A note on a class of rings found as \(G_a\)-invariants for locally trivial actions on normal affine varieties (Q1767349)

``Let \(k\) be an algebraically closed field and \(x_{1},\dots ,x_{n}\) algebraically independent elements over \(k\). Let \(L\) be a subfield of \( k( x_{1},\dots ,x_{n}) \) containing \(k\). Is the ring \(L\cap k [ x_{1},\dots ,x_{n}] \) finitely generated over \(k\)?'' This is Hilbert's fourteenth problem. In the paper under review, the author is interested in the case where \( L\cap k[ x_{1},\dots ,x_{n}] \) is the ring of invariants for a group action. The group of interest here is the additive group \(G_{a}.\) For \(X\) an affine variety, \(G_{a}\) actions on \(X\) are in bijective correspondence with locally nilpotent derivations \(k[ X] \to k[ X] \) of the coordinate ring of \(X\). Under this correspondence the invariant ring \(k[ X] ^{G_{a}}\) under some \(G_{a}\)-action is the kernel of its corresponding derivation. The author provides a locally trivial action on a normal affine variety of dimension 4 whose ring of invariants is not finitely generated, thereby giving an example of a \(G_{a}\)-action on an affine variety which locally can be expressed as a translation but does not admit an equivariant trivialization. This example is not explicitly given but can be constructed from the main result which says the following. Let \(A\) be an affine normal domain over \(k\), and \(L\) a field with \(k\subset L\subset K( A) ,\) where \(K\) is the quotient field of \(A.\) Then there is a normal affine variety \(X\) such that \(k[ X] \) is a proper ring extension of \( L\cap A\) and a locally trivial \(G_{a}\)-action on \(X\) such that \(L\cap A\) is the ring of invariants for this action. Taking \(k=\mathbb{C}\) and a counterexample of Rees gives the desired result.