Mixed multiplicities and Rees Theorems for graded families and their converses

Abstract

Let $(R,\mathfrak m)$ be a Noetherian local ring and let $\mathcal I^{(1)},\ldots,\mathcal I^{(s)}$ be graded $\mathfrak m$-primary families. We introduce weak asymptotic and strong homogeneous joint reductions and prove corresponding extensions of Rees's mixed multiplicity theorem relating joint reductions and mixed multiplicities. Actually, for weak asymptotic joint reductions, mixed multiplicities are realized as normalized limits of Hilbert--Samuel multiplicities, yielding an asymptotic Rees theorem. For multigraded admissible collections, a strong homogeneous joint reduction $\{x_{ij}\}$ of type $\mathbf d=(d_1,\ldots,d_s)$, with $x_{ij}\in I_{a_{ij}}^{(i)}$, satisfies the formula \[ e((x_{ij});R) = \left(\prod_{i,j}a_{ij}\right) e\!\left( \mathcal I^{(1)[d_1]},\ldots, \mathcal I^{(s)[d_s]} \right). \] We also prove a localized converse under formal equidimensionality, recovering the classical converse of Rees in the adic case. As an application, we derive a Böger-type theorem for graded families, whose adic specialization recovers Böger's classical theorem. In contrast, the converse to the weak asymptotic Rees theorem fails even for an adic family on a one-dimensional regular local ring.