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History

Let $\Cat_\infty^\perf$ denote the $\infty$-category of small idempotent complete stable $\infty$-categories. A typical example of an object is $\Perf(R)$ for a commutative ring $R$. The $\infty$-category $\Cat_\infty^\perf$ admits a natural symmetric monoidal structure which has the property that if $R$ and $S$ are commutative rings, then $\Perf(R)\otimes\Perf(S)\we\Perf(R\otimes_\bS S)$, where $\bS$ denotes the sphere spectrum. If $R$ is an $\bE_\infty$-ring spectrum, for example if it is a commutative ring, then $\Perf(R)$ is an $\bE_\infty$-algebra in $\Cat_\infty^\perf$.

A modern perspective on the Brauer group of a ring $R$ is that it is the Picard group of the symmetric monoidal category $\Cat_R$, which is defined to be the $\infty$-category of $\Perf(R)$-modules in $\Cat_\infty^\perf$. This definition was given by Toën in [4] over animated commutative rings using dg categories and extended by myself and David Gepner in [1] to handle the case when $R$ is an $\bE_\infty$-ring spectrum.

We will write $\bBr(R)$ for the space of invertible objects in $\Cat_R$. Specifically, there is an inclusion of $\CGp(\Sscr)\subseteq\CMon(\Cat_\infty)$ of the $\infty$-category of grouplike $\bE_\infty$-spaces1 into the $\infty$-category of symmetric monoidal $\infty$-categories. This inclusion admits a right adjoint $\bPic$. We let $\bBr(R)=\bPic(\Cat_R)$. Toën’s Bruer group is $\Br(R)=\pi_0\bPic(R)$.

If $R$ is connective, meaning that $\pi_iR=0$ for $i<0$, every $\Cscr\in\Br(R)$ is étale-locally trivial. We also have $\bPic(R)=\bPic(\Perf(R))$, which is the space of automorphisms of the $\Perf(R)$. This implies that $\bBr(R)$ can also be described as $\R\Gamma_\et(\Spec R,\B\bPic)$. This is a positive answer to a form of Grothendieck’s $\Br=\Br’$ question.

Let $\Pr^{\L,\dual}_\st$ be the $\infty$-category of dualizable stable presentable $\infty$-categories and left adjoint functors whose right adjoints are right adjoints. A result of [2] implies that $\Pr^{\L,\dual}_\st$ is $\omega_1$-compactly generated. We define $\Cat_R^\dual$ to be the $\infty$-category of $\D(R)$-modules in $\Pr^{\L,\dual}$. We let $\bBr^\dual(R)=\bPic(\Cat_R^\dual)$. There is an inclusion of spaces $\bBr(R)\subseteq\bBr^\dual(R)$.

Toën’s question was whether or not $\Br(R)=\Br^\dual(R)$, say for $R$ a commutative ring. (More generally, there is a related definition for derived stacks.) At the time, as Toën remarks, this question seemed a bit exotic. However, in light of the recent focus on continuous $K$-theory and dualizable stable $\infty$-categories, thanks especially to the work of Efimov [2], there has been renewed interest in Toën’s question.

Results

Germán Stefanich’s paper [3] was first posted in 2023 and proved that $\Br(R)=\Br^\dual(R)$ for $R$ a field and more generally for $R$ a connective truncated $\bE_\infty$-ring $R$ with $\pi_0R$ Artinian. Maxime Ramzi and I then observed in unpublished work that we could bootstrap using Stefanich’s results to get that $\Br(R)=\Br^\dual(R)$ also for noetherian commutative rings $R$. In particular, $\Br(\bZ)=\Br^\dual(\bZ)$, which is one result we particularly wanted.

However, the recent update to Stefanich’s paper now establishes that $\Br(R)=\Br^\dual(R)$ for all commutative rings and more generally for all connective truncated $\bE_\infty$-ring spectra $R$. There are several similarities in the arguments, but Stefanich is able to remove the noetherian hypotheses using a more clever argument to reduce to the Artin local case he had already proven.

While not stated in his paper, Stefanich’s result also implies that $\Br(X)=\Br^\dual(X)$ for $X$ a qcqs scheme. Indeed, one can prove this by induction on the number of affines needed to cover $X$, giving a standard argument as in [1, Thm. 6.11].

Let $\Cscr$ denote an invertible dualizable $\D(R)$-linear category, which defines a point of $\bBr^\dual(R)$. Stefanich shows that it is compactly generated, and hence is in $\bBr(R)$. The argument used by Stefanich, which I very much like, is to look at the poset $P$ of ideals $I\subseteq R$ such that $\Cscr\otimes_R R/I$ is not compactly generated. The goal is to show it is empty. No maximal ideal is in $P$ by the fact that $\Br(k)=\Br^\dual(k)$ for a field $k$ (as proven in the first version of the paper). One argues that $I$ is closed under filtered colimits inside the poset of all ideals of $R$. In particular, if $I$ is nonempty, then it contains a maximal element, say $J$. Stefanich then shows that $R/J$ is a domain. Over the fraction field $K$ one has that $\Cscr\otimes_R K$ does admit a compact generator. So, by another filtered colimit argument, there is some $0\neq x\in R$ such that $\Cscr$ admits a compact generator over $R/J[x^{-1}]$. But, now $\Cscr\otimes_R R/(J,x)$ does admit a compact generator too by the maximality hypothesis of $J$. By an arithmetic fracture square argument, one now concludes to see that $\Cscr\otimes_R R/J$ admits a compact generator, which is a contradiction. This means that $P$ is empty and that $\Cscr$ is compactly generated.

The paper contains a lot of other interesting results, especially about Grothendieck abelian categories.

References

[1] Antieau and Gepner, Brauer groups and étale cohomology in derived algebraic geometry, Geom. Topol. 18 (2014), no. 2, 1149–1244.

[2] Efimov, K-theory and localizing invariants of large categories, arXiv:2405.12169.

[3] Stefanich, Classification of fully dualizable linear categories, arXiv:2307.16337.

[4] Toën, Derived Azumaya algebras and generators for twisted derived categories, Invent. Math. 189 (2012), no. 3, 581–652.

Footnotes

  1. The $\infty$-category $\CGp(\Sscr)$ is equivalent to the $\infty$-category $\D(\bS)_{\geq 0}$ of connective spectra. ↩