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

1. The rank of $\varphi$ is the maximal $r$ such that $\wedge ^ r \varphi : \wedge ^ r R^ m \to \wedge ^ r R^ n$ is nonzero.

2. 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.

Comment #6598 by WhatJiaranEatsTonight on

Does $I(\phi)$ means the ideal generated by the determinants of $r\times r$ minors?

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• 2 comment(s) on Section 10.102: What makes a complex exact?

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