= Flatness criterion over dual numbers
{title2=$M\text{ flat over }k[\varepsilon]/(\varepsilon^2)\iff\ker\varepsilon=\varepsilon M$}
For $D=k[\varepsilon]/(\varepsilon^2)$, a <module> $M$ is a <flat module> exactly when $\ker(\varepsilon:M\to M)=\varepsilon M$. Flatness implies this by tensoring the exact sequence $0\to(\varepsilon)\to D\to k\to0$. Conversely lift a $k$-basis of $M/\varepsilon M$ to $M$. Every element is a finite combination of the lifts plus $\varepsilon$ times another such combination, proving generation over $D$. A relation first reduces to zero coefficients modulo $\varepsilon$; the kernel condition then reduces its remaining coefficients to zero as well. The lifts form a free $D$-basis, proving flatness.
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