2004/06/19 by Hagen Knaf, Knaf, Hagen
Mathematics · #13D05 #13H05 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:13D05 #msc:13H05
paper · pdf · doi:10.48550/arxiv.math/0406385
25 pages, 1 figure
arxiv created 2004/06/19 · openalex publication_date 2004/06/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 pRp is not finitely generated, then O/pO itself is regular.