nLab
flat morphism in derived geometry

Let AA be an E E_\infty-ring and let MM be an AA-module. MM is flat if

  1. π 0M\pi_0 M is a flat module over π 0A\pi_0 A in the classical sense (i.e. π 0AM\otimes_{\pi_0 A} M is an exact functor);

  2. For each nn, the induced map

    π nA π 0Aπ 0Mπ nM \pi_n A \otimes_{\pi_0 A} \pi_0 M \to \pi_n M

    is an isomorphism.

A map ABA \to B of E E_\infty-rings is flat if BB is flat when regarded as an AA-module.

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

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

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