Mathematische Zeitschrift

, Volume 271, Issue 3, pp 1193-1210

First online:

Criteria for flatness and injectivity

  • Neil EpsteinAffiliated withInstitut für Mathematik, Universität Osnabrück Email author 
  • , Yongwei YaoAffiliated withDepartment of Mathematics and Statistics, Georgia State University

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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.


Injective module Flat module Torsion-free module Divisible module h-divisible module Associated prime Coassociated prime

Mathematics Subject Classification (2010)

13C11 13C05