Remarks on a conjecture of Nakai
Let R be a geometric local domain over a field k of characteristic zero, Der(R/k) the algebra of high-order derivations of R/k and der(R/k) the subalgebra, generated by the first-order derivations. Nakai's problem asks, whether \(Der(R/k)=der(R/k)\) implies the regularity of R. The author investigates conditions under which every high-order derivation can be extended to the integral closure of R. From these he can prove that the condition \(Der(R/k)=der(R/k)\) implies regularity for the localizations of R with respect to the prime ideals of height one. Generalizing results of Bernstein et al. on derivations of graded rings he then proves a graded version of Nakai's conjecture: Let R be a two-dimensional weighted complete intersection over an algebraically closed field k of characteristic zero. If \(Der(R/k)=der(R/k)\) then R is a polynomial ring over k.
scientific article; zbMATH DE number 3950672
scientific article