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Let $R$ be a commutative ring (noetherian if needed) and $n,m$ be two nonnegative integers. Consider a map

$\varphi: R^n\rightarrow R^m$

Is there a characterisation, e.g. in terms of the matrix representation of $\varphi$ of the cokernel of this map being free?

remark: this question seems to be quite similar and I just learned from it that there is a criterion for when the module is projective in terms of minors of the matrix representing $\varphi$. So maybe this question is to strong as it asks essentially to classifiy the free modules among the projectives, but is there at least a sufficient criterion in nice situations?

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jorst
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1 Answers1

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You need ALL $(k+1)\times (k+1)$ minors for some $k$ to be zero and one $k\times k$ minor to be a unit.

Mohan
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  • You again :) Thanks, that is exactly the kind of thing I was hoping for! – jorst Jul 29 '15 at 09:10
  • just to fill in some details for other people reading this (correct me if I'm wrong please): From the condition described it follows that there are basis (does this have a plural?) such that $\varphi$ takes the form of a block matrix with one entry the $k\times k$ identity matrix and everything else $0$'s. – jorst Jul 29 '15 at 09:48
  • The given conditions ensure us that the module is f.g. projective of constant rank. I don't think that's enough to conclude the freeness. (The argument in the above comment works only for local rings.) However, the freeness follows for semilocal noetherian rings. – user26857 Jul 29 '15 at 17:32
  • @user26857, the condition I state is more than necessary. In fact, the second condition is not really necessary and can be replaced with the weaker condition that it be satisfied after multiplying by the appropriate size non-singular matrices on the left and right. This is in fact necessary and sufficient. – Mohan Jul 29 '15 at 20:30
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    Still not convincing. Please try to provide a full proof instead of an one-liner answer. (Meanwhile, I suggest you to have a look at Proposition 1.4.10 in Bruns and Herzog.) – user26857 Jul 30 '15 at 05:41