Timeline for "Spec" of graded rings?
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25 events
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| Jul 11, 2022 at 18:28 | history | edited | LSpice | CC BY-SA 4.0 |
While this is on the front page, inline question link; ... -> …
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 28, 2015 at 18:28 | comment | added | Ben Wieland | @user40276 I did my exercise and I must restrict the claim: The free CDGA on an acyclic complex supported in degrees $2n$ and $2n+1$ is equivalent to the ground ring, as expected, but the the free CDGA on an acyclic complex supported in degrees $2n-1$ and $2n$ has nontrivial homology. I am using a homological differential that lowers degree; with the cohomological convention, the two cases switch. | |
| Jul 27, 2015 at 0:27 | comment | added | user40276 | @SeanTilson Ok!Thanks!I will check this reference. | |
| Jul 25, 2015 at 9:54 | comment | added | Sean Tilson | ...asking about the vanishing of operations in characteristic 0, this is a simple computation in group homology of symmetric groups with coefficients in $\mathbb{Q}$, there isn't any outside degree 0. Mike Mandell has a paper on comparisons of TAQ in many different settings that might be relevant. In particular, he spells out when things are equivalent to $E_{\infty}$-dgas. | |
| Jul 25, 2015 at 9:52 | comment | added | Sean Tilson | The definition of the DL operations has built in to it the choices of homotopies that make things commutative. Therefore, if the the algebra were already commutative you could take the constant homotopy and it would be evident that the operation vanished. The operations are also homotopy invariant in a certain sense (a weak equivalence between suitably bifibrant algebras etc.). So they non-vanishing of the operations tells you that it is not equivalent. This does not mean that vanishing gives you a strictification a priori. I would be happy for a reference for that as well. If you are ... | |
| Jul 25, 2015 at 9:26 | comment | added | user40276 | @SeanTilson Thanks, this helped me. But let me ask for a clarification. Do you mean that the category of DGAs and $E_{\infty}$-DGAs are not even weak equivalent (by some suitable equivalence, for instance, the usual one on simplicial categories)? Anyway, do you know have a reference for this condition for the strictification using Dyer-Lashof operations? | |
| Jul 25, 2015 at 9:16 | comment | added | user40276 | @BenWieland Thanks for the response. | |
| Jul 16, 2015 at 22:08 | comment | added | Ben Wieland | @user40276 a good exercise is to see that the free functor from chain complexes to CDGAs does not preserve weak equivalences. I think that it fails even with the simplest acyclic complex. Tyler Lawson uses a different failure of a Quillen adjunction to show that a particular model structure does not exist. | |
| Jul 16, 2015 at 12:00 | comment | added | Sean Tilson | @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? | |
| Jul 15, 2015 at 22:49 | comment | added | user40276 | @BenWieland I've never heard about this badness of dga in positive characteristic. Could you elaborate on this (references)? | |
| Jul 15, 2015 at 21:27 | answer | added | pro | timeline score: 2 | |
| S Jul 15, 2015 at 20:55 | history | suggested | user 1 |
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| Jul 15, 2015 at 19:44 | review | Suggested edits | |||
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| Sep 6, 2010 at 19:22 | answer | added | DamienC | timeline score: 3 | |
| Jul 15, 2010 at 3:11 | comment | added | Kevin H. Lin | For what it's worth, I am personally only interested in characteristic zero things (for the most part). | |
| Jul 15, 2010 at 2:19 | comment | added | Ben Wieland | Sometimes Lurie deals with $E_\infty$ rings, sometimes simplicial commutative rings. He never deals with commutative dgas, because they work badly (except in characteristic zero, where they're all the same). | |
| Jul 15, 2010 at 2:12 | comment | added | Tyler Lawson | ...but even having said that, it really does sound like what you're looking for is derived algebraic geometry. | |
| Jul 15, 2010 at 1:50 | comment | added | Tyler Lawson |
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.
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| Jul 15, 2010 at 0:16 | comment | added | Aaron Bergman | It's all the same stuff rationally. | |
| Jul 15, 2010 at 0:12 | comment | added | Kevin H. Lin | @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... | |
| Jul 15, 2010 at 0:01 | comment | added | Marty | To me, it looks like $G_m$-equivariant scheme theory... though maybe this is not useful for the more specific questions above. | |
| Jul 14, 2010 at 23:58 | comment | added | S. Carnahan♦ | 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. | |
| Jul 14, 2010 at 23:51 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Jul 14, 2010 at 23:39 | history | asked | Kevin H. Lin | CC BY-SA 2.5 |