$ \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Mscr{\mathcal{M}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}} \newcommand\LBr{\mathrm{LBr}} \newcommand\TMF{\mathrm{TMF}} \newcommand\Pic{\mathrm{Pic}} \newcommand\shpic{\mathbf{pic}} \newcommand\KO{\mathrm{KO}} \newcommand\KU{\mathrm{KU}} \newcommand\iso{\cong} $

Lennart Meier, Vesna Stojanoska, and I have uploaded our paper [2] on the Brauer group of topological modular forms $\TMF$ to the arXiv.

Brauer groups in derived algebraic geometry and of ring spectra have been introduced and studied by Toën, Baker–Lazarev, Baker–Richter–Szymik, Gepner and myself, and Hopkins–Lurie. The upshot is that for connective $\bE_\infty$-ring spectra $R$ one has $\Br(R)\iso\Br(\pi_0R)$, where the latter group is not the classical Brauer group of $\pi_0R$, but rather its ``derived’’ Brauer group, which fits into a natural extension

\[0\rightarrow\H^2(\Spec\pi_0R,\Gm)\rightarrow\Br(\pi_0R)\rightarrow\H^1(\Spec\pi_0R,\bZ)\rightarrow 0.\]

In this setting there is no $\Br=\Br’$ problem. Indeed, Toën showed that every class in $\Br(\pi_0R)$ is represented by a derived Azumaya algebra, and this extends to connective $\bE_\infty$-ring spectra as well.

The situation in the nonconnective case is much more difficult, although Hopkins and Lurie are able to compute the $K(n)$-local analogue of the Brauer group for Lubin–Tate spectra. The main problem in the nonconnective case is that one does not know if Brauer classes are 'etale-locally trivial or at least 'etale-locally trivial to some kind of Brauer–Wall construction as in Gepner–Lawson. For example, we construct an $\bE_\infty$-ring in our paper where there are lots of Brauer classes which are not 'etale-locally trivial, although the example is not of a chromatic flavor.

We show first of all that for the derived moduli stack of elliptic curves $(\Mscr,\Oscr)$, one has $\Br(\Mscr,\Oscr)=\Br’(\Mscr,\Oscr)$. We then prove that, thanks to the $0$-affineness results established previously by Mathew–Lennart Meier [4], taking global sections induces an equivalence $\Br(\Mscr,\Oscr)\we\Br(\TMF)$.

Then, we study and mostly compute two subgroups of $\Br(\TMF)$: the subgroup $\LBr(\TMF)$ consisting of elements which are étale locally trivial on $\Spec\TMF$ and the subgroup $\LBr(\Mscr,\Oscr)\subseteq\Br(\Mscr,\Oscr)=\Br(\TMF)$ of elements which are étale locally trivial on the moduli stack of elliptic curves. There are many more étale covers of $(\Mscr,\Oscr)$ than there are of $\Spec\TMF$ because the étale site of the latter is that of $\Spec\bZ[j]$.

The groups $\LBr(\TMF)$ and $\LBr(\Mscr,\Oscr)$ are torsion and have only $2$ and $3$-primary torsion. We have


The $2$-torsion is more complicated. We have

  • the inclusion \(\LBr(\TMF)_{(2)}\rightarrow\LBr(\Mscr,\Oscr)_{(2)}\) has finite ($2$-torsion) cokernel;
  • both groups \(\LBr(\TMF)_{(2)}\) and \(\LBr(\Mscr,\Oscr)_{(2)}\) are extensions a countably infinite $\bF_2$-vector space and a group of order dividing $8$;
  • in particular, both groups can be written as a direct sum of $\bF_2^{\oplus\bN}$ and a group of order dividing $16$.

To make these calculations, we use a sheafified version of the $\Pic$ spectral studied by Mathew–Stojanoska [5]. Specifically, we consider the sheaf of picard spectra $\shpic_{\Oscr_\Mscr}$ on the derived moduli stack of elliptic curves and push this down along the $j$-homomorphism $j\colon(\Mscr,\Oscr)\rightarrow\Spec\TMF$. Then, we attempt to compute \(\pi_0 j_*\shpic_{\Oscr_\Mscr}\), which we are able to do except for some very long differentials ($d_{13}$, $d_{23}$, and $d_{25}$). This uncertainty prevented us from getting at the exact group structure as well as the precise difference between the local Brauer groups of $\TMF$ and of the derived moduli stack. We hope someone else might be able to figure them out, the problem being that they are outside of the ``exponentiable range’’, where the differentials in the $\Pic$ spectral sequence can be directly compared to the differentials in the (known) additive spectral sequence using the comparison tool of [5].

We also use the Brauer group of the classical moduli stack $(\Mscr,\pi_0\Oscr)$, which was studied by myself and Meier and proved to be zero in [1]. This gives us a single term in one of our spectral sequences. We were aware already at the HIM program on homotopy theory in 2015 that we wanted to make the calculation in the present paper, and we would need the classical calculation to do it. Seven years later…

Along the way, we prove that the local Brauer group of $\KO$ is $\bZ/2$ and that the local Brauer group of $\KU$ is $0$. This shows that the non-trivial Brauer class in $\Br(\KU|\KO)\iso\bZ/2$ discovered by David Gepner and Tyler Lawson in [3] is in fact étale-locally trivial over $\Spec\KO$.


[1] Antieau, Meier, The Brauer group of the moduli stack of elliptic curves, AGT 14:9 (2020), arXiv:1608.00851.

[2] Antieau, Meier, Stojanoska, Picard sheaves, local Brauer groups, and topological modular forms, arXiv:arXiv:2210.15743.

[3] Gepner, Lawson, Brauer groups and Galois cohomology of commutative ring spectra, Compositio 157 (2021), arXiv:1607.01118.

[4] Mathew, Meier, Affineness and chromatic homotopy theory, JTop 8 (2015), arXiv:1311.0514.

[5] Mathew, Stojanoska, The Picard group of topological modular forms via descent theory, G&T 20 (2016), arXiv:1409.7702.