# nLab flat morphism in derived geometry

Let $A$ be an $E_\infty$-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

$\pi_n A \otimes_{\pi_0 A} \pi_0 M \to \pi_n M$

is an isomorphism.

A map $A \to B$ of $E_\infty$-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 $E_\infty$-rings.

Revised on October 14, 2009 15:09:02 by Zoran Škoda (161.53.130.104)