Local Criteria for u-S-flat Modules

Résumé

Let R be a commutative ring and S a multiplicative subset of R. Following Zhang’s notion of
uniformly S-flat modules, we investigate local and homological properties of flatness in this relative setting.
Wefirst establish finite local criteria for uniformly S-flat modules under the assumption that a finite family
(f1, . ..,fp) of elements of R meets S. In particular, we show that an R-module M is uniformly S-flat if and only if its localizations Mfi
are uniformly Sfi-flat. We also obtain characterizations in terms of prime idealsdisjoint from S.
In the second part, we study the uniformly S-flat dimension of modules and the uniformly S-weak global
dimension of rings. Using the above finite local techniques, we establish prime ideal formulas and finite local
characterizations for these dimensions, extending classical results on weak global dimension to the uniformly
S-flat framework.