In Prof. Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, there is in Appendix 6.2 a proof of Govorov & Lazard theorem that seems to me slightly wrong.

It is written (last line of page 712 and first line of page 713 of my corrected third printing 1999 edition of the book) that because of Corollary 6.6 there is a map $\gamma : C\to B$ that cancels the kernel of $\beta$. I do not think it is true because there is no reason for this kernel to be finitely generated.

I think it is sufficient to say that there is a map $\gamma : C\to B$ that cancels the finitely generated submodule $\{(\phi_1(b),-\phi_2(b) ) ; b\in B'\}$ of this kernel to finish the proof identically.

Am I correct ?

Edit: Below is attached an excerpt of the text from the book of the proof of the theorem, as I should have done earlier as @rschweib kindly reminded me (but I am lazy ...)

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and here is the text of Corollary 6.6 referred to in the previous proof

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    $\begingroup$ @rschwieb You are obviously right. I will try asap to do as you say. $\endgroup$ – brunoh Sep 3 '13 at 16:40

I think that you are right in both points (the proof is not correct, but it can be corrected easily). Eisenbud collects errata for his book, you can mail him.


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