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 ?



(Co)tangent complexes of quotient stacks
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: $(\mathfrak g \otimes \mathcal O\_X\to T\_X)$. 


