2009. 52. 4. 535–543. 10.4153/CMB-2009-054-5. free.
14. Bass. H.. A non-triangular action of G _{a} on A ^{3} . Journal of Pure and Applied Algebra. 1984. 33. 1. 1–5. 10.1016/0022-4049(84)90019-7. free.
15. Book: Popov. V. L.. Algebraic Groups Utrecht 1986 . On actions of G _{a} on A ^{n} . 1271. 237–242. 10.1007/BFb0079241. Lecture Notes in Mathematics. 1987. 978-3-540-18234-4.
16. Kaliman. S.. Makar-Limanov. L.. On the Russell-Koras contractible threefolds. J. Algebraic Geom.. 1997. 6. 2. 247–268.