This chapter deals with central techniques to study the arithmetic of Rees algebras and certain symmetric algebras. It treats normality and divisor class groups, describes canonical modules and contains several constructions to generate new algebras from old ones. The more novel material are some techniques to effect a bridge between the special theory of ideals of linear type with more general ideals.

A natural limitation of the methods pursued in Chapter 3 is the requirement that one of the Fκ–condition holds. To help span the gap it is appropriate to view the Rees algebra of a ‘general’ ideal as an extension of the Rees algebra of an ideal of linear type. The vehicle for this theory is the notion of reduction of an ideal, an object first isolated by Northcott and Rees ([216]), and which plays a premier role in the study of Rees algebras. An ideal J ⊂ I is a reduction of I if JIr = Ir + 1 for some integer r; the least such integer, rj(I), is the reduction number of I with respect to J. Phrased otherwise, J being a reduction means that

R[Jt] ↪ R[It]

is a finite morphism of graded algebras. We shall introduce several structures where the two algebras interact. One of these is the notion of reduction modules, which while reflecting properties of R[It] are more easily studied over R[Jt].