Is the cotangent complexes of groupoids bounded above by degree $1$? Let $\mathcal{X}$ be a stack given by a groupoid $X_1\rightrightarrows X_0$, where $X_0$ and $X_1$ are smooth $k$-varieties. Let $\mathbb{L}_{\mathcal{X}/k}$ be the  cotangent complex of $\mathcal{X}$.
I've heard the fact that the amplitude of $\mathbb{L}_{\mathcal{X}/k}$ is bounded above by degree $1$, i.e. the cohomology of this complex vanishes for degree $>1$. I think this fact is well-known to experts but I cannot figure out a proof by myself. 
$\bf{My question}$ is: how to prove the above property? Is there any written-down proofs in the literature?
 A: Here's a proof assuming basic properties of the cotangent complexes of stack. At least for the proof to work I need to assume that $X_0$ is smooth over $k$ and the source/target maps $s, t: X_1 \to X_0$ of the groupoid structure are smooth (implying that $X_1$ is also smooth over $k$).  
Note: when I say "concentrated in degree 0" I mean "has cohomology only in degree 0." 
There's a cartesian diagram of the form 
$$
\begin{equation}
\begin{matrix}
X_1 & \xrightarrow{t} & X_0 \\
 s \downarrow & & \downarrow \pi \\
X_0 & \xrightarrow{\pi} &\mathcal{X} \\
\end{matrix}
\end{equation}
$$
By the construction of $\mathcal{X}$, $\pi$ is a smooth cover. It follows that 
$$
\begin{equation}
t^* L_{\pi} = L_{s}
\end{equation}
$$
Since $s$ is smooth by hypothesis, $L_s \simeq \Omega_{X_1/X}[0]$. So in particular $L_s$ is concentrated in degree 0. Since $\pi$ (and hence $t$) are smooth covers, by descent $L_\pi$ is concentrated in degree 0. 
Now consider the composition 
$$
\begin{equation}
X_0 \xrightarrow{\pi} \mathcal{X} \to \mathrm{Spec} \, k
\end{equation}
$$ and the resulting distinguished triangle 
$$
\pi^* L_{\mathcal{X}/k} \to L_{X_0/k} \to L_{\pi} \to \pi^* L_{\mathcal{X}/k}[1]
$$ in $\mathrm{D}(X_0)$. Since $X_0$ is smooth over $k$, $L_{X_0/k}$ is concentrated in degree 0, and we just showed $L_{\pi}$ is concentrated in degree 0. Taking the long exact sequence of cohomology sheaves completes the proof. 
