Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 3 / 5 / 1 / Solution

A flat module is one for which tensoring preserves injections. This is equivalent to exactness of the tensor functor, since the tensor product of modules is always right exact: its universal balanced maps identify tensoring a cokernel with the cokernel of the tensored map.

Suppose $M$ is flat. For every inclusion $N^{\prime }\hookrightarrow N$, the map $N^{\prime }{\otimes }_{A}M\to N{\otimes }_{A}M$ is injective. The detection property proved above therefore makes its character-dual map surjective. Under the character Tensor-hom adjunction, this is exactlyEvery map from a submodule into $M^{\ast }$ thus extends, proving