Regular local algebras over a Pruefer domain: weak dimension and regular sequences

Abstract

A not necessarily noetherian local ring O is called regular if every finitely generated ideal I of O possesses finite projective dimension. In the article localizations O of a finitely presented, flat algebra A over a Pruefer domain R at a prime q 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 of O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim(O). If height(p) is finite, where p is the intersection of q with R, then this sequence can be choosen such that the radical of the ideal generated by the members of the sequence equals M. As a consequence it is proved that if O is regular, then the (noetherian) factor ring O/pO is Cohen-Macaulay. If pR_p is not finitely generated, then O/pO itself is regular.