Finite Tor dimension and failure of coherence in absolute integral closures Original Research Article

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 the second case, then, under some additional conditions, the radical of a finitely generated ideal has finite Tor dimension, and bounds are obtained. Let R+ denote the integral closure of the domain R in an algebraic closure of its fraction field. The results are applied to show that R+ is not coherent when R is Noetherian of dimension at least 3, and, under additional restrictions, when the dimension is 2. Motivation for this question connected with tight closure theory is discussed.