Derived algebraic geometry via dg rings? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:07:37Z http://mathoverflow.net/feeds/question/37202 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37202/derived-algebraic-geometry-via-dg-rings Derived algebraic geometry via dg rings? Kevin Lin 2010-08-31T00:44:54Z 2010-08-31T23:43:15Z <p>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 <a href="http://mathoverflow.net/questions/31928/spec-of-graded-rings" rel="nofollow">this question</a> 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.</p> <p>So my question is:</p> <blockquote> <p>Is the theory of derived algebraic geometry via dg rings or dg algebras in characteristic zero developed anywhere? If not, why not?</p> </blockquote> <p>My motivations:</p> <ol> <li><p>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.</p></li> <li><p>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.</p></li> <li><p>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).</p></li> </ol> http://mathoverflow.net/questions/37202/derived-algebraic-geometry-via-dg-rings/37210#37210 Answer by Emerton for Derived algebraic geometry via dg rings? Emerton 2010-08-31T02:54:49Z 2010-08-31T02:54:49Z <p>Dear Kevin,</p> <p>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.</p> <p>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!)</p> <p>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.</p> <p>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.)</p>