A Free Module
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A Free Module
Let $A$ be a noetherian local domain with residue field $k$ and quotient field $K$. If $M$ is a finitely generated $A$-module and if \[ \dim_{k} M \otimes_{A} k = \dim_{K} M \otimes_{A} K = r \]then $M$ is a free $A$-module of rank $r$.
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