Do notions of algebraic cycles and motives extend to Lurie's derived and/or spectral schemes? Would one expect to construct them more easily in these more "flexible" settings, and then carry them over to ordinary schemes with $\pi_0$?

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    $\begingroup$ I don't think spectral algebraic geometry is ready to work with Chow groups: it doesn't even have a good notion of closed subscheme. Dunno about DAG but that looks a little bit more promising $\endgroup$ – Denis Nardin Sep 22 '18 at 8:07
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    $\begingroup$ I don't know the answer, but this thesis preschema.com/thesis/thesis.pdf and this paper arxiv.org/abs/1705.03340 seem relevant. $\endgroup$ – Phil Tosteson Sep 22 '18 at 15:51
  • $\begingroup$ @DenisNardin do you think you know a reference where the difficulty with closed immersions in spectral algebraic geometry is discussed? I am just interested in a question where this may be relevant. $\endgroup$ – Aknazar Kazhymurat Apr 3 at 7:20
  • $\begingroup$ @AknazarKazhymurat I don't know of any place where this particular difficulty is discussed, but if you look at the literature you'll find there's no definition anywhere of a "closed subscheme" of a derived scheme or of a "surjective map" of ring spectra. One approach is using Smith ideals, but you'll notice that in that point of view every map of ring spectra is a quotient map. You can do more sophisticated things but on the whole there's no theory I am aware of (cont.). $\endgroup$ – Denis Nardin Apr 3 at 7:38
  • $\begingroup$ (cont) There's this document but to my knowledge, the notion developed there has not been used anywhere and it doesn't satisfy some of the properties I'd expect from "closed immersions of spectral schemes" (e.g. for q-compact closed subschemes to be in bijection with finitely presented idempotent algebras) $\endgroup$ – Denis Nardin Apr 3 at 7:39

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