Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities

Abstract

In this paper, we explore mixed Buchsbaum--Rim multiplicities of finite-colength submodules of free modules over Noetherian local rings. We extend the Hilbert-polynomial construction to submodules in arbitrary homogeneous degrees and prove a product formula for the resulting coefficients. We then examine module analogues of Teissier's coefficient inequalities and the Minkowski inequality. Explicit examples show that the coefficientwise power inequalities, log-convexity, and the product-root inequality can all fail; several failures occur over regular local rings. We also prove that equality of integral closures forces equality of all mixed Buchsbaum--Rim multiplicities, while a one-dimensional example shows that the converse fails for arbitrary modules. A converse is recovered when the module closures are the integral closures of modules of the form $IF$ for $\mathfrak m$-primary ideals $I$. The general converse problem in dimension at least two is stated at the end.