Non-Finitely Generated Kernels

Abstract

The Maurer-Weitzenbr̈ock Theorem for linear $\mathbb{G}_{a}$-actions on affine space does not generalize to non-linear $\mathbb{G}_{a}$-actions. In 1990, Paul Roberts [359] gave the first examples of non-affine invariant rings for $\mathbb{G}_{a}$-actions on an affine space. These examples involved actions of $\mathbb{G}_{a}$ on $\mathbb{A}^{7}$ over a field k of characteristic zero, and are counterexamples to Hilbert’s Fourteenth Problem. Subsequent examples of $\mathbb{G}_{a}$-actions of non-finite type were constructed in Freudenburg [163] and in Daigle and Freudenburg [79] for $\mathbb{A}^{6}$ and $\mathbb{A}^{5}$, respectively.

Appendix: Nagata’s Problem Two

Nagata posed the following question in his 1959 paper [320] as “Problem 2”.

Let K be a subfield of the field k(x ₁, …, x _n ) such that tr. deg. _k K = 3. Is K ∩ k[x ₁, …, x _n ] always finitely generated?

The example in Nagata’s paper has tr. deg. _k K = 4, as does the example of Steinberg (Thm. 1.2 of [392]). Also, the kernels of the derivations D _(r, s) in Sect.  7.14 have tr. deg _k ( ker D _(r, s)) = 4. On the other hand, Rees’s counterexample to Zariski’s Problem has fixed ring of transcendence degree three; see Sect.  6.4 .

Nagata’s question was answered in the negative by Kuroda in [261, 263]. Kuroda’s first counterexamples are subfields of k ⁽⁴⁾. Let γ and δ _ij be integers (1 ≤ i ≤ 3, 1 ≤ j ≤ 4) such that γ ≥ 1 and:

$$\displaystyle{\delta _{ij} \geq 1\,\,\text{if}\,\,1 \leq j \leq 3\quad \text{and}\quad \delta _{i,4} \geq 0\,\,\text{if}\,\,1 \leq i \leq 3}$$

Let k(Π) denote the subfield of k(x) = k(x ₁, x ₂, x ₃, x ₄) = k ⁽⁴⁾ generated by:

$$\displaystyle\begin{array}{rcl} \varPi _{1}& =& x_{4}^{\gamma } - x_{ 1}^{-\delta _{1,1} }x_{2}^{\delta _{1,2} }x_{3}^{\delta _{1,3} }x_{4}^{\delta _{1,4} } {}\\ {}\\ \varPi _{2}& =& x_{4}^{\gamma } - x_{ 1}^{\delta _{2,1} }x_{2}^{-\delta _{2,2} }x_{3}^{\delta _{2,3} }x_{4}^{\delta _{2,4} } {}\\ {}\\ \varPi _{2}& =& x_{4}^{\gamma } - x_{ 1}^{\delta _{3,1} }x_{2}^{\delta _{3,2} }x_{3}^{-\delta _{3,3} }x_{4}^{\delta _{3,4} } {}\\ \end{array}$$

Let k[x] = k[x ₁, x ₂, x ₃, x ₄].

Theorem 7.30 (Kuroda [261], Thm 1.1)

If

$$\displaystyle{ \frac{\delta _{1,1}} {\delta _{1,1} +\min \{\delta _{2,1},\delta _{3,1}\}} + \frac{\delta _{2,2}} {\delta _{2,2} +\min \{\delta _{3,2},\delta _{1,2}\}} + \frac{\delta _{3,3}} {\delta _{3,3} +\min \{\delta _{1,3},\delta _{2,3}\}} <1}$$

then k(Π) ∩ k[x] is not finitely generated over k.

Kuroda also shows that k(Π) ∩ k[x] cannot be the kernel of any locally nilpotent derivation of k[x].

Nonetheless, there does exist D ∈ Der _k (k[x]) with ker D = k(Π) ∩ k[x]. For the simplest symmetric example, take k(Π) = k( f, g, h) for:

$$\displaystyle{\,f = x_{4} - x_{1}^{-1}x_{ 2}^{3}x_{ 3}^{3}\,\,,\,\,g = x_{ 4} - x_{1}^{3}x_{ 2}^{-1}x_{ 3}^{3}\,\,,\,\,h = x_{ 4} - x_{1}^{3}x_{ 2}^{3}x_{ 3}^{-1}}$$

The jacobian derivation Δ _{( f, g, h)} ∈ Der _k (k(x)) restricts to k[x], namely, Δ _{( f, g, h)} = 4x ₁ x ₂ x ₃ D, where:

$$\displaystyle\begin{array}{rcl} Dx_{i}& =& x_{i}(5x_{i}^{4}-\varphi )\qquad \text{for}\,\,\varphi = x_{ 1}^{4} + x_{ 2}^{4} + x_{ 3}^{4}\quad (1 \leq i \leq 3) {}\\ Dx_{4}& =& -20(x_{1}x_{2}x_{3})^{3} {}\\ \end{array}$$

That ker D = k(Π) ∩ k[x] can be proved using [264].

Kuroda’s second family of examples have members which are subfields L of K = k(x ₁, x ₂, x ₃) = k ⁽³⁾, i.e., K is an algebraic extension of L, but L ∩ k[x ₁, x ₂, x ₃] is not finitely generated. These appear in [263]. By the result proved in section “Appendix 1: Finite Group Actions” of Chap.  6 together with the Finiteness Theorem, it follows that L ∩ k[x ₁, x ₂, x ₃] cannot be the ring of invariants of any algebraic group action on $\mathbb{A}^{3}$.

Given positive integers γ and δ _ij (i, j = 1, 2), let k(H) denote the subfield of K generated by:

$$\displaystyle\begin{array}{rcl} H_{1}& =& x_{1}^{\delta _{2,1} }x_{2}^{-\delta _{2,2} } - x_{1}^{-\delta _{1,1} }x_{2}^{\delta _{1,2} } {}\\ {}\\ H_{2}& =& x_{3}^{\gamma } - x_{ 1}^{-\delta _{1,1} }x_{2}^{\delta _{1,2} } {}\\ {}\\ H_{3}& =& 2x_{1}^{\delta _{2,1}-\delta _{1,1} }x_{2}^{\delta _{1,2}-\delta _{2,2} } - x_{1}^{-2\delta _{1,1} }x_{2}^{2\delta _{1,2} } {}\\ \end{array}$$

Theorem 7.31 (Kuroda [263], Thm. 1.1)

If

$$\displaystyle{ \frac{\delta _{1,1}} {\delta _{1,1} +\delta _{2,1}} + \frac{\delta _{2,2}} {\delta _{2,2} +\delta _{1,2}} <\frac{1} {2}}$$

then K(H) ∩ k[x ₁, x ₂, x ₃] is not finitely generated over k.

Kuroda uses the theory of locally nilpotent derivations in his proofs. See also [265, 266]. These examples bear study in the effort to decide whether kernels of locally nilpotent derivations of k ^[4] are finitely generated.