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I want to learn more about differential graded algebras so that I can construct explicit examples of derived schemes over characteristic 0, compute smooth resolutions of morphisms of schemes, and compute examples of cotangent complexes of morphisms of schemes.

Where can I learn about differential graded algebras which will provide the computational tools to tackle these questions?

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  • $\begingroup$ If you want to learn differential homological algebra (I'm not familiar enough with algebraic geometry to know if this is what you need), I recommend Neisendorfer's "Algebraic methods in unstable homotopy theory", chapters 10 and 12. $\endgroup$ Jun 3, 2016 at 8:30

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In the same vein as zyx's answer, if you want to learn about cdgas and their applications to rational homotopy theory (before trying to use them in derived AG, which would make sense IMO), I can recommend these two books:

  • Yves Félix, Stephen Halperin, and Jean-Claude Thomas. Rational homotopy theory. Graduate Texts in Mathematics 205. New York: Springer-Verlag, 2001, pp. xxxiv+535. ISBN: 0-387-95068-0. DOI: 10.1007/978-1-4613-0105-9. MR1802847.
  • Yves Félix, John Oprea, and Daniel Tanré. Algebraic models in geometry. Oxford Graduate Texts in Mathematics 17. Oxford: Oxford University Press, 2008, pp. xxii+460. ISBN: 978-0-19-920651-3. MR2403898.

There is also this introduction that you can find online:

  • Kathryn Hess. “Rational homotopy theory: a brief introduction”. In: Interactions between homotopy theory and algebra. Contemp. Math. 436. Providence, RI: Amer. Math. Soc., 2007, pp. 175–202. DOI: 10.1090/conm/436/08409. MR2355774.

You also have the original papers:

  • Dennis Sullivan. “Infinitesimal computations in topology”. In: Inst. Hautes Études Sci. Publ. Math. 47.47 (1977), 269–331 (1978). ISSN: 0073-8301. Numdam: PMIHES_1977__47__269_0. MR0646078.
  • Daniel Quillen. “Rational homotopy theory”. In: Ann. of Math. (2) 90 (1969), pp. 205–295. ISSN: 0003-486X. JSTOR: 1970725. MR0258031.

But note that the two books I mentioned at the beginning have the advantage of having been written 20+ years after these things were discovered, the subject matter was a bit more "settled" and the exposition is a bit clearer.

The book by Félix/Halperin/Thomas also has a sequel, Rational homotopy theory. II (same authors, published last year) that goes over the same material as the first book but more quickly, and then explain how to apply the same techniques to non-simply connected spaces. The original paper by Sullivan already deals with non-simply connected spaces, he does the whole theory at once; I find it's a bit simpler to learn first about rational homotopy theory of simply connected spaces, then learn about non-simply connected ones.

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  • $\begingroup$ Felix-Halperin-Thomas also published a smaller article called "Differential graded algebras in topology" in the Handbook of Algebraic Topology. $\endgroup$ Jun 3, 2016 at 8:25
  • $\begingroup$ This algebraic models in geometry book looks amazing! $\endgroup$ Jun 15, 2016 at 1:11
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Here is a more recent and concise version on this: Jenny August, Matt Booth, Juliet Cooke, Tim Weelinck, "Differential Graded Algebras and Applications," Jan 2016, available at http://www.maths.ed.ac.uk/~mbooth/files/hodgeproject.pdf

This also has 31 references, that can help go deep in the sub-part where you need more details.

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DGA is usually understood first from the point of view of rational homotopy theory. The application to deformation problems and the interpretation of that as derived geometry came later.

Some references from the "classical period":

Griffiths and Morgan notes Rational Homotopy and Differential Forms

the 4-author paper on formality of Kaehler manifolds (Deligne-Griffiths-Morgan-Sullivan)

and the original papers by Sullivan and Quillen.

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  • $\begingroup$ I don't know about the notes by Griffiths and Morgan, but I really wouldn't start with the DGMS paper... It only contains a very terse account of cdgas in rational homotopy theory; if you don't already know what they're talking about, I think you would get lost rather quickly. $\endgroup$ Jun 3, 2016 at 7:37
  • $\begingroup$ DGMS was the paper that made DGA known to the wider world outside topology. It does the very valuable service of providing a condensed exposition that was sufficient for people with enough other background to get the high-level idea of what the theory is about, some examples, and a demonstration of how it can be used. It shows something about how people thought about things before the textbooks of the field were written, and why they were building the theory. $\endgroup$
    – zyx
    Jun 3, 2016 at 14:35
  • $\begingroup$ All of this is certainly true, but surely we can agree that a paper being historically significant doesn't mean that one's first exposure to the topic should be through it... I wouldn't recommend Poincaré's papers to someone trying to learn about the fundamental group. $\endgroup$ Jun 5, 2016 at 6:35
  • $\begingroup$ Reading it after, or in addition to, other material (such as Griffiths-Morgan) is not a "first exposure". I assume you were not recommending to use Sullivan's paper as a first textbook when including it in your answer, and DGMS is cited here for similar reasons. It is not a paper that one might immediately find and add to the reading list by starting from a modern account and tracing backward in the literature. Rather it is a useful exposition embedded in a paper on another subject. $\endgroup$
    – zyx
    Jun 5, 2016 at 14:52
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I think that the Manetti's Lectures on deformations of complex manifolds is sufficiently complete about differential graded algebras theory.

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