TITLE: (Co)tangent complexes of quotient stacks
QUESTION [6 upvotes]: Let $X$ be an algebraic variety over a field $\mathbb{K}$ equipped with a right action of a smooth algebraic group $G$.
One can form the quotient stack $[X/G]$. My question is probably quite elementary for algebraic geometers: what is the tangent complex of such a quotient stack over a given $\mathbb{K}$-point $Spec(\mathbb{K})\rightarrow [X/G]$ ? Is there a good reference where such an example is well explained ?

REPLY [13 votes]: First of all remember that differentiating the action of $G$ at the identity gives you a Lie algebra morphism $\mathfrak{g}\to\Gamma(T_X)$, and thus, for any point $x\in X$, a map $\mathfrak g\to T_xX$. 
Now pick a lift $x:Spec(\mathbb{K})\to X$ of your point $[x]:Spec(\mathbb{K})\to [X/G]$. 
Claim: $\mathbb{T}_{[x]}[X/G]$ is (quasi-isomorphic to) the two term complex $(\mathfrak{g}\to T_xX)$. 
Note that when the action is formally locally free at $x$ then $\mathfrak g\to T_xX$ is injective and thus $\mathbb{T}_{[x]}[X/G]=T_xX/\mathfrak g$ is the naive tangent space to the "local orbit space" at $x$. 
More generally, the tangent complex of $[X/G]$ is the follwing two term complex of $G$-equivariant $\mathcal O_X$-modules: $(\mathfrak g \otimes \mathcal O_X\to T_X)$.