FINITE TOR DIMENSION AND FAILURE OF COHERENCE IN ABSOLUTE INTEGRAL CLOSURES

It is shown both in characteristic p>0 and in mixed characteristic p>0 that if R is a perfect ring in the first case or R/pR is perfect in t he second case, then, under some additional conditions, the radical of a finitely generated ideal has finite Tor dimension, and bounds are o btained. Let R+ denote the integral closure of the domain R in an alge braic closure of its fraction field. The results are applied to show t hat R+ is not coherent when R is Noetherian of dimension at least 3, a nd, under additional restrictions, when the dimension is 2. Motivation for this question connected with tight closure theory is discussed. ( C) 1997 Elsevier Science B.V.