Regular local algebras over valuation domains: Weak dimension and regular sequences

A local ring O is called regular if every finitely generated ideal I Delta O possesses finite projective dimension. In the article localizations O = A(q), q is an element of SpecA, of a finitely presented, flat algebra A over a Prufer 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 wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M Delta O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q boolean AND R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q boolean AND R)O, which is noetherian, is Cohen-Macaulay. If in addition (q boolean AND R) R-q boolean AND R is not finitely generated, then O/(q boolean AND R)O itself is regular.