$ \newcommand\A{\mathrm{A}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\we{\simeq} \newcommand\qc{\mathrm{qc}} \newcommand\id{\mathrm{id}} \newcommand\Sp{\mathrm{Sp}} \newcommand\Cat{\mathrm{Cat}} \newcommand\perf{\mathrm{perf}} \newcommand\Mot{\mathrm{Mot}} \newcommand\loc{\mathrm{loc}} \newcommand\unit{\mathbf{1}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Fun{\mathrm{Fun}} \newcommand\fin{\mathrm{fin}} \newcommand\Ac{\mathrm{Ac}} $

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 $M$.


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


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


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!


[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.