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.