Classical Deligne–Mumford stack and cotangent complex

Let $k$ be a field. A classical Deligne–Mumford stack $X$ over $k$ is an algebraic stack with representable unramified diagonal and a surjective étale atlas $u:U\to X$ from a scheme. Its atlas groupoid is $R=U{\times }_{X}U\rightrightarrows U$, with source and target maps $s,t:R\to U$. The cotangent complex $L_{X\mathrm{/}k}$ is the quasi-coherent complex on the étale site of $X$ whose pullback to each scheme atlas $U$ is the scheme cotangent complex $L_{U\mathrm{/}k}$. The scheme cotangent complex is the sheaf-ring complex whose restriction to an affine chart $\mathrm{Spec}B\subseteq U$ is $L_{B\mathrm{/}k}$, representing derived $k$-derivations. Atlas descent data are induced by the canonical comparison along the two étale maps $s,t$. Indeed $L_{R\mathrm{/}U}=0$ for an étale morphism, so transitivity identifies $s^{\ast }{L}_{U\mathrm{/}k}\simeq L_{R\mathrm{/}k}\simeq t^{\ast }{L}_{U\mathrm{/}k}$; the groupoid identities give the cocycle condition. Étale descent then glues these local complexes, independently of the chosen atlas.

This convention is the cotangent complex used in the Behrend–Fantechi obstruction theory for classical Deligne–Mumford stacks. The definition does not assert a particular amplitude or perfectness; those are separate hypotheses on a map $E^{\bullet }\to L_{X\mathrm{/}k}$.

The cotangent complex of a morphism of schemes