# Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the setting of (infinity, 1)-categories, the stable (infinity, 1)-category Spt of spectra is the analogue of Ab. Hence in derived algebraic geometry one studied spaces locally modelled on E-infinity rings, i.e. commutative algebra objects in Spt.

Spt is by definition the stabilization of the category of spaces. In motivic homotopy theory (Morel-Voevodsky), one considers motivic spaces (A^1-homotopy invariant Nisnevich sheaves on the category of smooth k-schemes, say) and their stabilization MSpt (motivic spectra). Hence one could probably consider what might be called "motivic derived algebraic geometry", where spaces are locally modelled on commutative algebra objects in MSpt.

(Then one could consider derived motivic spaces, derived motivic spectra, and motivic 2-derived algebraic geometry, etc...)

I apologize for the rather ridiculous question, but: could this possibly result in something useful? Maybe another way to put the question is: how do we know that E-infinity rings give us the most complete picture? Or perhaps there are technical reasons that prevent one from using other categories than Spt?

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I have no idea what I'm talking about... but if such a thing existed where your generalized ring objects lived in motivic spectra, then perhaps you could build a motivic tmf a la Lurie? – Dylan Wilson Feb 9 '14 at 22:56
I guess everything depends on what your goal is. Derived or homotopical algebraic geometry started with the purpose of building a framework to prove some theorems, rather than a per se generalization of classical stuff. – Fernando Muro Feb 9 '14 at 22:56
I believe this is a sensible construction, since there are numerous objects in algebraic geometry (algebraic K-theory, algebraic cobordism etc) that seem to be formal counterparts of similar objects in classical homotopy theory, which respectively are often studied via algebraic geometry. However, such objects are very big and complicated sheaves, so I doubt that there you would be able to prove any complex geometric statements about them, except for some general abstract nonsense and calculations at specific points (which may be still very non-trivial and interesting). – Anton Fetisov Feb 10 '14 at 21:54
Is there a nice realization functor from motivic spectra to spectra that you could use to view these objects as derived schemes with extra structure? One might expect this to roughly be a derived version of the category of schemes with an action of $G$ for $G$ an algebraic group, where $G$ in this case is the motivic Galois group. Then adjoint functors between the new sort of schemes and usual schemes are critical in understanding the categories. These would be related to adjoint functors between spectra and motivic spectra going in the other direction. – Will Sawin Feb 11 '14 at 21:07
There is definitely an interest in determining what "motivic cohomology for derived schemes should be. I believe Rognes asks about this in both his ICM address and MSRI talk. There are typed notes (by Rognes) and videos for both. His reason for desiring such a thing relates to the usefulness of motivic homotopy theory in computing the K-theory of schemes. – Sean Tilson Sep 25 '14 at 13:44