A Geometric Theory of the Buchsbaum-Rim Multiplicity

Mixed buchsbaum-rim multiplicities

“…In fact, we define the multiplicities as the degrees of certain zerodimensional "mixed twisted" Segre classes, and we develop an encompassing general theory of these new rational equivalence classes in all dimensions. In parallel, we develop a theory of pure "twisted" Segre classes, and we recover the main results in [6] about the pure Buchsbaum-Rim multiplicities, the polar multiplicities, and so forth. Moreover, we identify the additivity theorem [6, (6.7b)(i), p. 205] as giving a sort of residual-intersection formula, and we show its (somewhat unexpected) connection to the mixed-multiplicity formula.…”

“…It turned out that these generalized multiplicities are nothing but the multiplicities introduced a decade before Teissier's work by Buchsbaum and Rim, who established many of their fundamental properties, but no mixed-multiplicity formula. Recently, the authors gave a general treatment of the Buchsbaum-Rim multiplicity, based on blowups and intersection numbers, in [6] (that paper also contains a more extensive history of the subject). On p. 225, the authors announced a mixed-multiplicity formula for an arbitrary pair of submodules.…”

A criterion for integral dependence of modules

“…Hence U is a reduction of E because N is faithful. If on the other hand d = (U ) − e + 1, one can use [21, 5.6] to conclude that again U is a reduction of E (see also [10], [12], [14], [20]). Remark 4.6.…”

“…Conversely, whenever this equivalence holds in a Noetherian local ring R, then R has to be equidimensional and universally catenary [16]. Rees' result has been generalized in various directions, most notably to modules of finite colength in a free module and, more generally, to ideals and then modules whose quotients have finite length or, yet more generally, have 'sufficiently high' codimension [2], [3], [8], [10], [11], [12], [17], [19], [21] ( [8] works for arbitrary modules U ⊂ E, but the codimension conditions are built into the definition of the multiplicity which uses a codimension filtration descending to the integral closure of the module). These criteria are all based on the Hilbert-Samuel or Buchsbaum-Rim 150 BERND ULRICH AND JAVID VALIDASHTI multiplicity, and variations thereof.…”

“…The purpose of the present article is to fill this gap and at the same time provide a self-contained, fairly short proof that implies the earlier algebraic results. In the complex-analytic setting Gaffney and Gassler [9] had given a criterion for integral dependence of arbitrary ideals that precedes [6] and uses the Segre numbers defined in [12].…”