show/hide this revision's text 2 minor edits

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).
show/hide this revision's text 1

Derived algebraic geometry via dg rings?

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 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 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, which I understand is one of Lurie's motivations).