The divisor class group of ordinary and symbolic blow-ups
Given a (multiplicative) filtration of ideals in a ring R, one associates to it a ``Rees algebra'' S and a graded ring G (``the cone''). This paper is concerned with the arithmetical properties of these objects. Firstly, if S is a Krull domain (not necessarily noetherian) then its divisor class group Cl(S) is shown to be an extension of Cl(R) by a finitely generated free group. In case the filtration happens to be that of the symbolic powers of an ideal then, under minor requirements, the rank of the above free group is easily established. Clearly, this takes care of many instances of the filtration by (ordinary) powers of an ideal. The question, however, in general, seems far from being settled. Another related question is that whether the above extension is actually a split extension; again, here, whether G is torsion free over R/I \((I:= leading\) ideal of the filtration) or not seems to be pertinent. A recently announced result of Huneke, Vasconcelos and the present first author deals with the question of when actually happens that G is torsion free. A third problem dealt with is the determination of the canonical class of S and G. In particular, it is said under which (approximate) conditions S is a Gorenstein or quasi-Gorenstein ring. Finally, similar questions are posed for the extended Rees algebra. In particular, one derives a calculation of Shimoda of 1979. Needless to say, the present results extend and unify previous results of various authors - among others, J. Herzog, W. Vasconcelos, R. Villarreal and W. Bruns jointly with the present first author.
scientific article; zbMATH DE number 4017047
scientific article