Mixed multiplicities and Rees Theorems for graded families and their converses
Abstract
Let (R,m) be a Noetherian local ring and let ℐ(1),…,ℐ(s) be graded 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 {xij} of type d=(d1,…,ds), with xij∈ Iaij(i), satisfies the formula e((xij);R) = (∏i,jaij) e ( ℐ(1)[d1],…, ℐ(s)[ds] ). 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.
Related
- Hopkins-Levitzki Type Theorems for Groupoid Graded Rings
- Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities
- Asymptotic multiplicity sequences of graded families
- Normality of ideals beyond the standard graded setting: families from numerical semigroup rings
- S-integrality for families of ordinary K3 surfaces and algebraicity theorems
BibTeX
@misc{freitas2026mixed,
title = {{Mixed multiplicities and Rees Theorems for graded families and their converses}},
author = {T. H. Freitas and V. H. Jorge Pérez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23202},
eprint = {2609.23202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23202},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mixed multiplicities and Rees Theorems for graded families and their converses
AU - T. H. Freitas
AU - V. H. Jorge Pérez
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23202
UR - https://arxiv.org/abs/2609.23202
AB - Let (R,m) be a Noetherian local ring and let ℐ^(1),…,ℐ^(s) be graded 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 d=(d_1,…,d_s), with x_ij∈ I_a_ij^(i), satisfies the formula e((x_ij);R) = (∏_i,ja_ij) e ( ℐ^(1)[d_1],…, ℐ^(s)[d_s] ). 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.
ER -