  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.
