Some \(K\)-theoretic properties of the kernel of a locally nilpotent derivation on \(k[X_1, \dots , X_4]\) (Q2826762)

Let \(k\) be an algebraically closed field of characteristic zero and \(R\) an integral domain containing \(k\). A derivation \(D:R\rightarrow R\) is called \textit{locally nilpotent} if for every \(x\in R\), there exists a positive integer \(s\) such that \(D^{s}(x) = 0\). \(D\) is said to \textit{fixed point free} if the ideal generated by \(DR\) is the unit ideal. If \(S\) is a subring of \(R\), the set of all \(S\)-linear locally nilpotent derivations of \(R\) is denoted by \(\mathrm{LND}_{S}(R)\). The paper under review studies properties of kernels of locally nilpotent derivations of \(R = k[X_{1},\dots, X_{4}]\) in connection with the following question of \textit{M. Miyanishi} [in: Affine algebraic geometry. Dedicated to Masayoshi Miyanishi on the occasion of his retirement from Osaka University. Osaka: Osaka University Press. 307--378 (2007; Zbl 1124.14309)]: If \(D\in\mathrm{LND}_{k}(R)\) and \(A = \mathrm{Ker} D\), is it true that every projective \(A\)-module is free? Implicit is a subquestion: Is the Grothendieck group \(K_{0}(A)\) trivial?NEWLINENEWLINEThe paper under review presents an explicit \(k[X_{1}]\)-linear fixed point free locally nilpotent derivation of \(k[X_{1},\dots, X_{4}]\) whose kernel \(A\) has an isolated singularity and whose Grothendieck group \(K_{0}(A)\) is not finitely generated. The authors also show that even though the Miyanishi's question does not have an affirmative answer in general, suitably modified versions of the question do have affirmative answers when the derivation annihilates a variable.NEWLINENEWLINEThe main results of the paper are the following three statements (we use the above notation):NEWLINENEWLINEI. If \(D(X_{1}) = 0\) and \(A = \mathrm{Ker} D\) is regular, then \(A\) is a polynomial ring over \(k[X_{1}]\). In particular, every projective \(A\)-module is free.NEWLINENEWLINEII. If \(D\in\mathrm{LND}_{k[X_{1}]}(R)\), then the canonical maps \(G_{j}(k)\rightarrow G_{j}(A)\) (\(j=0, 1\)) are isomorphisms. (If \(\mathcal{M}(R)\) denotes the category of finitely generated modules over a commutative Noetherian ring \(R\), then \(G_{0}(R)\) and \(G_{1}(R)\) denote the Grothendieck and Whitehead groups of \(\mathcal{M}(R)\), respectively.) In particular, \(G_{0}(A) = \mathbb{Z}\) and \(G_{1}(A) = k^{\ast}\).NEWLINENEWLINEIII. Let \(F(Z, T)\) be an irreducible polynomial in \(k[Z, T]\), \(C = k[Z, T]/(F)\) and \(B = k[U, V, Z, T]/(U^{m}V - F(Z,T))\) (\(m\geq 1\)). Then: (i) If the ring \(C\) is not regular, then \(K_{0}(B)\) is not finitely generated; in particular, there exists an infinite family of non-isomorphic projective \(B\)-modules of rank two which are not even stably isomorphic. (ii) For some \(D\in\mathrm{LND}_{k[X_{1}]}(k[X_{1},\dots, X_{4}])\), \(B\) is isomorphic (as a \(k\)-algebra) to \(\mathrm{Ker} D\) if and only if \(C\) is a \(k\)-subalgebra of a polynomial ring \(k[W]\).