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I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived category.

However, in Lemma 2.1 and Proposition 2.2 of Zhang's paper(https://arxiv.org/pdf/1111.6294), the cotangent complexes of Artin stacks are elements of their derived category, where we consider their lisse-etale sites.

Is this correct ? (Did the author take the limit?) I am confused...

Any comment welcome! Thank you!

Edit:I would appreciate if you could tell me any references to cotangent complexes of Artin stacks。

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  • $\begingroup$ This is covered in Toën-Vezzosi Homotopical Algebraic Geometry II, at least. but I don't know of any books or papers that redevelop this in completely modern language (HAG II is a nice paper, but it is written in a language that people don't really use anymore). Jacob Lurie's book «Spectral Algebraic Geometry» specifically avoids dealing with Artin stacks (mentioned in the introduction). Anyway, the key thing to note about the cotangent complex is that it only makes sense as an object of the derived category. It is a fundamentally homotopical object (even back to Quillen). $\endgroup$
    – Harry Gindi
    Commented Jul 13, 2020 at 10:54

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There is an upgrade of Olsson's result in Laszlo-Olson: https://math.berkeley.edu/~molsson/article-Iweb.pdf The relevant part is Example 2.2.5, which applies Theorem 2.2.3 to the case of a hypercovering of Artin stacks. This gives an equivalence of the derived category of quasicoherent sheaves on the stack with the one of the hypercover (rather than just between the categories of projective systems), which together with Olsson's original paper gives the cotangent complex as an object in the derived category of quasi-coherent sheaves (in the lisse-étale site).

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  • $\begingroup$ Maybe I'm missing something, but I don't see how one passes from the pro-system which is the original definition of the cotangent complex to an element of the derived category of quasi-coherent sheaves on the hypercover. Is this also covered in Olsson's original paper somewhere, or is this something generally known? $\endgroup$
    – Stahl
    Commented Apr 2 at 0:54
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    $\begingroup$ In the original paper, he constructs a complex on the hypercover, and the pro-system is obtained by truncating this complex. This is only needed to use the descent result in that paper which doesn't work for the unbounded derived category. With the improved result, you simply omit passing to the pro-system. $\endgroup$
    – Nikolas Kuhn
    Commented Apr 3 at 13:52

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