Hi everyone, I hope this isn't too basic a question, but I'm looking for a citation or quick proof that localization of a simplicial $k$-algebra is flat.

If $R$ is a simplicial $k$-algebra, my definition of localization at an $f \in \pi_0 R$ is $R otimes_{k[x]}^L k[x]_f$ where $R$ is a $k[x]$-algebra via $x \mapsto f$. My definition of a flat map $A \to B$ is that $\pi_0 B$ is flat over $\pi_0 A$ and $\pi_i A \otimes_{\pi_0 A} \pi_0 B \to \pi_i B$ is an isomorphism. Let me know if these definitions are wrong too!