Criteria for flatness and injectivity
arXiv:1103.4726 · doi:10.1007/s00209-011-0910-y
Abstract
Let be a commutative Noetherian ring. We give criteria for flatness of -modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if has characteristic , or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of -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.
19 pages