# arXiv reviews 8: no nonconnective theorem of the heart

Ramzi, Sosnilo, and Winges have proved a lovely result in [6], showing that every spectrum $M$ is equivalent to $\K(\Cscr)$ for some idempotent complete stable $\infty$-category $\Cscr$. In fact, they prove that there is a functor $\Cscr_{(-)}\colon\Sp\rightarrow\Cat_\infty^\perf$ such the composition $\K\circ\Cscr_{(-)}\we\id_{\Sp}$. The stable $\infty$-category $\Cscr_M$ is a categorification of $M$.

# History

I will not go into the interesting proof of this result which uses trace methods and Goodwillie calculus. Rather, I want to discuss their application to a conjecture of Gepner, Heller, and mine.

In [2] we made three conjectures, building on previous conjectures of Schlichting [7].

One was that for $X$ a noetherian scheme of finite Krull dimension, if $X$ admits a local ring which is not regular, then $\Perf(X)$ does not admit a bounded $t$-structure. This was proved by Smith in [8] using [1] in the affine case and then in general by Neeman in [5]. The second was that if $\Cscr$ is a stable idempotent complete stable $\infty$-category with a bounded $t$-structure, then $\K_{-n}(\Cscr)=0$ for all $n\geq 1$. When $n=1$, this was the main theorem of our paper which also proved the result for all $n\geq 1$ when $\Cscr^\heart$ is noetherian. These results were established by Schlichting in the case when $\Cscr\we\D^b(\Ascr)$. However, Neeman disproved this vanishing conjecture in [4].

Only one conjecture remained open: that the natural map $\K(\D^b(\Cscr^\heart))\rightarrow\K(\Cscr)$ is an equivalence. In non-negative degrees, this is Barwick’s theorem of the heart [3]. In degree $-1$, this followed from our work. But, it remained open, despite Neeman’s counterexamples. Ramzi, Sosnilo, and Winges disprove it as a consequence of their main theorem.

# The counterexample

Given their theorem that every spectrum is a $\K$-theory spectrum, it is very easy to describe the counterexample. Choose $M$ to be a spectrum which is not $\K(\bZ)$-local and let $\Cscr=\Cscr_M$. For example, $K(n)$ works for $n\geq 2$, where $K(n)$ denotes some Morava $\K$-theory spectrum. Note however that every connective spectrum is $\K(\bZ)$-local as is the $\K$-theory spectrum of every $\bZ$-linear stable $\infty$-category, such as $\D^b(\Ascr)$ if $\Ascr$ is an abelian category.

Let $\Cscr^\times=\Fun^\times(\Cscr^\op,\Sp)$ be the $\infty$-category of additive presheaves on $\Cscr$ and let $\Cscr^\fin\subseteq\Cscr^\times$ be the smallest idempotent complete stable subcategory containing the image of the Yoneda embedding $\Cscr\hookrightarrow\Cscr^\times$. The functor $y\colon\Cscr\rightarrow\Cscr^\fin$ is additive (and is in fact the universal additive functor into a small stable $\infty$-category), but it is not exact in general. There is also a colimit map $\Cscr^\fin\rightarrow\Cscr$, which is a Verdier localization. Let $\Ac(\Cscr)$ be the kernel, so there is an exact sequence

\[\Ac(\Cscr)\rightarrow\Cscr^\fin\rightarrow\Cscr\]of small idempotent complete stable $\infty$-categories. The stable $\infty$-category $\Ac(\Cscr)$ is generated by cofibers of the natural maps $y(b)/y(a)\rightarrow y(b/a)$ for morphisms $f\colon a\rightarrow b$ in $\Cscr$.

Now, $\Ac(\Cscr)$ admits a natural bounded $t$-structure, which I will not describe here. This $t$-structure has been observed in various forms before; Neeman uses it and the exact sequence above in [4]. If $\K(\D^b(\Ac(\Cscr)^\heart))\rightarrow\K(\Ac(\Cscr))$ is an equivalence, as asserted by our conjecture, then $\K(\Ac(\Cscr))$ is $\K(\bZ)$-local as $\D^b(\Ac(\Cscr))$ is $\bZ$-linear. If this is the case, then the cofiber sequence

\[\K(\Ac(\Cscr))\rightarrow\K(\Cscr^\fin)\rightarrow\K(\Cscr)\]implies that $\K(\Cscr^\fin)$ cannot be $\K(\bZ)$-local, since we assume that $\K(\Cscr)\we M$ is not $\K(\bZ)$-local. So, it suffices to prove that $\K(\Cscr^\fin)$ is $\K(\bZ)$-local to obtain a contradiction.

However, $\Cscr^\fin$ admits a weight structure and Sosnilo’s theorem of the heart implies that the map $\K(\Cscr^\fin)\rightarrow\K(\Ho(\Cscr^\fin))$ is an equivalence in non-positive degrees; that is to say that the fiber is connective and hence $\K(\bZ)$-local. Additionally, $\Ho(\Cscr^\fin)$ is an additive $1$-category and hence $\K(\Ho(\Cscr^\fin))$ is $\K(\bZ)$-local since it admits the structure of a $\K(\bZ)$-module. Thus, $\K(\Cscr^\fin)$ is $\K(\bZ)$-local, and we are done!

# References

[1] Alonso Tarrio, Jeremias Lopez, and Saorin, *Compactly generated t-structures on
the derived category of a Noetherian ring*, J. Algebra **324** (2010), no. 3, 313-346.

[2] Antieau, Gepner, and Heller, *K-theoretic obstructions to bounded t-structures*, Invent. Math. **216** (2019), no. 1, 241-300.

[3] Barwick, *On exact $\infty$-categories and the theorem of the heart*, Compos. Math. **151** (2015), no. 11, 2160-2186.

[4] Neeman, *A counterexample to some recent conjectures*, `arXiv:2006.16536`.

[5] Neeman, *Bounded $t$–structures on the category of perfect complexes*,
`arXiv:2202.08861`.

[6] Ramzi, Sosnilo, Winges, *Every spectrum is the K-theory of a stable $\infty$-category*,
`arXiv:2401.06510`.

[7] Schlichting, *Negative K-theory of derived categories*, Math. Z. **253** (2006), no. 1, 97–134.

[8] Smith, *Bounded t-structures on the category of perfect complexes over a
Noetherian ring of finite Krull dimension*,
Adv. Math. **399** (2022) 108241.