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Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X_{\alpha} \rightarrow X$, such that for each $\alpha$, there is a scheme $U_{\alpha}$ and a finite group $\Gamma_{\alpha}$ acting on $U_{\alpha}$ with the property that $\mathcal{X} \times_X X_{\alpha}$ is isomorphic to the stack theoretic quotient $[U_{\alpha}/ \Gamma_{\alpha}]$. Is something like this true for the Deligne-Mumford geometric derived stacks of Toen-Vezzosi? Vaguely, I understand the above statement to be true because of how etale maps are defined when your affine objects are actual schemes. (If $V$ is a scheme and $V \rightarrow \mathcal{X}$ is etale, the pullback of $V$ and the henselization of $X$ at a geometric point is finite over the henselization). If you have a much more general definition of etale, can this property be salvaged?

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Yes, these type of local structure theorems also hold for derived stacks, even more general ones such as Theorem 1.2 in https://arxiv.org/abs/1504.06467. If $\mathcal{X}$ is a derived stack (+ adjectives), apply such a result to the classical truncation $t\mathcal{X}$ to get an etale covering $(X_\alpha^0 \to t\mathcal{X})_\alpha$ where $X^0_\alpha$ is the quotient of an affine by $\Gamma_\alpha$. The etale site is invariant under derived nilpotent thickenings such as $t\mathcal{X} \to \mathcal{X}$, so this lifts uniquely to an etale covering $X_\alpha \to \mathcal{X}$. The affine map $X_\alpha^0 \to B\Gamma_\alpha$ uniquely extends to an affine map $X_\alpha \to B\Gamma_\alpha$ (which means that $X_\alpha$ is the quotient of an affine by $\Gamma_\alpha$). This is because the obstructions to this extension are controlled by the cotangent complex $L_{B\Gamma_\alpha}$ which is 0 if $\Gamma_\alpha$ is a finite group (or more generally $L_{B\Gamma_\alpha}$ is concentrated in degree $1$ at most which is enough to conclude if $\Gamma_\alpha$ is linearly reductive as in the Alper-Hall-Rydh theorem).

One place such a structure theorem is used (without actually justifying..) is Corollary 5.1(3) in Toen's Inventiones paper on derived Azumaya algebras.

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  • $\begingroup$ This is what I was looking for - thanks! $\endgroup$ – John Rached May 22 at 19:05

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