Functorial characterizations of flat modules
2017
We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism. We give functorial characterizations of finitely generated projective modules, flat modules and flat Mittag-Leffler modules.
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