Definition 10.102.5. Let $R$ be a ring. Suppose that $\varphi : R^ m \to R^ n$ is a map of finite free modules.

The

*rank*of $\varphi $ is the maximal $r$ such that $\wedge ^ r \varphi : \wedge ^ r R^ m \to \wedge ^ r R^ n$ is nonzero.We let $I(\varphi ) \subset R$ be the ideal generated by the $r \times r$ minors of the matrix of $\varphi $, where $r$ is the rank as defined above.

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