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I am reading the section on differentials in Eisenbud's book (Commutative Algebra), and I'm just wondering what he means in sentences like this one:

"Suppose that $J:R^t \rightarrow R^r$ is a map of free modules over a ring $R$ whose rank is less than or equal to $c$, as for the Jacobian matrix of an ideal of codimension $c$..." (Chapter 16.7, Page 407)

I'm not sure what "rank" stands for in this generality (where the image need not be free). Vanishing of minors?

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You might check out section 1.4 of Bruns and Herzog's "Cohen Macaulay Rings", starting at page 20. Definition 1.4.2 and Proposition 1.4.3 might be helpful. – mbrown Sep 2 '13 at 3:20
up vote 2 down vote accepted

Presumably it means the largest $k$ such that the induced map $\Lambda^k(J) : \Lambda^k(R^t) \to \Lambda^k(R^r)$ on exterior powers doesn't vanish. (This is a coordinate-free restatement of a condition on vanishing of minors.) At least, that would be my guess. Does the rest of the statement make sense with this interpretation?

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I would have to brush up on my exterior powers (and I will!) to see if your interpretation fits as well, but I do think mbrown's lines up pretty nicely with what Eisenbud is doing. The $R$s that Eisenbud actually deals with wind up being domains, so it seems reasonable that he could mean the rank of the map induced by localization at 0 (he does mention vanishing of minors though). – Cass Sep 2 '13 at 4:04
@Cass: if $R$ is a domain, the two conditions should be equivalent. – Qiaochu Yuan Sep 2 '13 at 4:28

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