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I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).

Let A,B be two hearts of bounded t-structures in a triangulated category D (which we may assume to be the homotopy category of a dg-category or $\pi_0$ of a stable $\infty$-category). Let E be an object. Then there is an $E_2$ (or $E_1$?) spectral sequence $H^p_A( H^q_B(E)) \Rightarrow H_A^{p+q}(E)$ -- or possibly with $p$ and $q$ swapped on the LHS.

Where $H^i_\bullet$ is the cohomology with respect to the t-structure $\bullet$. The point of the spectral sequence is that you can filter E in two ways, according to A or to B and then A.

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  • $\begingroup$ I guess the right book where to find such a thing would have been Kashiwara-Schapira's "Cats and Shvs", but they don't like spectral sequences so they don't use them. $\endgroup$
    – bananastack
    Commented Jun 17, 2014 at 16:55

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The t-structure corresponding to $B$ will give you a filtration of any object $X$

$$ \ldots \to X_{-1} \to X_0 \to X_{1} \to \ldots $$

such that the cofiber of the map $f_i: X_{i-1} \to X_i$ is equivalent to $H_B^i(X)[i]$.

Given any filtration of an object $X$ so that $ colim_i \, X_i \cong X$, under some hypotheses (stated below) you get a spectral sequence

$$ E_{r}^{p,q} \implies H_A^{p+q} colim_i \, X_i $$

whose $E_1$ page is given by

$$ E_1^{p,q} \cong H_A^{p+q} cofib(f_p). $$

The only hypotheses you need for this to hold are that the filtration $X_i$ is equivalent to the zero object for $i<<0$, and that the t-structure $A$ is compatible with sequential colimits. (This just means that the aisle $\mathcal{C}_A^{\leq 0}$ is stable under colimits of $\mathbb{Z}_{\geq 0}$-indexed diagrams.)

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  • $\begingroup$ thanks. do you have a reference I can point to? $\endgroup$
    – bananastack
    Commented Jun 17, 2014 at 17:46
  • $\begingroup$ I learned this from Chapter I of Lurie's Higher Algebra, but there may be a better introduction for people interested in a less $\infty$-categorical setting.... $\endgroup$
    – Hiro Lee Tanaka
    Commented Jun 17, 2014 at 17:49
  • $\begingroup$ I personally don't mind, but it's not to everyone's tastes... But maybe I can ask you a question: is there a simple way to rephrase the condition that the aisle be stable under colimits? And by rephrase I mean, say it in 1-categorical terms (like, it's closed under finite direct sums and taking cones). $\endgroup$
    – bananastack
    Commented Jun 17, 2014 at 18:10
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    $\begingroup$ A colimit in a triangulated category? $\endgroup$
    – Fernando Muro
    Commented Jun 17, 2014 at 18:40
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    $\begingroup$ @FernandoMuro: I'm sure Hiro means $\infty$-colimit or homotopy colimit. That's why I was hoping he would decategorify it a bit. (or am I misunderstanding your cryptic remark?) $\endgroup$
    – bananastack
    Commented Jun 17, 2014 at 18:58

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