Welcome to MathOverflow

Question

8

4

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? .....)

edited Jul 14 2010 at 23:51

asked Jul 14 2010 at 23:39

Kevin Lin
13k●2●34●115

6

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 2010 at 23:58

5

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 2010 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 Lin Jul 15 2010 at 0:12

4

It's all the same stuff rationally. – Aaron Bergman Jul 15 2010 at 0:16

1

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 2010 at 1:50

show 3 more comments

DamienC · Answer 1 · 2010-09-06 19:22:03Z UTC

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).

“Spec” of graded rings?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.