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I'm trying to see my way around the following

False Claim: Bounded spectra are the same as bounded chain complexes (as a triangulated category, say).

Dubious Proof: Consider the standard $t$-structure on $Spectra$. The bounded objects in this $t$-structure are those spectra with finitely many nonzero homotopy groups, which forms a full triangulated subcategory with an induced $t$-structure, which is bounded. The heart of this $t$-structure is the category $\mathbb Z \text{-}Mod$ of abelian groups. The functor from bounded $t$-structures to abelian categories sending a $t$-structure to its heart is an equivalence. Therefore we have an equivalence between bounded spectra and the bounded derived category of abelian groups.

Questions:

  1. Where did I go wrong?

  2. Is there some triangulated-category-theoretic concept which "measures" the difference between chain complexes and spectra?

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  • $\begingroup$ What does "the functor from bounded $t$-structures to abelian categories sending a $t$-structure to its heart is an equivalence" mean? As you correctly observe, chain complexes vs. spectra is a clear counterexample. $\endgroup$ Oct 20 '18 at 19:07
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    $\begingroup$ @Qiaochu Ah, thanks. I think I simply read a statement of the form "A bounded $t$-structure on a fixed triangulated category $\mathcal T$ is determined by its heart" and misinterpreted it as saying that "Any bounded $t$-structure is determined by its heart". However there are conditions under which a $t$-structure must be equivalent to the derived category of the heart -- cf HA 1.3.3.7. I suppose I should learn what that theorem actually says and see which conditions fail in spectra. $\endgroup$
    – tcamps
    Oct 20 '18 at 19:19
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As Qiaochu points out, I'm simply wildly wrong in believing that a bounded $t$-structure is determined by its heart -- although if you fix an ambient triangulated category $\mathcal T$ it is true that the bounded part of any $t$-structure on $\mathcal T$ is determined by its heart.

As alluded to in the comments, there is a recognition principle for determining when a $t$-structure is the derived category of its heart, given in Lurie's Higher Algebra, Prop 1.3.3.7 (used to spectacular effect by Gheorghe, Wang, and Xu). There is always a functor from the derived category of the heart to the original triangulated category. Lurie gives a recognition principle for when this functor is fully faithful and identifies its essential image. In the case of spectra, it boils down to asking whether $Ext_{Spectra}^\ast(\mathbb Z, \mathbb Z)$ vanishes for $\ast > 0$. Which it doesn't, because there exist nontrivial stable integral cohomology operations. Stable integral cohomology operations are not that familiar, so I'm a bit more comfortable $p$-completing and observing that the forgetful functor from $H\mathbb F_p$-modules to $p$-complete spectra is not fully faithful because the Steenrod algebra at $p$ is nontrivial.

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