Rees Algebras of Modules - Purdue Universitybulrich/papers/ram.pdf · 2002. 5. 11. · the Rees...
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Rees Algebras of Modules ARON SIMIS Departamento de Matem´ atica Universidade Federal de Pernambuco 50740-540 Recife, PE, Brazil e-mail: [email protected] BERND ULRICH † Department of Mathematics Michigan State University E. Lansing, Michigan 48824 e-mail: [email protected] WOLMER V. V ASCONCELOS ‡ Department of Mathematics Rutgers University 110 Frelinghuysen Rd Piscataway, New Jersey 08854-8019 e-mail: [email protected] Dedicated to David Rees on his eightieth birthday Abstract 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 IE 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 IE . 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 exam- ined 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. 0 1991 Mathematics Subject Classification. Primary 13H10; Secondary 13A30, 13H15. 0 Keywords. Analytic spread, Bourbaki ideal, Buchsbaum–Rim multiplicity, Cohen–Macaulay ring, minimal reduction, reduction number, Rees algebra Partially supported by CNPq, Brazil. † Partially supported by the NSF. ‡ Partially supported by the NSF. 1
