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arXiv · 1103.4726

Criteria for flatness and injectivity

Neil Epstein·Yongwei Yao
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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.

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Neil Epstein, Yongwei Yao. 2011-03-24. Criteria for flatness and injectivity. https://doi.org/10.1007/s00209-011-0910-y

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