nLab
flat morphism in derived geometry

Let $A$ be an $E_{\mathrm{\infty }}$-ring and let $M$ be an $A$-module. $M$ is flat if

1. ${\pi }_{0}M$ is a flat module over ${\pi }_{0}A$ in the classical sense (i.e. ${\otimes }_{{\pi }_{0}A}M$ is an exact functor);

2. For each $n$, the induced map

   $${\pi }_{n}A{\otimes }_{{\pi }_{0}A}{\pi }_{0}M\to {\pi }_{n}M$$\pi_n A \otimes_{\pi_0 A} \pi_0 M \to \pi_n M

   is an isomorphism.

A map $A\to B$ of $E_{\mathrm{\infty }}$-rings is flat if $B$ is flat when regarded as an $A$-module.

The same definitions work for some other contexts of derived local algebra, e.g. dg-algebras.

A morphism $X\to Y$ of derived schemes is flat if for all affine subsets $U\subset X$ and $f(U)=V\subset Y$ the induced map on global sections $\Gamma (V)\to \Gamma (U)$ is flat as a map of $E_{\mathrm{\infty }}$-rings.