This is the first post in a series on my favorite work of 2020.

Schlichting’s conjecture

Let $R$ be a regular noetherian commutative ring. It has been known since the time of Bass and the definition of negative K-groups that $\K_n(R)=0$ for $n<0$. More generally, $\G_n(R)=0$ for $n<0$ when $R$ is a noetherian ring. Here, $\K(R)$ is short-hand for the algebraic K-theory of $\Perfscr(R)$, the $\infty$-category of perfect complexes of $R$-modules, while $\G(R)$ denotes the algebraic $K$-theory of $\Dscr^b(R)$, the $\infty$-category of complexes of $R$-modules with bounded and finitely presented homology.

When $R$ is regular and noetherian, the natural inclusion $\Perfscr(R)\rightarrow\Dscr^b(R)$ is an equivalence so we have $\K(R)\simeq\G(R)$ and in particular the more general vanishing for negative G-theory implies vanishing for negative K-theory, in this case.

In [5], Schlichting proved that for any small abelian category $\Ascr$ one has

\[\K_{-1}(\Dscr^b(\Ascr))=0.\]

If moreover $\Ascr$ is noetherian, then an inductive argument is used to prove that

\[\K_n(\Dscr^b(\Ascr))=0\]

for all $n<0$. Schlichting made the following conjecture.

Conjecture. If $\Ascr$ is any small abelian category, then $\K_{n}(\Dscr^b(\Ascr))=0$ for $n<0$.

Stable $\infty$-categories

Let $\Escr$ be a stable $\infty$-category equipped with a bounded $t$-structure. Neeman had long ago realized that there was a connection between the K-theory of the triangulated homotopy category $\Ho(\Escr)$ and of the heart $\Escr^\heart$ of the $t$-structure, although this was phrased in the language of his K-theory of triangulated categories. For example, it is easy to check that $\K_0(\Escr^\heart)\cong\K_0(\Escr)$.

Barwick proved the following fantastic theorem in [3].

Theorem. If $\Escr$ is a stable $\infty$-category with a bounded $t$-structure, then the natural map $\K^{\cn}(\Escr^\heart)\rightarrow\K^{\cn}(\Escr)$ is an equivalence.

Here, $\K^{\cn}$ denotes connective K-theory. Barwick’s Theorem of the Heart implies that $\K_n(\Escr^\heart)\cong\K_n(\Escr)$ for all $n\geq 0$. It is natural to wonder if Barwick’s theorem extends to negative degrees. Together with David Gepner and Jeremiah Heller, I proved the following analogs of Schlichting’s theorems in [2].

If $\Escr$ is a stable $\infty$-category with a bounded $t$-structure, then

\[\K_{-1}(\Escr)=0.\]

If $\Escr^\heart$ is noetherian, then

\[\K_{n}(\Escr)=0\]

for all $n<0$.

In particular, in the case where the heart is noetherian, one gets a nonconnective theorem of the heart $\K(\Escr^\heart)\simeq\K(\Escr)$,¹ simply by combining the vanishing results in negative degrees and Barwick’s theorem of the heart.

We then proposed a generalization of Schlichting’s conjecture.

Conjecture. If $\Escr$ is a stable $\infty$-category with a bounded $t$-structure, then $\K_n(\Escr)=0$ for $n<0$.

Amnon Neeman’s counterexample

Both conjectures are false! Here is the idea.

Neeman considers in [3] a general idempotent complete exact category $\Escr$ and the associated localization sequence

\[\Acscr^b(\Escr)\rightarrow\Kscr^b(\Escr)\rightarrow\Perfscr(\Escr)\]

of idempotent complete stable $\infty$-categories, where $\Kscr^b(\Escr)$ models bounded chain complexes of objects of $\Escr$ up to chain homotopy and $\Acscr^b(\Escr)$ is the full subcategory of acyclics.

The universal property of $K$-theory implies that there is a cofiber sequence

\[\K(\Acscr^b(\Escr))\rightarrow\K(\Kscr^b(\Escr))\rightarrow\K(\Perfscr(\Escr))\]

of nonconnective K-theory spectra.

Moreover, Neeman makes a basic but very clever observation: $\Acscr^b(\Escr)$ always admits a bounded $t$-structure! The heart is a certain subcategory of the abelian category of additive functors $\Escr^\op\rightarrow\Mod_{\bZ}$. It follows that to disprove the generalized Schlichting conjecture, one can find $\Escr$ such that $\K_{-1}(\Perfscr(\Escr))\neq 0$ and where the boundary map

\[\K_{-1}(\Perfscr(\Escr))\rightarrow\K_{-2}(\Acscr^b(\Escr))\]

is non-zero.

It is well-known if $X$ is a projective nodal (and non-smooth) curve over a field $k$, then $\K_{-1}(X)=\K_{-1}(\Perfscr(X))=\K_{-1}(\Perfscr(\mathrm{Vect}(X)))$ is non-zero. For instance, this can be seen by using the Nisnevich descent spectral sequence

\[\E_2^{s,t}=\H^{-s}_{\mathrm{Nis}}(X,\K_{t})\Rightarrow\K_{s+t}(X),\]

which degenerates for a $1$-dimensional scheme, together with the fact that $\H^1_{\mathrm{Nis}}(X,\K_0)\cong\H^1_{\mathrm{Nis}}(X,\bZ)\neq 0$. Here, $\K_t$ dentoes the Nisnevich sheafification of the presheaf $U\mapsto\K_t(U)$.

Amazingly, Neeman takes this simple case, where the exact category $\Escr$ is $\mathrm{Vect}(X)$, and shows that it possesses the desiderata above: we already have that $\K_{-1}(\Perfscr(\mathrm{Vect}(X)))\neq 0$. Neeman proves that $\K_{-1}(\Kscr^b(\mathrm{Vect}(X)))=0$ by analyzing idempotents on complexes of vector bundles on the curve using slope stability. This gives the counterexample.

Neeman also proves that in this case the natural map $\Dscr^b(\Acscr^b(\Escr)^\heart)\rightarrow\Acscr^b(\Escr)$ is an equivalence, so that he also gets a counterexample to Schlichting’s original conjecture.

There’s not much more to say about this simple and elegant argument.

Open problems

Problem 1. It remains possible that Barwick’s theorem of the heart could extend to give an equivalence $\K(\Escr^\heart)\rightarrow\K(\Escr)$ for all stable $\infty$-categories with bounded $t$-structures. Prove or disprove this.

Problem 2. Non-vanishing negative K-theory is an obstruction to regularity for schemes. 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 $\Perfscr(X)$.

Neeman’s counterexample complicates the search for a proof, but it is also the case that one can have singular schemes $X$ where $\K_{n}(X)=0$ for all $n<0$.² In other words, the vanishing of negative $K$-theory is not a perfect test for singularities: there are false negatives.

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 $\Perfscr(X)$!

Resolving the conjecture in the remaining cases is the second open problem. Smith’s methods, which rely heavily on the results of [1] classifying compactly generated $t$-structures on the derived $\infty$-categories of noetherian commutative rings, seem quite far from extending to the non-affine case, so a significant new idea is needed.

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

[6] Smith, Bounded t-structures on the category of perfect complexes over a Noetherian ring of finite Krull dimension, arXiv:1910.07697.

Footnotes

At this point, $\K(\Escr^\heart)$ stands for $\K(\Dscr^b(\Escr^\heart))$. This is the only known definition of nonconnective algebraic K-theory of an abelian category. ↩
For example, if $X=\Spec(k[x]/(x^2))$ where $k$ is a field, then $\K_n(X)=0$ for $n<0$. ↩