Flat epimorphisms and silting epimorphisms coincide for commutative rings
Jan Šťovíček
TL;DR
This work proves that for a commutative ring $A$, flat ring epimorphisms $\lambda: A\to B$ coincide with silting ring epimorphisms, unifying classical generalized localizations with silting theory. The authors develop a framework linking torsion pairs, Gabriel filters, and Thomason subsets to characterize silting classes as divisibility or localization data, and they show flatness is equivalent to silting in the commutative setting. They establish a chain of localization concepts (localizations, universal localizations, silting epimorphisms, flat epimorphisms) and prove that, in the commutative case, these notions collapse to a single class; Thomason theory provides a geometric parametrization of these epimorphisms via subsets of $\mathrm{Spec}\,A$. The results have implications for generalized localizations, torsion theories, and the classification of ring epimorphisms in commutative algebra, offering a cohesive view of how silting and flatness interact in algebraic geometry and module theory.
Abstract
We investigate the relation between partial silting modules, Gabriel topologies, and ring epimorphisms, with a particular emphasis on commutative rings. We show that a ring epimorphism of commutative rings is flat if and only if it is a silting ring epimorphism.