Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework. We introduce an approach through the notion of Bourbaki ideals that allow the use of deformation theory. One can talk about the (essentially unique) Bourbaki ideal I(E) of a module E which, in many situations, allows to reduce the nature of the Rees algebra of E to that of its Bourbaki ideal I(E). Properties such as Cohen–Macaulayness, normality and being of linear type are viewed from this perspective. The known numerical invariants of an ideal and its associated algebras are considered in the case of modules, such as the analytic spread, the reduction number, the analytic deviation. Corresponding notions of complete intersection, almost complete intersection and equimultiple modules are examined to some detail. A special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way. For these modules one is able to estimate the reduction number and other invariants in terms of the Buchsbaum–Rim multiplicity. 01991 Mathematics Subject Classification. Primary 13H10; Secondary 13A30, 13H15. 0