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In Rotman's book "Intro to homological algebra" Theorem 3.62 Let $0\rightarrow K\rightarrow\ F\rightarrow A\rightarrow 0$ be an exact sequence of right R-modules, where $F$ is free. The following are equivalent:

  1. $A$ is flat

  2. For every $v \in K$, there is an $R$-map $\theta:F\rightarrow K$ with $\theta(v)=v$.

My problem is that isn't the second condition implies that $A$ is a direct summand of free module $F$, hence projective. Then isn't this theorem is saying that flat implies projective? And we already know that projective modules are flat. So isn't this theorem saying that projective are equivalent to flat which is not true in general. For example $\mathbb Q$ as a $\mathbb Z$-module. So where am I wrong?

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    $\begingroup$ In TeX and in MathJax, there are special environments for enumerated lists, so don't abuse TeX's {align} environment (meant for math mode) to achieve the effect. $\endgroup$
    – LSpice
    Commented Nov 27, 2018 at 18:53
  • $\begingroup$ Why do you think that the second condition implies that $A$ is a direct summand of $F$? It would be if there is a single $\theta$ that works for all $v$; but if every $v$ requires different $\theta$ and you cannot find one $\theta$ that would work for all of them, then you cannot conclude that $A$ is a direct summand. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 5, 2018 at 9:09

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The point is that the map $\theta$ is allowed to depend on $v$. So this is not a splitting of the exact sequence.

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