What is the Rees algebra of a module?

Abstract

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to a free module as the image of the map induced by f on symmetric algebras. We show that the analytic spread and reductions of M can be determined from any embedding of M into a free module, and in characteristic 0-but not in positive characteristic!-the Rees algebra itself can be computed from any such embedding.