# Vanishing of the cotangent complex for non-perfect algebras

Let $k$ be a perfect field of characteristic $p>0$. If $R$ is a perfect $k$-algebra, then the cotangent complex $L_{R/k}$ vanishes (the Frobenius is zero and induces an isomorphism on $L_{R/k}$ since $R$ is perfect). Is the converse true? Namely, if $L_{R/k}$ vanishes, then is $R$ perfect over $k$? I suspect not, but I can't find a counterexample. What happens if $k$ is a field of characteristic $0$?