Criteria for flatness and injectivity

TL;DR

This paper establishes new criteria for flatness and injectivity of modules over Noetherian rings using associated and coassociated primes, torsion-freeness, and divisibility, with applications to regularity and endomorphisms.

Contribution

It introduces novel criteria for flatness and injectivity of modules based on prime-related properties, extending to regularity and endomorphism contexts.

Findings

Criteria for flatness via associated primes and torsion-freeness.

Criteria for injectivity via coassociated primes and divisibility.

Development of tools for analyzing divisibility and base change effects.

Abstract

Let $R$ be a commutative Noetherian ring. We give criteria for flatness of $R$-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if $R$ has characteristic $p$, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of $R$-modules in terms of coassociated primes and (h-)divisibility of certain $\Hom$-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a $\Hom$-module base change, and a local criterion for injectivity.