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
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
It's all the same stuff rationally. – Aaron Bergman Jul 15 '10 at 0:16
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
@user40276 In positive characteristic, (graded) commutative DGAs are not the same as $E_{\infty}$-DGAs. Dyer-Lashof/Steenrod operations are obstructions to strictification and they do not generically vanish unless you are in characteristic $0$. If you want a model structure, you have to consider these weaker notions as they are homotopical and the strict notion is not. Does this help? – Sean Tilson Jul 16 '15 at 12:00

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).