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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N. Epstein was partially supported by a grant from the DFG (German Research Foundation). Y. Yao was partially supported by the National Science Foundation DMS-0700554.
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Epstein, N., Yao, Y. Criteria for flatness and injectivity. Math. Z. 271, 1193–1210 (2012). https://doi.org/10.1007/s00209-011-0910-y
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DOI: https://doi.org/10.1007/s00209-011-0910-y
Keywords
- Injective module
- Flat module
- Torsion-free module
- Divisible module
- h-divisible module
- Associated prime
- Coassociated prime