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Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) no such conditions are known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in that generality.

I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!

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    $\begingroup$ At the cost of working with stable oo-categories with t-structure, Proposition 1.3.3.7 of Higher Algebra might interest you. The result doesn't specify when t-structures exist, but does give some criteria to check to see if C is the derived category of its heart. In particular, you need a left-complete t-structure on C with the property that every object is right-bounded, enough projectives in the heart, and you need certain Ext groups of projective objects to vanish. I'm sure there's some more powerful result out there, but this is the first one that comes to mind. $\endgroup$
    – Liam Keenan
    Commented Aug 20, 2020 at 1:50
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    $\begingroup$ As long as you accept to work with stable ∞-categories (which $DM$ is) t-structures are cheap. What we want is a t-structure with some special properties (e.g. that interacts well with Betti/étale realization), and that's a much harder problem. $\endgroup$
    – Denis Nardin
    Commented Aug 20, 2020 at 6:47
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    $\begingroup$ You might be interested in knowing that the problem of finding $t$-structures on a triangulated/stable category $\cal D$ is as difficult as the one of finding suitable factorization systems on $\cal D$: arxiv.org/abs/2005.14295 but there's also a similar result for old-fashioned, triangulated cats: sciencedirect.com/science/article/abs/pii/S0021869320300296 (the journal version has amended some typos). $\endgroup$
    – fosco
    Commented Aug 20, 2020 at 8:20

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A silly remark is that "trivial" $t$-structures always exist. You should probably say that you want a bounded or a non-degenerate $t$-structure. As far as I remember, non-zero negative $K$-groups of $T$ should give an obstruction for the former condition if you believe that the heart is noetherian or something like this. Note also that these groups for $DM_{gm}$ are isomorphic to that of Chow motives; see Sosnilo, Vladimir, Theorem of the heart in negative K-theory for weight structures. Doc. Math. 24 (2019), 2137–2158.

As for comparing $DM_{gm}$ with $D^b(MM)$: try to read (the introduction to?) Positselski, Leonid, Mixed Artin-Tate motives with finite coefficients. Mosc. Math. J. 11 (2011), no. 2, 317–402.

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    $\begingroup$ The K-theoretic obstruction for bounded t-structures can be found in an article of Antieau, Gepner and Heller: arxiv.org/abs/1610.07207 $\endgroup$
    – Lennart Meier
    Commented Aug 20, 2020 at 12:53
  • $\begingroup$ @MikhailBondarko: Thank you all for the wonderful answers, clarifications and references! These will be very helpful guidelights in what to me is a daunting selva oscura. $\endgroup$
    – user163840
    Commented Aug 20, 2020 at 15:29
  • $\begingroup$ You are welcome! $\endgroup$
    – Mikhail Bondarko
    Commented Aug 21, 2020 at 8:09

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