Timeline for Which derived categories of coherent sheaves are equivalent (or "$t$-related") to derived categories of rings?
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6 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2020 at 23:46 | comment | added | SashaP | If such $R_X$ exists then Hochschild homology of $X$ is concentrated in non-negative homological degrees. In characteristic zero HKR theorem (for smooth $X$) $HH_i(X)=\bigoplus\limits_{p-q=i}H^q(X,\Omega^p)$ then implies that $H^q(X,\Omega^p)=0$ for $q>p$. If $X$ is smooth proper this implies that $H^{p,q}(X)=0$ when $p\neq q$, so the cohomology of such smooth proper variety would have to be Tate. | |
| Dec 17, 2020 at 20:13 | comment | added | Aaron Bergman | Well, for what it's worth, I believe all the del Pezzos have tilting objects and I believe Craw has proved it true for a class of toric varieties. I know more has been done, but I'd just end up googling "tilting object" and "tilting sheaf" to get a more up-to-date answer. | |
| Dec 17, 2020 at 19:09 | comment | added | Mikhail Bondarko | I am not willing to go to dg algebras since this will not yield a t-structure.:) And I have a trivial family of examples: one can certainly take $X$ to be an affine variety. | |
| Dec 17, 2020 at 19:07 | comment | added | Aaron Bergman | If you're willing to go to dg-algebras, I think smooth and proper is enough by Bondal and Van Den Bergh arxiv.org/abs/math/0204218 | |
| Dec 17, 2020 at 19:01 | comment | added | Leo Alonso | It may well be that this is only possible when the varieties are cohomologically simple, projective spaces, grassmannians, quadrics. I doubt there is such description in "general type" varieties, but this is just a guess. | |
| Dec 17, 2020 at 16:42 | history | asked | Mikhail Bondarko | CC BY-SA 4.0 |