Groups Acting on Hereditary Rings

Abstract

Theorem 4.2 of my paper is incorrect as stated. A counterexample was given 30 years ago by Asano and Nakayama ([1]). In slightly simplified form, it is this. Let k be a field, K = k(X, Y), Lhe the subfield of symmetric elements, i.e. k(XY, X+Y), i.e. the fixed field of the automorphism α interchanging X and Y. Take for R the principal ideal domain k(X)[Y]. Then R′ = L∩R ⊆ R∩αR = k[X, Y], so in fact R′ = k[XY,X+Y], which is a polynomial ring in two indeterminates, and not a Dedekind domain. The error in the proof given is the assumption that R will be the integral closure in K of R′ = R∩L. If we add this as a hypothesis, the result becomes valid. I find, incidentally, that the proof I gave is essentially the same as that of [1] Theorem 1 (though the authors there were interested only in the property of being a Dedekind domain, and not in the invertibility of arbitrary ideals). I am indebted to Adrian Wadsworth for pointing out this error and this reference. The validity of the other results of the paper is not affected.