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In derived algebraic geometry there are several different setting,i.e., sometimes we use $E_{\infty}$ring,somethings we use dg-algebra,... It is for different situations. But could someone give some examples illustrating under what problem we use relevant "derived structure" ($E_{\infty}$ ring, dg-algebra...)? Some motivations?

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In mirror symmetry one of the things we are interested in is deformations of the relevant dg/A-infty algebras/categories. The relevant deformation spaces are, or should be, related to, for instance, quantum cohomology (see Barannikov-Kontsevich or Kontsevich's original homological mirror symmetry paper). As for E-infty rings, see Lurie's survey on elliptic cohomology for some motivation. – Kevin H. Lin Jul 26 '10 at 8:53
At least in the context of Toen-Vezzosi's HAG (vol 2, if I remember correctly), derived algebraic geometry is specifically simplicially-enriched algebraic geometry, while they use "complicial algebraic geometry" and "brave new algebraic geometry" for enrichment in complicial sets and spectra respectively. – Harry Gindi Jul 26 '10 at 8:58
A frequent remark is that dgas are only good in characteristic zero, while in positive characteristic one has to use simplicial rings or E_k or E_\infty rings. See for a case, where this thought is worked out. – Lennart Meier Jul 26 '10 at 12:46
up vote 5 down vote accepted

Here are the two motivations I know of:

Number 1 comes from algebraic topology. The definitive reference is , which explains it much better than I can. Very roughly, complex oriented cohomology theories are represented by E-oo rings and are classified by their group-laws. The group law is what the cohomoly theory does to the tensor product of line bundles. Some interesting group laws come from elliptic curves. So if you are very daring, you can take the moduli space of elliptic curves, and assign to each point the E-oo ring representing the cohomology theory fitting to the group law of the elliptic curve. The result should be a sheaf of E-oo rings on the moduli space of elliptic curves, or in other words a derived structure of E-oo rings.

Number 2 comes from virtual fundamental classes in algebraic geometry. Kontsevich in section 1.4 of suggested that for specific types of moduli spaces (those with a perfect obstruction theory) their should exist a derived structure of dg-algebras. The dg-algebra structure should come from local presentations as intersection of submanifolds as discussed in this question: Serre intersection formula and derived algebraic geometry? . Kontsevich suggested that via a Riemann-Roch formula you should get the virtual fundamental class needed in Donaldson-Thomas theory and Gromov-Witten Theory.

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I minor point is that complex orientations are not necessarily represented by $E_{\infty}$ ring spectra. Having an $E_{\infty}$ complex orientation is something one has to work for. See recent work of Niles Johnson and Justin Noel or the thesis of Matt Ando or Jim McClure. – Sean Tilson Oct 17 '12 at 21:56

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