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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?

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    $\begingroup$ The standard references for cotangent complexes of stacks are Laumon - Moret-Bailly and Olsson. $\endgroup$ – Jason Starr Sep 9 '14 at 11:17
  • $\begingroup$ @JasonStarr Thank you very much! Is there any illustrative description in easy cases? $\endgroup$ – Zhaoting Wei Sep 9 '14 at 19:55
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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.

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    $\begingroup$ Thanks @DavidRoberts for fixing the diagrams. $\endgroup$ – cgodfrey Mar 12 '19 at 20:14

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