Mixed Multiplicities, Joint Reductions and Quasi‐Unmixed Local Rings

Joint reductions and mixed multiplicities are relatively new concepts and there is very little about them in the literature. We provide some background in the first two sections. Some of the properties about joint reductions and mixed multiplicities which we prove have not appeared elsewhere, but are probably folklore to the people who have worked on mixed multiplicities or joint reductions. This makes it very difficult to attribute them to their rightful discoverers. The story of mixed multiplicities starts in 1957 with Bhattacharya's paper [1], where he described Hilbert polynomials for two ideals and their highest degree coefficients. The name mixed multiplicities first appeared in Teissier's paper [15], published in 1973, in which he and Risler studied Hilbert polynomials and their coefficients for any finite number of ideals by using superficial elements (of a then new kind). The connection between mixed multiplicities and joint reductions was first observed by Rees in 1983 during a research symposium associated with the Nordic Summer School. This was also the first time that joint reductions were denned. Rees' approach to mixed multiplicities, as displayed in his 1984 paper [11] and later in his book [13], is different from Teissier's; Rees studies mixed multiplicities via multiplicities of certain ideals in 'general' extensions of the original ring. Other authors have used Rees algebras and blow-ups for studying joint reductions and mixed multiplicities (see for example [3, 17]). A motivation for this work was Rees' multiplicity theorem (cf. [11, Corollary 3.8]) which relates multiplicities to reductions in quasi-unmixed local rings. A similar relation holds between mixed multiplicities and joint reductions, and that is the main theorem of this paper (Theorem 3.7). I am indebted to Jugal Verma for suggesting this problem and for supplying a proof in dimension 2 (see [16]). Rees' original theorem and Boger's generalization of it [2, Corollary 3.9] both follow from Theorem 3.7. For very different proofs of Rees' multiplicity theorem see also Rees and Sharp [14], Katz [3] and Kirby and Rees [4]. Kirby and Rees use Buchsbaum-Rim multiplicities and an elaboration of Rees' machinery from [12]. Kirby communicated to me that in a future version of [4] there will be a generalization of Theorem 3.7 of this paper. Section 1 gives the definition of joint reductions. The original concept of joint reductions, given by Rees in [12], was defined for d m-primary ideals in a local ring (R,m) of dimension d. O'Carroll's and Verma's definitions of joint reductions [8 and 18] are not as restrictive, and the definition of joint reductions in Section 1 here works in even more general settings. Section 1 also gives the machinery which is necessary for the main results. This machinery comes mostly from Teissier's rich paper [15]. Section 2 proves a few lemmas about joint reductions and mixed multiplicities. Some