From the discussion at Hochschild cohomology and A-infinity deformations, 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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    $\begingroup$ 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. $\endgroup$
    – S. Carnahan
    Jul 14, 2010 at 23:58
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    $\begingroup$ To me, it looks like $G_m$-equivariant scheme theory... though maybe this is not useful for the more specific questions above. $\endgroup$
    – Marty
    Jul 15, 2010 at 0:01
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    $\begingroup$ It's all the same stuff rationally. $\endgroup$ Jul 15, 2010 at 0:16
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    $\begingroup$ 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. $\endgroup$ Jul 15, 2010 at 1:50
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    $\begingroup$ @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? $\endgroup$ Jul 16, 2015 at 12:00

2 Answers 2


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 one has to deal with simplicial commutative rings or $E_\infty$-ring spectra, like in Lurie's work).

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    $\begingroup$ The link to the paper From HAG to DAG seems to be dead - but the paper is available as arXiv:math/0210407. The author's website now seems to be here: math-beb.net/paper $\endgroup$ Jul 11, 2022 at 7:17
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    $\begingroup$ Thanks. Link updated in the answer. $\endgroup$
    – DamienC
    Jul 11, 2022 at 8:24

maybe these notes by Vezzosi can be helpful for some



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