This question is similar to this one because it's asking about a possible roadmap towards learning some derived algebraic geometry (DAG). But it's also different, because the goal is not to form a research level maturity in DAG but just to go through an introduction to the main basic ideas of the subject.

Through this question I would like to gather some suggestions about running a cycle of a few self-learning seminars on DAG, with further info as follows:

1) It should be a cycle of around ten seminars (but this is kind of flexible), one per week, each of the duration of 50 minutes (with people belonging to the same institution).

2) Each seminar should be generally given by a different person. It should be reasonably independent from the technical details of the previous one, but it can assume the results and ideas outlined in all the previous seminars.

3) The background of the audience: "Hartshorne style" algebraic geometry, homological algebra, derived categories, derived functors, spectral sequences, some basics of algebraic stacks (not the theory of the cotangent complex), moduli spaces and some enumerative geometry, perhaps some elementary deformation theory. No background should be assumed in: homotopical algebra and simplicial methods, homotopy theory, model categories, higher category theory, topos theory, dg-categories, spectra (in the sense of algebraic topology). This doesn't mean that the subjects outside the above common background should not be present in the seminars: just that they can't be considered as understood (some seminars could be entirely devoted to introducing the basics of these prerequisites).

4) The seminars should emphasize the basic intuition, motivation, methods, and applications, rather than aiming at the greatest possible generality.

5) The "flavor" of DAG should be the most suitable for applications to "geometric" algebraic geometry (mainly over $\mathbb{C}$, as opposed to "arithmetic" algebraic geometry), not algebraic topology or homotopy theory.

6) It would be nice if at least one of the last seminars presented some actual application(s) of the theory: some instance in which DAG has been useful or even "unavoidable" to obtain some result(s). Applications to moduli spaces and enumerative geometry would be especially appreciated (I'm thinking maybe of the context of Joyce et al.'s construction of generalized DT invariants, and "shifted symplectic structures", but I'm not qualified to say more about that).

  • 2
    $\begingroup$ As you're in characteristic $0$, you can base things on dg algebras. Dg schemes are a fairly natural place to start - see Ciocan-Fontanine and Kapranov's papers on derived Quot and Hilb. Infinitesimally, there are Manetti's "Extended deformation functors" and Hinich's "dg coalgebras as formal stacks", the latter introducing simplicial stuff. My "Notes characterising higher and derived stacks concretely" give a quick characterisation of the modern approach, and "Constructing derived moduli stacks" links the older and newer approaches (but might feature too much homotopy theory for you). $\endgroup$ – Jon Pridham Jan 19 '18 at 9:47
  • $\begingroup$ @ Jon Pridham: thank you for your suggestions. $\endgroup$ – Qfwfq Jan 19 '18 at 15:29
  • $\begingroup$ Probably the best way to do it is not to do it, as it has really little applications to actual geometry. $\endgroup$ – Bernie Jan 21 '18 at 13:40
  • $\begingroup$ @Nostradamus JR: well,it would be nice to have a "non derived" application, but let's count as applications also a better understanding of phenomena and structures, such as Serre's intersection formula or the cotangent complex... $\endgroup$ – Qfwfq Jan 21 '18 at 14:10

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