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? .....)

`$E_\infty$`

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 anassociativeDGA, not a commutative one. – Tyler Lawson Jul 15 '10 at 1:50