Regular local algebras over a Prüfer domain: weak dimension and regular sequences
نویسنده
چکیده
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. A formula to compute wdimO is provided. Furthermore regular sequences within the maximal ideal M O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdimO. If height (q ∩ R) 6= ∞, 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/(q∩R)O is CohenMacaulay. If (q∩R)Rq∩R is not finitely generated, then O/(q∩R)O itself is regular.
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