I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably correspond to the study of $p$-torsion of (co)homology. As far as I remember, $S^0/p$ is not a ring spectrum by a result of Schwede (so, one cannot consider modules over it); yet does there exist any method for "overcoming" this difficulty (and still obtaining a "non-algebraic" triangulated category) somehow? Here non-algebraic means that the category should not be "closely related" to the derived category of any abelian category. I suspect that there should exist certain categories of this sort since certain cohomology theories (including $K$-theory) cannot be defined in terms of complexes.
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3$\begingroup$ The endomorphism ring $E$ of $S^0/p$ is an associative ring spectrum, and is $p$-torsion (for odd $p$). Is that not "interesting for topologists"? $\endgroup$– Charles RezkCommented Sep 17, 2015 at 14:49
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1$\begingroup$ I would like to have a "topological" example of a triangulated category not admitting a dg-enhancement (i.e., it cannot be "close to" derived categories of abelian categories). Can you produce such an example using this ring? $\endgroup$– Mikhail BondarkoCommented Sep 17, 2015 at 15:13
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$\begingroup$ You may want to look up Shwede's papers The p-order of topological triangulated categories and The n-order of algebraic triangulated categories. There he introduced the notion of n-order which measures how badly a topological triangulated category fails to be algebraic. I don't think your question is answered in these papers, but the methods are certainly relevant. $\endgroup$– Karol SzumiłoCommented Sep 17, 2015 at 16:19
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$\begingroup$ Yes, I have read this paper of Schwede; yet I should (and probably would) have one more look at it. $\endgroup$– Mikhail BondarkoCommented Sep 17, 2015 at 16:39
2 Answers
Depending on exactly what you mean by "killed by $p$", the answer may be no. Let $\mathcal{C}$ be a stable $\infty$-category and let $\iota_{\mathcal{C}}$ be the identity functor from $\mathcal{C}$ to itself. If the ``multiplication by $p$'' map $p: \iota_{\mathcal{C}} \rightarrow \iota_{\mathcal{C}}$ is nullhomotopic, then $\mathcal{C}$ is $\mathbf{F}_p$-linear (and can therefore be obtained from a pretriangulated differential graded category over $\mathbf{F}_p$).
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$\begingroup$ Thank you!! Could you give some reference(s) for this argument? $\endgroup$ Commented Sep 17, 2015 at 18:14
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8$\begingroup$ If $\mathcal{C}$ is a stable $\infty$-category, then the endomorphisms of $\iota_{\mathcal{C}}$ has the structure of an $E_2$-ring spectrum (the "Hochschild cohomology" of $\mathcal{C}$). Making $\mathcal{C}$ $R$-linear is equivalent to giving an $E_2$-map from $R$ into this endomorphism ring. And $\mathbf{F}_p$ has a very simple presentation as an $E_2$-ring spectrum: you need only the single relation ``$p=0$'' (result of Mahowald when $p=2$, Hopkins for odd primes). $\endgroup$ Commented Sep 17, 2015 at 18:18
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1$\begingroup$ Though I am somewhat disappointed by not obtaining an example I want, I am certainly deeply grateful for your answer! $\endgroup$ Commented Sep 17, 2015 at 18:29
Yes. For $p$ a prime and $n > 0$, the Morava $K$-theories $K(n)$ and their connective versions $k(n)$ are associative (but not commutative) ring spectra with coefficient rings $\Bbb F_p[v_n^{\pm 1}]$ and $\Bbb F_p[v_n]$ respectively, and the homotopy categories of their module categories are triangulated categories in which $p=0$. These homotopy categories are equivalent to the derived categories of modules over their coefficient rings (this was studied by Franke and in this paper of Patchkoria, and is roughly a consequence of the coefficient rings having small homological dimension).
There has been quite a lot of work on this recently, though I'm having trouble finding references to more recent developments. I believe that it is still an open question whether these equivalences of categories lift to equivalences of triangulated categories, but that this problem is most difficult for $p$ and $n$ small.
(There are further variants of this for which we let the finite field, or a formal group law over it, vary.)
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$\begingroup$ Thank you; that's interesting! I should have a look at these things. Yet I am (currently) rather interested in categories not admitting a dg-enhancement (i.e., that are not "close to" derived categories of abelian categories). $\endgroup$ Commented Sep 17, 2015 at 15:11
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$\begingroup$ A related question: can one prove that the "Morava category" is not isomorphic to the corresponding derived category if $p=2$ and $n=1$ (or $n=2$)? $\endgroup$ Commented Sep 17, 2015 at 15:40
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$\begingroup$ @MikhailBondarko it is, as Tyler says. $\endgroup$ Commented Sep 17, 2015 at 16:04
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5$\begingroup$ @MikhailBondarko Certainly the category does admit a dg-enhancement, but it may be that this dg-enhancement only works for a different triangulated structure. $\endgroup$ Commented Sep 17, 2015 at 16:25
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2$\begingroup$ @TylerLawson I always thought it was a triangulated equivalence, but as you say Remark 5.2.4 in Patchkoria's paper says it's unknown. Interesting. $\endgroup$ Commented Sep 17, 2015 at 16:31