Rees algebras of conormal modules

We study relationships among algebras associated to the pair ( R, p ), where R is a Noetherian ring and p is a prime ideal of R, by examining normality conditions in the algebras.; This thesis focuses on the study of the Rees algebra Rp/p2 of a conormal module p/p2 as an R/p -module. There is a surjection from the symmetric algebra SR/pp/p2 of the conormal module p/p2 onto the Rees algebra Rp/p2 of p/p2 . If (R, m ) is a Noetherian local ring and Rp is a regular local ring, then the Rees algebra Rp/p2 is a homomorphic image of the associated graded ring grpR . In general, the Rees algebra Rp/p2 depends only on the module p/p2 over the ring R/p , not on the ring R itself. To ensure R has a role, we must force a relationship between p and R, which is the case when we assume, for example, that p has finite projective dimension over R.; We deal with special classes of ideals p whose associated graded ring grpR is isomorphic to the Rees algebra Rp/p2 in order to describe the divisor class group of Rp/p2 and to examine normality of the conormal module p/p2 .; When the associated graded ring grpR is integrally closed, we show that there is a group isomorphism between the divisor class group Cl (G) of G and that Cl ( R/p ) of R/p . If G is a domain which is not integrally closed, this result is used to describe the divisor class group of the integral closure G¯ of the associated graded ring G. There is an exact sequence of divisor class groups. 0→H→ClG →ClR/p →0, where H is a finitely generated group (conjecturally free group). The technique works equally well for other algebras derived from the associated graded ring G.; Suppose that R/p is integrally closed. Normality of the associated graded ring G = ⊕ n≥0Gn means that Gn is an integrally closed R/p -module for every n ≥ 0. One of our objectives is to obtain a statement ensuring that Gn is an integrally closed R/p -module for n ≤ n0 implies completeness for all n. We prove that, if R is a Gorenstein local ring, p is a prime ideal generated by a strongly Cohen-Macaulay d -sequence, p has finite projective dimension, R/p is an integrally closed domain of dimension 2, and the associated graded ring G = grpR is a domain, then completeness of G1 implies that G is normal. Moreover we obtain a result which shows a relationship between normality of the conormal module and reflexivity of other components of the associated graded ring G. Underlying our examination of the integral closure of the associated graded ring G, there are several issues from the general theory of complete modules. Given an integrally closed domain R and a submodule E of Rr, we develop some general techniques connected to the notion of divisors to decide when E is integrally closed.