A not necessarily noetherian local ring O is called regular if every finitely generated ideal I O possesses finite projective dimension. In the article localizations O = Aq, q ∈ SpecA, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdimO. ...

In my paper [F3] I more or less explicitly conjectured that if A is a complete local integral domain with maximal ideal m and if I = (tί9 , tn) is an ideal in A with n < dim (A) — 2, then Spec (A/7) — m is G3 in Spec (A) — m. This will be proved in this paper. Unfortunately the algebraisation-theorems for subsheaves in [F3] make great difficulties in this context, and I only can prove similar t...