Semir Mujevic | Searcharxiv

arXiv2023

Uniform arithmetic in local rings via ultraproducts

We reinterpret various properties of Noetherian local rings via the existence of some $n$-ary numerical function satisfying certain uniform bounds. We provide such characterizations for seminormality, weak normality, generalized Cohen-Macaulayness, and $F$-purity, among others. Our proofs that such numerical functions exist are nonconstructive and rely on the transference of the property in question from a local ring to its ultrapower or catapower.

math.AC↗

arXiv2022

Homotopical models for metric spaces and completeness

Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories $\mathbb{R}_+\text-\mathbf{Cat}$ and $\mathbb{R}_+\text-\mathbf{Cat}^{\mathrm{sym}}$ of Lawvere metric spaces and symmetric Lawvere metric spaces, each of which captures different features pertinent to the study of metric spaces. More precisely, in the three model structures we construct, the fibrant-cofibrant objects are the extended metric spaces (in the usual sense), the Cauchy complete Lawvere metric spaces, and the Cauchy complete extended metric spaces, respectively. Finally, we show that two of these model structures are unique in a similar way to the canonical model structure on $\mathbf{Cat}$.

math.CT↗