Mittag-Leffler modules and variants on intersection flatness

Abstract

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.