Let $A$ be a commutative ring, and $\alpha:F_1\to F_2$ a map of free modules.
Question: is the image $\text{im}\alpha$ flat?
I've looked at the standard references, e.g. Stacks Project, and can't find anything.
2026-03-28 02:42:57.1774665777
Is the image of a map of free modules flat?
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Consider the ideal $I=(x)/(x^2)$ in $\mathbb R[x]/(x^2)=R$.
It is the image of the homomorphism $R\to R$ given by multiplication by $x$.
Now if $R$ is a commutative local ring, and we know that a finitely generated flat module over such a ring is free. But that means $I$ cannot be flat: it is not free because it is annihilated by $x$.