The Atiyah class of DG manifolds of amplitude $+1$

A DG manifold of amplitude $+1$ encodes the derived intersection of a section $s$ and the zero section of a vector bundle $E$. In this paper, we compute the Atiyah class of DG manifolds of amplitude $+1$. In particular, we show that the Atiyah class vanishes if and only if the intersection of $s$ with the zero section is a clean intersection. As an application, we study the Atiyah class of DG manifolds that encodes the derived intersection of two smooth manifolds.