The MacRae invariant and the first local Chern character

Abstract

Let R be a commutative Noetherian ring, and let M be a finitely generated module of finite projective dimension. In his investigation of invariants of these modules, MacRae [6] constructed an invertible ideal G(M) associated to M which describes th; part of the support of M of codimension 1. The fact that G(M) is invertible implies many properties of the support of M, and it has recently been used by Foxby [3] to prove some conjectures on intersection multiplicities of modules of finite projective dimension with modules of Krull dimension one. In this paper we generalize the construction of MacRae to a bounded complex E* of locally free sheaves on a Noetherian scheme Y and show that this can be used to describe the first local Chern character of E*. A bounded complex of locally free sheavesoof finite rank will be called a perfect complex. For technical reasons, we assume that Y is connected and quasi-projective over an affine scheme. Let X be the support of E*, denoted Supp(E*); this can be defined as the set of points of Y where E* is not exact, or, equivalently, as the union of the supports of the homology modules Hi(E*). We assume that X is contained in some Cartier divisor. Locally, this means that the ideal defining X contains a non-zero-divisor, and if E* is a resolution of a module, our assumption follows whenever X is a proper subset of Y. An equivalent formulation of this condition is that X contains no points y of Y such that the local ring dy has depth zero; such a point will be called a point of depth zero. In this situation we construct a Cartier divisor G(E*) on Y generalizing the MacRae invariant. If Y is quasi-projective over a regular local ring (this includes, among others, the case where Y= SpecR and R is a complete local ring), there is a theory of local Chern characters defined for perfect complexes on Y. For a complex E* with support X as above, for any scheme Y' together with a map of finite type f: Y' Y,