Mathematische Zeitschrift

, Volume 271, Issue 3, pp 1193–1210

Criteria for flatness and injectivity

Authors

    • Institut für MathematikUniversität Osnabrück
  • Yongwei Yao
    • Department of Mathematics and StatisticsGeorgia State University
Article

DOI: 10.1007/s00209-011-0910-y

Cite this article as:
Epstein, N. & Yao, Y. Math. Z. (2012) 271: 1193. doi:10.1007/s00209-011-0910-y

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.

Keywords

Injective moduleFlat moduleTorsion-free moduleDivisible moduleh-divisible moduleAssociated primeCoassociated prime

Mathematics Subject Classification (2010)

13C1113C05
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Copyright information

© Springer-Verlag 2011