INFINITELY GENERATED DERKSEN AND MAKAR-LIMANOV INVARIANT

Abstract

AbstractIn this paper, we give an example of a finitely generated 3-dimensional C-algebrawhich has infinitely generated Derksen invariant as well as Ma kar-Limaonv invariant. 1. Introduction and toolsThe Derksen invariant and Makar-Limanov invariant are useful tools to recognize iftwo varieties or rings are not isomorphic. Both invariants use locally nilpotent deriva-tions: if A is a commutative k-algebra (wherek is a field of characteristic zero), thenD is a derivation if D is k-linear and satisfies the Leibniz rule: D(ab) = aD(b)+bD(a).A derivation is locally nilpotent if for each a 2 A we can find some n N such thatD n (a) = 0. The kernel of a derivation, denoted by A D , is the set of all elements thatare mapped to zero under the derivation D. The Makar-Limanov invariant is definedas the intersection of all kernels of locally nilpotent derivations, while the Derksen in-variant is defined as the smallest algebra containing the ker nels of all nonzero locallynilpotent derivations.In the paper [4] the question was posed if the Derksen invariant could be infinitelygenerated. In this paper we give an example of an infinitely ge nerated Derksen in-variant of a finitely generated C-algebra. It will be at the same time an example of aninfinitely generated Makar-Limanov invariant, as in this example, the Derksen invariantis equal to the Makar-Limanov invariant. By now, there are many examples of casesof “nice” subrings that are not finitely generated [1, 3, 5, 6, 7]. In regard of this,the author would like to remark that it will pay off to consider theorems as general aspossible (with respect to not restricting to finitely genera ted algebras).N