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From the discussion here, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have nonzero degree.

So I have some naive and maybe stupid questions:

How can I interpret this geometrically? What is the "base space" of the deformation? What kind of object is it?

In other words, what is the "Spec" of a graded ring or a graded algebra (e.g. $k[t]$ or $k[[t]]$ or $k[t]/(t^n)$ with the variable $t$ having some nonzero degree)?

(..... maybe what I'm really asking is: Is there a theory of "schemes" where the "affine schemes" correspond to graded commutative rings rather than commutative rings? .....)

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This looks like derived algebraic geometry (for which there seems to be more than one school of thought), but where the differentials vanish. In this setting I think one usually views Spec of a graded ring as ringed space whose underlying topological space is Spec of the degree zero subring, but with a graded sheaf of functions. –  S. Carnahan Jul 14 '10 at 23:58
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To me, it looks like $G_m$-equivariant scheme theory... though maybe this is not useful for the more specific questions above. –  Marty Jul 15 '10 at 0:01
    
@Scott: I was going to guess that it had something to do with "derived algebraic geometry" -- whatever that means... But in for instance Lurie's stuff (which is the only thing under the heading of "DAG" that I've read), it seems that he deals with $E_\infty$ ring spectra, rather than dg rings or dg algebras... –  Kevin H. Lin Jul 15 '10 at 0:12
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It's all the same stuff rationally. –  Aaron Bergman Jul 15 '10 at 0:16
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To amplify Aaron's comment, in characteristic zero $E_\infty$ ring spectra are an equivalent notion to commutative DGAs. One needs to be slightly careful tossing around the derived-algebraic-geometry or $\mathbb{G}_m$-equivariant monikers, though, because Hochschild cohomology (or something like it) really classifies deformations as an associative DGA, not a commutative one. –  Tyler Lawson Jul 15 '10 at 1:50

1 Answer 1

One possible answer is in Toën-Vezzozi paper From HAG to DAG, who were themselves inspired by Ciocan-Fontanine and Kapranov (Derived Quot schemes and Derived Hilbert schemes).

This approach works well in characteristic zero (otherwise ne has to deals with simplicial commutative rings or $E_\infty$-ring spectra, like in Lurie's work).

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