# What is the difference between $D(\mathbb Z)$ and $Spectra$ in terms of $t$-structures?

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?

• 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. Oct 20 '18 at 19:07
• @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. Oct 20 '18 at 19:19

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.