I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.

  • I understand that the standard reference is Illusie's two books, unfortunately my French abilities are lacking

  • I'm aware of the stacks project treatment. It looks quite terse. Is there a more elementary treatment?

  • I think the nlab article is amazing, shame it's rather brief.

People who understand the cotangent complex: how did you first learn it?

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    $\begingroup$ I think Quillen's "Homology of Commutative Rings" is beautiful and extremely clear. You can find a link at the bottom of en.wikipedia.org/wiki/Daniel_Quillen. A concise and useful summary of the basic properties around the cotangent complex can be found in section 3.1 of Bhargav Bhatt's AWS2017 notes, found swc.math.arizona.edu/aws/2017/2017BhattNotes.pdf. You may also learn a lot trying to work out exercises 8 through 12 there. Finally, a quick treatment can be found in section 2 of the notes arxiv.org/abs/1606.01921 by Tamás Szamuely and Gergely Zábrádi. $\endgroup$ – Raymond Cheng Jan 30 '18 at 5:34

The homotopy-theoretic way to look at the cotangent complex is as the left derived functor of the Kahler differentials functor. Now:

  • This statement makes sense just in the context of homological algebra. i.e. in some category of chain complexes of modules. This route to constructing the cotangent complex is the subject of section 8.8 in Weibel's homological algebra book.

  • More generally we have notions of derived functors in any model category, a good reference is Dwyer and Spalinski. Constructing the cotangent complex at this level of generality is the subject of a very nice note by Zhou.

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  • $\begingroup$ It seems the link to the note is broken. Do you have a different link or another way I could access the paper? $\endgroup$ – Stahl Nov 16 '18 at 21:48
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    $\begingroup$ Have updated link. $\endgroup$ – zzz Nov 17 '18 at 0:11

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