Jacob Lurie's stuff seems to develop derived algebraic geometry via $E_\infty$ rings and/or maybe something like simplicial commutative rings. Ben Wieland's comment in this question indicates that Lurie never deals with commutative dg algebras. However, it is supposed to be true that all of these different things are the same (meaning more precisely that their model categories are Quillen equivalent) in characteristic zero.

So my question is:

Is the theory of derived algebraic geometry via dg rings or dg algebras in characteristic zero developed anywhere? If not, why not?

My motivations:

  1. I feel like there must be a good reason why Lurie does not use dg rings/algebras, other than the fact that they apparently don't work well in positive characteristic. So I wonder what the reasons are.

  2. I don't know very much about homotopy theory, so I find the $E_\infty$ rings approach to DAG a bit daunting. I am personally more comfortable with dg algebras.

  3. I am personally more interested in things involving "sheaves of dg algebras" than things involving "sheaves of $E_\infty$ rings" (such as elliptic cohomology (and TMF), which I understand is one of Lurie's motivations).

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    $\begingroup$ You may want to take a look at the work of Toen and Vezzosi, e.g. arxiv.org/abs/math/0210407 $\endgroup$ – Greg Stevenson Aug 31 '10 at 1:37
  • $\begingroup$ Thanks. That (and the papers of Ciocan-Fontanine and Kapranov) seems to be precisely what I am looking for. $\endgroup$ – Kevin H. Lin Aug 31 '10 at 1:58
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    $\begingroup$ You may find that many homotopy theorists really will not think twice about casually conflating "sheaves of DG-algebras" and "sheaves of $E_\infty$ rings" in characteristic zero, precisely because of the strong way in which they're known to have equivalent homotopy theories. I can't put words in Lurie's mouth, but I think that the answer to (1) really is that dg-objects don't cover many of the contexts homotopy theorists are interested in. You may just need to wait a little for references to come out that specialize Lurie's work to dg-algebras. $\endgroup$ – Tyler Lawson Aug 31 '10 at 2:19
  • $\begingroup$ This equivalence is a Quillen equivalence of model categories, or ...? $\endgroup$ – Kevin H. Lin Aug 31 '10 at 2:26
  • $\begingroup$ @Kevin: Yes, exactly. $\endgroup$ – Tyler Lawson Aug 31 '10 at 4:51

Dear Kevin,

This is more or less an amplification of Tyler's comment. You shouldn't take it too seriously, since I am certainly talking outside my area of expertise, but maybe it will be helpful.

My understanding is that homotopy theorists are extremely (perhaps primarily) interested in torsion phenomena. (After all, homotopy groups are often non-trivial but finite.) TMF, for example, involves quite subtle torsion phenomena. Coupled with Tyler's remark that homotopy theorists have no fear of $E_{\infty}$ rings, and so are (a) happy to identify them with dg-algebras in char. zero, and (b) don't feel any psychological need to fall back on the crutch of dg-algebras, this makes me suspect that your assumption (1) is likely to be wrong. (I share your motivation (2), but this is a psychological weakness of algebraists that homotopy theorists seem to have overcome!)

In particular, one of Lurie's achievements is (I believe) constructing equivariant versions of TMF, which (as I understand it) involves (among other things) studying deformations of $p$-divisible groups of derived elliptic curves. It seems hard to do this kind of thing without having a theory that can cope with torsion phenomena.

Also, when Lurie thinks about elliptic cohomology, he surely includes under this umbrella TMF and its associated torsion phenomena. (So your (3) may not include all the aspects of elliptic cohomology that Lurie's theory is aimed at encompassing.)

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    $\begingroup$ It's not quite true that homotopy theorists are only interested in torsion (and integrality) phenomena; rational homotopy theory is a thriving field (though with techniques that are rather different from the rest of homotopy theory). However, <em>stable</em> rational homotopy theory is completely trivial (or as trivial as the theory of graded rational vector spaces is). $\endgroup$ – Torsten Ekedahl Aug 31 '10 at 4:06
  • $\begingroup$ I think this is a fairly good characterization. I can't help but comment on two things. First, as in ordinary characteristic zero algebraic geometry, the derived elliptic curves and their p-divisible groups are not themselves torsion objects, but are merely representing objects for torsion, and so this part of the story still doesn't strictly require $E_\infty$-algebras. $\endgroup$ – Tyler Lawson Aug 31 '10 at 4:56
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    $\begingroup$ Second, $E_\infty$-algebras are often studied in models where they are strictly commutative monoids - we use the same terminology to reflect that there is an underlying structure independent of which model is chosen. So far as "fear" of $E_\infty$-algebras, I might add an analogy with little mathematical content. Injective modules might be initially less intuitive than projective ones. However, this does not mean that algebraic geometers are necessarily brave when basing homological algebra on injective sheaves; the alternative is simply not always available to them! $\endgroup$ – Tyler Lawson Aug 31 '10 at 5:02
  • $\begingroup$ Dear Torsten and Tyler, Thank you for these clarifying remarks. $\endgroup$ – Emerton Aug 31 '10 at 11:48
  • $\begingroup$ @TylerLawson I like the example with injective/projective! Not only do we need injective sheaves but in retrospect one realizes that injective modules are in fact much easier to classify then projective ones (each splitting as a sum of injective hulls of residue fields) and so are in this sense even simpler then their more intuitive counterparts. $\endgroup$ – Saal Hardali Oct 11 '17 at 21:50

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