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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). $\endgroup$ Jul 13, 2020 at 10:54

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