Locally nilpotent derivation

In mathematics, a derivation $\partial$ 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.^[vague]^[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 $\partial _{1}+\partial _{2}\in \operatorname {LND} (A)$ ^[3] and if $\partial \in \operatorname {LND} (A)$ , $h\in \ker \partial$ then $h\partial \in \operatorname {LND} (A)$ .

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

Let $A$ be an algebra over a field $k$ of characteristic zero (e.g. $k=\mathbb {C}$ ). Then there is a one-to-one corresponence between the locally nilpotent $k$ -derviations on $A$ and the actions of the additive group $\mathbb {G} _{a}$ of $k$ on the affine variety $\operatorname {Spec} A$ , as follows.^[3] A $\mathbb {G} _{a}$ -action on $\operatorname {Spec} A$ corresponds to an $k$ -algebra homomorphism $\rho \colon A\to A[t]$ . Any such $\rho$ determines a locally nilpotnent 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 $\rho =\exp(t\partial )=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}\partial ^{n}.$

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 $\exp(t\cdot \alpha \circ \partial \circ \alpha ^{-1})=\alpha \circ \exp(t\partial )\circ \alpha ^{-1}$

The kernel algorithm

The algebra $\ker \partial$ consists of the invariants of the corresponding $\mathbb {G} _{a}$ -action. It is algebraically and factorially closed in $A$ .^[3] A special case of Hilbert's 14th problem asks whether $\ker \partial$ is finitely generated, or, if $A=k[X]$ , whether the quotient $X/\!/\mathbb {G} _{a}$ 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 $X=\mathbb {C} ^{n}$ , $n\geq 4$ . For $n\geq 5$ the answer, in general, is negative.^[5] The case $n=4$ is open.^[3]

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 $A=k[g_{1},\dots ,g_{n}]$ 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 $\pi _{r}(a)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\partial ^{n}(a){\frac {r^{n}}{f^{n}}}$ . Now for every $i=1,\dots ,n$ , chose a minimal integer $m_{i}$ such that $h_{i}\colon =f^{m_{i}}\pi _{r}(g_{i})\in \ker \partial$ , put $B_{0}=k[h_{1},\dots ,h_{n},f]\subseteq \ker \partial$ , 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 $B_{0}\subset B_{1}\subset \dots \subset \ker \partial$ 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.^[7]

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^[3] 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 $A_{\partial r}$ , and thus obtain that $A$ is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient $\pi \colon \,X\to X/\!/\mathbb {G} _{a}$ 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 $U\times \mathbb {A} ^{1}$ , where $\mathbb {G} _{a}$ acts by translation on the second factor.

However, in general it is not true that $X\to X//\mathbb {G} _{a}$ is locally trivial. For example,^[8] let $\partial =u{\tfrac {\partial }{\partial x}}+v{\tfrac {\partial }{\partial y}}+(1+uy^{2}){\tfrac {\partial }{\partial z}}\in \operatorname {LND} (\mathbb {C} [x,y,z,u,v])$ . 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 $\mathbb {G} _{a}$ -action, it is important to understand the goemetry $\Gamma$ . Assume further that $k=\mathbb {C}$ 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^[10] that each irreducible component of $\Gamma$ is a polynomial curve, i.e. its normalization is isomorphic to $\mathbb {C} ^{1}$ . The curve $\Gamma$ for the action given by Freudeburg's (2,5)-derivation (see below) is a union of two lines in $\mathbb {C} ^{2}$ , so $\Gamma$ may not be irreducible. However, it is conjectured that $\Gamma$ is always contractible.^[11]

Examples

Example 1

The standard coordinate derivations ${\tfrac {\partial }{\partial x_{i}}}$ of a polynomial algebra $k[x_{1},\dots ,x_{n}]$ are locally nilpotent. The corresponding $\mathbb {G} _{a}$ -actions are translations: $t\cdot x_{i}=x_{i}+t$ , $t\cdot x_{j}=x_{j}$ for $j\neq i$ .

Example 2 (Freudenbug's (2,5)-homogeneous derivation^[12])

Let $f_{1}=x_{1}x_{3}-x_{2}^{2}$ , $f_{2}=x_{3}f_{1}^{2}+2x_{1}^{2}x_{2}f_{1}+x^{5}$ , and let $\partial$ be the Jacobian derivation $\partial (f_{3})=\det[{\tfrac {\partial f_{i}}{\partial x_{j}}}]_{i,j=1,2,3}$ . 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 $\mathbb {G} _{a}$ -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 $\mathbb {G} _{a}$ on $Sl_{2}(k)$ via right multiplication of upper triangular matrices. This action gives a nontrivial $\mathbb {G} _{a}$ -bundle over $\mathbb {A} ^{2}\setminus \{(0,0)\}$ . However, if $k=\mathbb {C}$ then this bundle is trivial in the smooth category^[13]

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 $k=\mathbb {C}$ ^[14] ) and let $A=k[x_{1},\dots ,x_{n}]$ be a polynomial algebra.

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

Rentschler's theorem

Every LND of $k[x_{1},x_{2}]$ can be conjugated to $f(x_{1}){\tfrac {\partial }{\partial x_{2}}}$ for some $f(x_{1})\in k[x_{1}]$ . This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.^[15]

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

Miyanishi's theorem

The kernel of every nontrivial LND of $A=k[x_{1},x_{2},x_{3}]$ is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial $\mathbb {G} _{a}$ -action on $\mathbb {A} ^{3}$ is isomorphic to $\mathbb {A} ^{2}$ .^[16]^[17]

In other words, for every $0\neq \partial \in \operatorname {LND} (A)$ there exist $f_{1},f_{2}\in A$ such that $\ker \partial =k[f_{1},f_{2}]$ (but, in contrast to the case $n=2$ , $A$ is not necessarily a polynomial ring over $\ker \partial$ ). In this case, $\partial$ is a Jacobian derivation: $\partial (f_{3})=\det[{\tfrac {\partial f_{i}}{\partial x_{j}}}]_{i,j=1,2,3}$ .^[18]

Zurkowski's theorem

Assume that $n=3$ and $\partial \in \operatorname {LND} (A)$ is homogeneous relative to some positive grading of $A$ such that $x_{1},x_{2},x_{3}$ are homogeneous. Then $\ker \partial =k[f,g]$ for some homogeneous $f,g$ . Moreover,^[18] if $\deg x_{1},\deg x_{2},\deg x_{3}$ are relatively prime, then $\deg f,\deg g$ are relatively prime as well.^[19]^[3]

Bonnet's theorem

A quotient morphism $\mathbb {A} ^{3}\to \mathbb {A} ^{2}$ of a $\mathbb {G} _{a}$ -action is surjective. In other words, for every $0\neq \partial \in \operatorname {LND} (A)$ , the embedding $\ker \partial \subseteq A$ induces a surjective morphism $\operatorname {Spec} A\to \operatorname {Spec} \ker \partial$ .^[20]^[10]

This is no longer true for $n\geqslant 4$ , e.g. the image of a quotient map $\mathbb {A} ^{4}\to \mathbb {A} ^{3}$ by a $\mathbb {G} _{a}$ -action $t\cdot (x_{1},x_{2},x_{3},x_{4})=(x_{1},x_{2},x_{3}-tx_{2},x_{4}+tx_{1})$ (which corresponds to a LND given by $x_{1}{\tfrac {\partial }{\partial x_{4}}}-x_{2}{\tfrac {\partial }{\partial x_{3}}})$ equals $\mathbb {A} ^{3}\setminus \{(x_{1},x_{2},x_{3}):x_{1}=x_{2}=0,x_{3}\neq 0\}$ .

Kaliman's theorem

Every fixed-point free action of $\mathbb {G} _{a}$ on $\mathbb {A} ^{3}$ is conjugate to a translation. In other words, every $\partial \in \operatorname {LND} (A)$ such that the image of $\partial$ generates the unit ideal (or, equivalently, $\partial$ defines a nowhere vanishing vector field), admits a slice. This results answers one of the conjectures from Kraft's list.^[10]

Again, this result is not true for $n\geqslant 4$ :^[21] e.g. consider the $x_{1}{\tfrac {\partial }{\partial x_{2}}}+x_{2}{\tfrac {\partial }{\partial x_{3}}}+(x_{2}^{2}-2x_{1}x_{3}-1){\tfrac {\partial }{\partial x_{4}}}\in \operatorname {LND} (\mathbb {C} [x_{1},x_{2},x_{3},x_{4}])$ . The points $(x_{1},1,0,0)$ and $(x_{1},-1,0,0)$ are in the same orbit of the corresponding $\mathbb {G} _{a}$ -action if and only if $x_{1}\neq 0$ ; hence the (topological) quotient is not even Hausdorff, let alone homeomorphic to $\mathbb {C} ^{3}$ .

Principal ideal theorem

Let $\partial \in \operatorname {LND} (A)$ . Then $A$ is faithfully flat over $\ker \partial$ . Moreover, the ideal $\ker \partial \cap \operatorname {im} \partial$ is principal in $A$ .^[14]

Triangular derivations

Let $f_{1},\dots ,f_{n}$ be any system of variables of $A$ ; that is, $A=k[f_{1},\dots ,f_{n}]$ . A derivation of $A$ is called triangular with respect to this system of variables, if $\partial f_{1}\in k$ and $\partial f_{i}\in k[f_{1},\dots ,f_{i-1}]$ for $i=2,\dots ,n$ . 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 example

The derivation of $k[x_{1},x_{2},x_{3}]$ given by $x_{1}{\tfrac {\partial }{\partial x_{2}}}+2x_{2}x_{1}{\tfrac {\partial }{\partial x_{3}}}$ is not triangulable.^[22] Indeed, the fixed-point set of the corresponding $\mathbb {G} _{a}$ -action is a quadric cone $x_{2}x_{3}=x_{2}^{2}$ , while by the result of Popov,^[23] a fixed point set of a triangulable $\mathbb {G} _{a}$ -action is isomorphic to $Z\times \mathbb {A} ^{1}$ for some affine variety $Z$ ; and thus cannot have an isolated singularity.

Freudenburg's theorem

The above necessary geometrical condition was later generalized by Freudenburg.^[24] To state his result, we need the following definition:

A corank of $\partial \in \operatorname {LND} (A)$ is a maximal number $j$ such that there exists a system of variables $f_{1},\dots ,f_{n}$ such that $f_{1},\dots ,f_{j}\in \ker \partial$ . Define $\operatorname {rank} \partial$ as $n$ minus the corank of $\partial$ .

We have $1\leq \operatorname {rank} \partial \leq n$ and $\operatorname {rank} (\partial )=1$ if and only if in some coordinates, $\partial =h{\tfrac {\partial }{\partial x_{n}}}$ for some $h\in k[x_{1},\dots ,x_{n-1}]$ .^[24]

Theorem: If $\partial \in \operatorname {LND} (A)$ is triangulable, then any hypersurface contained in the fixed-point set of the corresponding $\mathbb {G} _{a}$ -action is isomorphic to $Z\times \mathbb {A} ^{\operatorname {rank} \partial }$ .^[24]

In particular, LND's of maximal rank $n$ cannot be triangulable. Such derivations do exist for $n\geq 3$ : the first example is the (2,5)-homogeneous derivation (see above), and it can be easily generalized to any $n\geq 3$ .^[12]

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 $\mathbb {G} _{a}$ -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 $\mathbb {C} ^{3}$ , it is not.^[25]

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