What Is the Rees Algebra of a Module?

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. The Rees algebra of an ideal I in a ring R, namely R = ⊕∞ n=0 I n = R[It] ⊂ R[t], plays a major role in commutative algebra and in algebraic geometry since Proj(R) is the blowup of Spec(R) along the subscheme defined by I. Several authors have found it useful to generalize this construction from ideals to modules; see for instance Gaffney and Kleiman [GK], Katz [K], Katz and Kodiyalam [KK], Kleiman and Thorup [KT], Kodiyalam [Ko], Liu [L], Rees [R], Simis, Ulrich, and Vasconcelos [SUV1], [SUV2], and Vasconcelos [V], who define the Rees algebra of a module satisfying one or another hypothesis. Usually this hypothesis was tailored to approach the problem(s) the authors were interested in solving. The goal of this paper is to clarify the definition for arbitrary finitely generated modules over a Noetherian ring R. Our interest in this clarification arose through our work on generalized prinicipal ideal theorems and the heights of ideals of minors, where we heavily use Rees algebras (see Eisenbud-Huneke-Ulrich [EHU1], [EHU2]). It seems worthwhile to understand the differences and similarities of the various approaches from the papers above, and to make the definition as functorial as possible. Even for ideals there is a problem: in the grade 0 case it is not clear from the definition above whether the Rees algebra depends on the embedding of I in R. A natural approach is to define the Rees algebra of a module as the symmetric algebra modulo R-torsion (that is, modulo elements killed by non-zerodivisors of R). This does not provide a satisfactory definition in all cases in the sense that it may give the wrong answer even for an ideal, if the ring is not a domain. In general, as was well-known, it is a good definition when the module M “has a rank”, i.e., when M is free of constant rank locally at the associated primes of R. This hypothesis is sufficient for many applications; however, for example, it is Received by the editors May 2, 2001 and, in revised form, October 19, 2001. 2000 Mathematics Subject Classification. Primary 13A30, 13B21; Secondary 13C12.