$ \newcommand\A{\mathrm{A}} \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}} $

Neeman is at it again: after disproving in [4] the conjectures of Schlichting and of myself, Gepner, and Heller on the vanishing of $K$-theory of stable $\infty$-categories with bounded $t$-structures, Neeman has proved in [5] the other conjecture from our paper, about the connection between regularity and $t$-structures on $\Perfscr(X)$ when $X$ is a reasonable scheme.

The conjecture

I quote from my previous post on Neeman’s counterexample.

In [2], we observed that if $X$ is a scheme with $\K_{-1}(X)\neq 0$, then there is no bounded $t$-structure on $\Perfscr(X)$. Similarly, if $\K_{n}(X)\neq 0$ for some $n<0$, then there is no bounded $t$-structure on $\Perfscr(X)$ with noetherian heart. We made the following conjecture.

Conjecture. If $X$ is a noetherian scheme which is not regular, then there is no bounded $t$-structure on $\Perfscr(X)$.

The idea here is that if $X$ is regular, then $\Perfscr(X)\simeq\Dscr^b(X)$, the bounded derived category of coherent sheaves, which has the usual canonical $t$-structure arising from the good truncation of complexes. When $X$ is singular, the inclusion $\Perfscr(X)\subseteq\Dscr^b(X)$ is proper and it is not hard to prove that the canonical $t$-structure on $\Dscr^b(X)$ does not restrict to a bounded $t$-structure on $\Perfscr(X)$. For example, it is easy to cook up perfect complexes whose homology sheaves do not (even locally) admit finite locally free resolutions.

My PhD student Harry Smith showed in his thesis [6] that the conjecture is true for noetherian affine schemes of finite Krull dimension. He even shows more strongly that for irreducible noetherian affine schemes of finite Krull dimension there are no non-trivial $t$-structures at all on $\Perfscr(X)$!

Smith’s methods rely heavily on the results of [1] classifying compactly generated $t$-structures on the derived $\infty$-categories of noetherian commutative rings; it is not known how to generalized these results to the non-affine case, so a significant new idea was needed and discovered by Neeman.

Neeman’s result

Theorem (Neeman). If $X$ is a separated noetherian scheme of finite Krull dimension, then $\Perfscr(X)$ supports a bounded $t$-structure if and only if $X$ is regular.

Neeman’s proof relies on some unpublished work of his, so he also gives a self-contained proof of the slightly weaker case when $X$ is in addition of finite type over a noetherian commutative ring.

The basic idea is to introduce an equivalence relation on the class of $t$-structures on the larger category $\Dscr_\qc(X)$. Two $t$-structures are equivalent if their connective objects are each uniformly bounded below with respect to the other $t$-structure.

Then, Neeman proves that when $X$ satisfies the hypotheses of the theorem, every $t$-structure on $\Dscr_\qc(X)$ whose aisle is generated by a set of perfect complexes is in the same equivalence class. The proof is basically to look at how badly a given compact projective generator of $\Dscr_\qc(X)$ can fail to be connective. Hence, if $\Perfscr(X)$ admits a bounded $t$-structure, then the induced $t$-structure on $\Dscr_\qc(X)$ is in the same equivalence class as the canonical $t$-structure on $X$. Using this, Neeman shows how a previous result of his and Lipman implies that in fact $\Perfscr(X)\we\Dscr^b(X)$, which implies regularity.

Fix a bounded $t$-structure on $\Perfscr(X)$. The idea is very elegant: given an object $C$ of $\Dscr^b(X)$ one can find a perfect complex $P\rightarrow C$ which is an equivalence in arbitrarily large degrees in the standard $t$-structure. Since all compactly generated $t$-structures on $\Dscr_\qc(X)$ are equivalent this is also possible in the $t$-structure induced by the bounded $t$-structure fixed on $\Perfscr(X)$ above. On the other hand, $C$ is also bounded above with respect to this induced $t$-structure. It follows that if $F$ denotes the fiber of $P\rightarrow C$, then the fiber sequence $F\rightarrow P\rightarrow C$ must be equivalent to \(\tau_{\geq N}P\rightarrow P\rightarrow\tau_{\leq N-1}P\) for $N$ sufficiently large. But, this means that $C$ is perfect!

The hypotheses on $X$ are mainly to guarantee that the canonical $t$-structure on $\Dscr_\qc(X)$ is in the same equivalence class as the compactly generated $t$-structures, which is in some sense a question related to the existence of enough vector bundles on $X$.


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

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