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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, 2018 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, 2018 at 15:51
  • $\begingroup$ @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, 2019 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, 2019 at 7:39
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    $\begingroup$ There is an interesting, but quite limited, version of quotienting in the world of E_oo-rings: if A is an E_oo-ring, and x in pi_0(A) is a non-zero-divisor such that the map S{t} -> A from the free E_oo-ring on a generator in degree zero detecting x factors through the map Σ^oo_+ Z -> A, then the usual quotient A/x (cofiber of x:A -> A) is another E_oo-ring. These "strictly commutative" elements are quite restrictive (and hard to compute), but you can work with these sort of closed immersions. $\endgroup$
    – skd
    Apr 12, 2019 at 1:17

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