Picard sheaves, local Brauer groups, and topological modular forms

We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some \'etale cover of the moduli stack, up to a finite 2-torsion group.


Introduction
The Brauer group Br( ) of an ∞ -ring spectrum was introduced by Baker-Richter-Szymik [8] following previous work of Baker-Lazarev [6] and Toën [67].The group classifies Azumaya algebras over up to Morita equivalence; equivalently it classifies invertible -linear stable ∞-categories.These can be seen as twisted versions of -modules and thus Br( ) classifies all possible twists of Mod .One can actually replace Mod here by any symmetric monoidal ∞-category, like quasi-coherent sheaves on a scheme or stack.In the most classical case of vector spaces over a field , Azumaya algebras are just central simple algebras (i.e.matrix algebras over a central division algebra) and the corresponding Brauer group was introduced by Brauer around 1930.
Classically, Brauer groups can often be computed as étale cohomology groups.They thus allow cohomological control of natural occurrences of Azumaya algebras (e.g. as endomorphism algebras of representations [60,Section 12.2]) or twisted sheaves (like in the theory of moduli of stable sheaves [16]).Another ∞-categorical example is given by the relevance of twists of parametrized spectra in Seiberg-Witten Floer homotopy theory [26].On the other hand, Brauer groups also allow algebraic or geometric interpretations of cohomology classes, as utilized e.g. in the classic Artin-Mumford example of a non-rational unirational variety [5] or the Merkurjev-Suslin theorem [31].Brauer groups give also one of the approaches to class field theory [58,69,54] and form the basis of the Brauer-Manin obstruction for rational points [21].Thus, the study of Brauer groups of ring spectra might be interesting for possible theories of étale cohomology on ∞ -ring spectra, and can be seen as a contribution to the nascent subject of arithmetic of ∞ -ring spectra.Moreover, the Brauer space provides a natural delooping of the Picard space, like the Picard space is a natural delooping of the space of units of an ∞ -ring spectrum.
When is a connective ∞ -ring spectrum, Br( ) depends only on 0 and Br( ) ≅ H1 (Spec 0 , ℤ) × H 2 (Spec 0 , ), where all cohomology is étale unless otherwise specified; see [3,67].For example, for a prime , we have Br( [1∕ ]) ≅ ℤ∕2, so there is a "twisted form" of finite spectra after inverting .These twisted forms are ∞-categories of modules for spherical quaternion algebras.In case that is a classical ring, Br( ) might actually be larger than the classical Brauer group of because of the presence of derived, non-classical Azumaya algebras.
The role of connectivity is to ensure (by an argument of Toën) that Brauer classes on connective ∞ -rings are étale-locally trivial.This fact enables the cohomological calculation of the Brauer group as in (1.1).We will show in Example 5.7 that this fails in general for nonconnective ring spectra.Thus we will differentiate between Br( ) and its subgroup LBr( ) of Brauer classes that are étale-locally trivial, i.e. become trivial after some faithful étale extension in the sense of [46,Definition 7.5.0.4].Two of our main themes are that LBr( ) is quite computable (up to the general difficulty of computing differentials), and that sometimes we may enlarge LBr( ) by allowing more general extensions to kill Brauer classes.We can say something about the resulting subgroups of Br( ), which may or may not coincide with LBr( ).
Our main examples are real K-theory and topological modular forms.Let us begin with the former.
The nontriviality of Br(KO) goes back to [30], where Gepner and Lawson compute the subgroup Br(KU|KO) ⊆ Br(KO) of classes split by the faithful ℤ∕2-Galois extension KO → KU to be ℤ∕2.It is not hard to check that LBr(KU) = 0 and thus we find in fact that LBr(KO) = Br(KU|KO) as subgroups of Br(KO) although one a priori might expect Br(KU|KO) to be bigger.In particular, we show that the non-trivial class ∈ Br(KU|KO) is split by the faithful étale extension KO → KO[ 1  2 , 4 ] × KO[ 1  3 , 3 ].Regarding the spectrum TMF of topological modular forms, we recall that Goerss, Hopkins and Miller have defined a sheaf of ∞ -ring spectra on the moduli stack ℳ of elliptic curves [25].The pair (ℳ, ) defines a nonconnective spectral Deligne-Mumford stack in the sense of [47] and TMF is the spectrum of global sections of .We may define Br(ℳ, ) as the Brauer group of QCoh(ℳ, ), 1 which coincides with Mod TMF as (ℳ, ) is 0-affine by [49].On the other hand, we may define LBr(ℳ, ) as the subgroup of Brauer classes that become trivial after pulling back to an étale cover of ℳ, and this group is potentially bigger than LBr(TMF).As by [48,Theorem 10.4], all faithful Galois extensions of localizations of TMF arise from étale covers of ℳ, this local Brauer group LBr(ℳ, ) is a natural analogue of Br(KU|KO) above.
After localizing at 2, the inclusion LBr(TMF) ⊂ LBr(ℳ, ) has finite cokernel and both groups admit surjections to (ℤ∕2) ∞ with kernel of order at most 8.In particular, Br(TMF) is an infinitely generated torsion abelian group.
For the (partial) determination of LBr(KO) and LBr(TMF) our most important tool is an exact sequence for LBr( ) (with mild assumptions on 0 ) of the form Br( 0 ) → LBr( ) → H 1 (Spec 0 , 0 ), which will be proven in a more precise form in Proposition 2.25.Here, 0 is the Picard sheaf of .It arises as the étale sheafification of the presheaf sending each étale extension of 0 to Pic( ), where → is the unique étale extension realizing 0 → .We determine the Picard sheaf of KO in Proposition 3.8 and give a partial determination of the Picard sheaf of TMF in Theorem 6.5.The main method is a sheafy version of the Picard spectral sequence of [50].The remaining uncertainties lie in our inability to compute long differentials in the sheafy Picard spectral sequence or, essentially equivalently, in our inability to compute Pic(TMF (2) [ 2 −1 ]) for ≥ 2. For possible subleties arising in such computations, we refer to Remark 3.12.
For the (partial) determination of LBr(ℳ, ) a crucial point is to compare the (local) Brauer groups of a nonconnective spectral Deligne-Mumford stack ( , ) with the following variant: The cohomological Brauer group Br ′ ( , ) is defined using descent from the affine case and we also obtain a subgroup LBr ′ ( , ).The Brauer group of ( , ) is the subgroup Br( , ) ⊆ Br ′ ( , ) of Brauer classes representable by Azumaya algebras.If ( , ) ≃ Spec is affine, then Br( ) ≅ Br(Spec ) ≅ Br ′ (Spec ) and likewise for LBr, but Br and Br ′ might be different in general.In • Pr L , the infinity category of presentable ∞-categories and left adjoint morphisms; • Ĉat ∞ , the infinity category of possibly large ∞-categories.

The local Brauer group in the affine case
After reminding the reader about the classical Brauer group of a commutative ring, we recall in this section the definition of the Brauer group and Brauer space of a commutative ring spectrum and introduce the notion of the local Brauer group.We will prove several basic properties (in particular that Brauer spaces define an étale hypersheaf) and provide basic tools for the computation of local Brauer groups.

The classical Brauer group
In this subsection, we will give a short introduction to the classical Brauer group.For more background we refer for example to [31], [21] and the series of articles starting with [32].
Let be a commutative ring.An -algebra is called Azumaya if one of the following equivalent conditions holds: 1. is finitely generated, faithful, and projective as an -module and the map is an isomorphism.
Two Azumaya algebras and are called Morita equivalent if their module categories are equivalent.
Definition 2.1.The classical Brauer group Br cl ( ) of is the set of Azumaya algebras over up to Morita equivalence.
Remark 2.2.Instead of working with Morita equivalence classes of Azumaya algebras, one can also directly define the Brauer groups via the module categories.This is the approach we will take in Definition 2.11.
In the case that is regular noetherian, Br cl ( ) coincides with what we later introduce as Br( ); thus we will drop the superscript in this case.Moreover, a result of Gabber identifies Br( ) in the regular noetherian case with H 2 (Spec ; ) [42,Corollary 3.1.4.2].As Pic( ) ≅ H 1 (Spec ; ), this gives one perspective on why Brauer groups are a higher variant of Picard groups.If = is a field, every finite-dimensional division -algebra with center is Azumaya.Conversely, every Azumaya -algebra is Morita equivalent to a unique such.Thus, Br( ) is in bijection with isomorphism classes of finite-dimensional division -algebras with center .For example, Br(ℝ) ≅ {[ℝ], [ℍ]} ≅ ℤ∕2 and Br(ℂ) = 0.In contrast, the Brauer group of a non-archimedean local field (like ℚ ) is isomorphic to ℚ∕ℤ.
It will be important for our later calculations to understand the Brauer groups of rings like ℤ or ℤ[ 1  2 , 4 ].More generally, we consider a number field and let be a localization of the ring of integers of .In this case, by [34,Proposition 2.1], there is an exact sequence 0 → Br( ) → Br( ) → ⨁ ∈Spec (1)   Br(Spec ), where Spec (1) denotes the set of closed points of Spec and denotes the completion.This exact sequence is compatible with the exact sequence of class field theory (see [56,Theorem 8.1.17]).The sum ranges over the finite and the infinite places of , and the map Br(Spec ) → ℚ∕ℤ is the isomorphism described above when is a finite place, the natural inclusion ℤ∕2 → ℚ∕ℤ when ≅ ℝ, and the natural map 0 → ℚ∕ℤ when ≅ ℂ.
Brauer groups have several nice properties, three of which we will summarize in the next theorem.
Theorem 2.5.Let be a regular noetherian ring. ( and is a non-zero divisor, we have more precisely a short exact sequence If there is a ring homomorphism right inverse of → ∕ , the sequence is split, sending [ ] to the Brauer class of the cyclic algebra ( , ). ( induced by the morphism [ ] → , sending to 0. The right vertical morphism is an isomorphism since The horizontal arrows are injections by the first point.Thus, Br( [ ]) ( ) → Br( ) ( ) must be an injection as well.On the other hand, it is a split surjection, using the map → [ ].This implies that it is an isomorphism.
In some of our examples below, we will also use the following classical results, which will help us compute Brauer groups of various ring spectra.The first is Grothendieck's rigidity result for the Brauer group [32, Corollaire 6.2] Theorem 2.7.Suppose is Hensel local with residue field ; then Br( ) ≅ Br( ).If is also regular, then Br( ) ≅ H 2 (Spec , ) so that H 2 (Spec , ) ≅ H 2 (Spec , ).
The next is a corollary of the affine analogue of proper base change as proved in Gabber-Huber [29,39], see also [11,Corollary 1.18(a)].

Brauer groups of ring spectra
In this subsection, we will recall the Brauer group and Brauer space of a commutative ring spectrum, which were first introduced by [8] and [65].For our purposes, an approach will be convenient that sees Brauer groups as categorified Picard groups.Let us thus first recall the definition of the Picard group and Picard space.
If is a symmetric monoidal ∞-category, its underlying ∞-groupoid naturally admits the structure of an ∞ -space and the counit map → is symmetric monoidal.We define the Picard space ( ) to be the maximal grouplike ∞ -groupoid in .In other words, ( ) is the space of ⊗-invertible objects of and equivalences.The Picard group of is Pic( ) = 0 ( ).We refer to [50] for more background on Picard groups and spaces.
Next we introduce the Brauer group Br( ) of a commutative ring spectrum as a categorification of the Picard group.In the case that is a regular noetherian ring, this will agree with the classical Brauer group (see Remark 2.26).Definition 2.11.Let be a commutative ring spectrum and let Cat denote the presentably symmetric monoidal ∞-category of compactly generated -linear stable ∞-categories and compact object-preserving left adjoint functors. 2a) We let ( ) = (Cat ) denote the Brauer space of .The Brauer group of a commutative ring spectrum is Br( ) = 0 ( ).
(b) If is an -algebra, we say that is an Azumaya algebra over if Mod defines a point of ( ).
(d) An Azumaya algebra is étale-locally trivial if there is an étale cover → such that ⊗ is trivial.
This definition of an Azumaya algebra is due to Toën [67].It agrees with the original definition of an Azumaya algebra in this setting due to [8]; see [3] for more details.Lemma 2.12.If is a commutative ring spectrum, then there is a natural equivalence ( ) ≃ Ω ( ), where Ω ( ) is computed via loops based at the trivial Brauer class.
Proof.By construction, Ω ( ) is the space of autoequivalences of the unit object of Cat .The unit object is Mod and the autoequivalences must be -linear, so they correspond to tensoring with invertible -modules.
We will prove in the next section that  → ( ) is a Postnikov complete étale sheaf.To do so, we first establish that  → Cat is an étale sheaf (with values in Pr L ).The result was discovered in the context of the present project, but appeared first in [2, Thm.

The local Brauer group
While in classical algebras, Azumaya algebras are always étale-locally Morita equivalent to the ground ring, this is no longer true in the spectral setting.In this subsection (and actually the whole article), we will concentrate on those which are étale-locally trivial.Definition 2.15.Let 0 denote the étale sheaf of connected components of .We let be the fiber of the natural map → 0 in étale sheaves.The space ( ) is the local Brauer space of and LBr( ) = 0 ( ( )) is the local Brauer group of .Remark 2.16.Thanks to Lemma 2.12, we could equivalently have defined as , the étale classifying space of , computed in étale sheaves.However, note that the functor  → B ( ), sending to the classifying space of its Picard space, is not a sheaf, and is its sheafification.
The name 'local Brauer group' is short-hand for 'locally-trivial Brauer group', which is justified by the following lemma.
(a) The natural map LBr( ) → Br( ) is an injection and hence ( ) → ( ) is the inclusion of a subspace of connected components.
(b) An element ∈ Br( ) is contained in LBr( ) if and only if there is a faithful étale map → such that maps to zero in Br( ).
Proof.For (a), use the fiber sequence of spaces.Recall here that Γ denotes the space (as opposed to the set) of global sections of an étale sheaf and 0 denotes the étale-sheafified homotopy group.Since Γ(Spec , 0 ) = 0 for > 0, the first claim follows from the long exact sequence in homotopy.Thus we can identify LBr( ) as a subgroup of Br( ).
If is a commutative ring spectrum, we will always equip Spec with the étale topology.The small étale sites of Spec and Spec 0 agree, so we can compute cohomology of sheaves of abelian groups on Spec via étale cohomology on Spec 0 : given a sheaf of abelian groups ∈ Shv Sp (Spec ) ♡ , we have Γ(Spec , ) ≃ Γ(Spec 0 , ) and thus − Γ(Spec , ) ≅ − Γ(Spec 0 , ) ≅ H (Spec 0 ; ) This will be used constantly below.
Specifically, to compute ( ), we can restrict and to étale sheaves and on the small étale site of .Note that these are already sheaves and no additional sheafification is necessary.While we may and will view the and as sheaves on Spec 0 , they certainly depend crucially on , not only on 0 .We use the notation also for the restriction of to Spec .
Lemma 2.18.Let be the local Brauer space sheaf constructed above on Spec for a commutative ring spectrum .The homotopy sheaves of are given by where is the étale sheaf ( ) ≅ ( 0 ) × for an étale commutative -algebra .In particular, is quasi-coherent for ≥ 3.
Proof.This follows from Lemma 2.12, using that étale sheafification commutes with restriction along the morphism CAlg ét → CAlg from commutative étale -algebras to all commutative ring spectra.
Remark 2.19.Analysis of 1 ≅ 0 is often the most difficult part of local Brauer group computations.
Definition 2.20.Let be a prestable ∞-category in the sense of [47, Appendix C] having all limits, which is automatically the nonnegative part of a t-structure on a stable ∞-category.We say that ∈ is ∞-connective if Map( , ) ≃ 0 for every truncated object .An object of is hypercomplete if Map( , ) ≃ 0 for every ∞-connective object .Finally, is Postnikov complete if the natural map → lim ≤ is an equivalence; this occurs if and only if lim ≥ +1 ≃ 0 as fiber sequences are closed under limits.
Postnikov complete objects are hypercomplete, but the converse is not always true.The significance of Postnikov completeness is that it allows us to compute global sections by using descent spectral sequences.As our prestable we will use sheaves with values in grouplike ∞ -spaces (i.e., connective spectra).Note that the forgetful functor from grouplike ∞ -spaces to spaces preserves and detects limits.[3,Section 7] in the special case of connective commutative ring spectra using a different argument, although the proof in the connective case and of [45, Proposition 6.5], which is used in the proof of Proposition 2.13, are closely related under the hood.The main point of [3] is that when is connective, every Azumaya -algebra is étale locally trivial.This is not true in general, as Example 5.7 below shows.
As the sheaves , and take values in grouplike ∞ -spaces, we can deloop them to presheaves of spectra.Sheafifying these results in sheaves , and and the restrictions , and when working on the étale site of Spec .Note that by construction, ≃ [1].Note moreover that ≅ , but the global sections can acquire additional negative homotopy groups.
Proof.This spectral sequence is the descent spectral sequence for the sheaf of spectra, associated to the tower of global sections of the truncations of .Convergence follows from the fact that is quasi-coherent for ≥ 3 by Lemma 2.18 so that E , 2 = 0 for ≥ 3 and ≥ 1 and hence the spectral sequence degenerates at the E 3 -page.
The next proposition is our main tool to attack the local Brauer group of a commutative ring spectrum.Recall to that purpose that a commutative ring spectrum is weakly 2 -periodic if 2 is an invertible 0 -module and
(iii) Fix ≥ 1.If is weakly 2 -periodic, with = 0 for not divisible by 2 , and such that 0 is regular noetherian, then there is a natural exact sequence Proof.Part (i) is the exact sequence of low-degree terms of the spectral sequence (2.24) using that 0 = 0 and that the spectral sequence degenerates at the E 3 -page.
Part (ii) is the content of [3,Theorem 5.11 and Corollary 7.13].Note that the exact sequence from part (i) splits in the case as ℤ is the free grouplike 1 -space, which lets us split the map → 1 ≃ ℤ in the connective case.Here we use that Pic( ) ≅ Pic( * ) ≅ Pic( 0 ) × ℤ by a result of [7,Theorem 21] and [50,Theorem 2.4.4] and that the functor  → Pic( 0 ) × ℤ sheafifies in the étale topology to the constant sheaf ℤ since every Picard group element is étale locally trivial.
Example 2.28.Let be a perfect field of positive characteristic and let be a 1-dimensional formal group law of height on .Let ( , ) be the Lubin-Tate spectrum associated to so that * ( , where ( ) denotes the ring of -typical Witt vectors of , each has degree 0, and has degree 2. We want to show that the local Brauer group LBr( ( , )) is typically non-zero.Note that this is related to but different than the results of Hopkins and Lurie in [38], who look at the Brauer group of ( )-local ( , )-modules, which is different from that of ( , )-modules.Moreover, they study the Brauer group and not just the local Brauer group.Since ( , ) is 2-periodic, part (iii) of Proposition 2.25 applies.To compute the groups that build LBr( ( , )), note first that H 2 (Spec 0 ( , ), ) ≅ H 2 (Spec , ) by Theorem 2.7 and H 2 (Spec , ) ≅ Br( ) is typically non-zero.Moreover, H 1 (Spec 0 ( , ), ℤ∕2) ≅ H 1 (Spec , ℤ∕2) by Theorem 2.8.
On the other hand, if we work over a separably closed (rather than finite) field , then H 2 (Spec , ) is again zero, but so is H 1 (Spec , ℤ∕2).This results in the fact that LBr( ( , )) is zero, and in particular, it implies that any non-trivial Brauer class in LBr( ( , ) is necessarily split by ( , ) → ( , ).
Remark 2.29.We do not in fact know an example where the differential from Proposition 2.25(iii) is non-zero.Similarly, it would be informative to know if there is a commutative ring spectrum such that Pic( ) → H 0 (Spec 0 , 1 ) is not surjective.
Remark 2.30.We are primarily interested in integral results as we want to understand contributions to the Brauer group for commutative ring spectra such as the various forms of topological modular forms.Nevertheless, when is even and weakly 2-periodic and if additionally 2 is a unit on , then there is an identifiable non-étale-locally trivial contribution to the Brauer group in general.If is actually 2-periodic and ∈ 2 a unit, let be the Azumaya algebra constructed in [30,Example 7.2]: it is an -algebra with * = * [ ] where | | = 1.We let be the sheafification of the components of containing 0 and the [ ] for units ∈ 2 for étale extensions of .Using that [ ] + [ ] lies in LBr( ) for any units , ∈ 2 , there is a natural fiber sequence of sheaves on CAlg ét .More generally, if 2 is not a unit on (but is still even and weakly periodic), then we can construct an extension where ∶ Spec 0 [ 1  2 ] → Spec 0 .An easy check using [30,Proposition 7.6] verifies that the algebraic Brauer group of , as defined in [30], is a subgroup of LBrW( ) = 0 ( ( )).

The Picard sheaf and local Brauer group of KO
The aim of this section is to show that the local Brauer group of KO is ℤ∕2.By the previous section, the key is to understand the étale Picard sheaf 0 KO on Spec KO.To achieve that, we essentially re-run the calculations of (KO) from Gepner-Lawson and Mathew-Stojanoska, but this time in sheaves of spaces on Spec KO.As an aside we will also compute Pic(KO ) for any étale extension of ℤ, where KO denotes the étale extension of KO lifting .(We will use similar notation for other ring spectra as well.) Recall that KO → KU is a 2 -Galois extension, and consequently (KO) ≃ ≥0 (KU) ℎ 2 by Galois descent.Similarly, if is an étale ℤ-algebra, then Thus, there is an equivalence of sheaves of connective spectra on Spec KO, which results in a homotopy fixed point descent spectral sequence with signature The notation ℰ , 2 ≅ ℋ ( 2 , KU ) means that the 2 -cohomology is taken in étale sheaves, and the differentials are Note that in the figures below, we will depict this spectral sequence with the Adams indexing convention, i.e. in the ( − , )-plane.
This allows us to compute 2 -cohomology and hence the ℰ 2 -page of (3.1).
Example 3.2.The action of 2 on 0 × is trivial, so the cohomology sheaves are where 2 and 2 fit into the exact sequence Note that on Spec ℤ, the sheaf 2 is isomorphic to the constant sheaf ℤ∕2; indeed, every étale extension of ℤ is a product of integral domains with 2 ≠ 0.
The following identification will not be necessary for our computation of LBr(KO), but we add it for completeness.
Proof.Since 2 is supported only at 2 with stalk given by × ∕( × ) 2 where = ℤ ℎ (2) is the strict Henselization, it is enough to compute the value of this group with its structure as a module over the absolute Galois group Ẑ of 2 (cf.[53,Corollary II.3.11]).Let = ( 2 ) be the ring of Witt vectors.There is an injection → and is the 2-adic completion of .We will see that the induced map × ∕( × ) 2 → × ∕( × ) 2 will turn out to be an isomorphism.
To prove that this map is injective, it suffices to show that if ∈ × is a square in × , then it is already a square in × .To see this, let = [ ]∕( 2 − ).This is a finite -algebra with 2-adic completion ∧ 2 ≅ [ ]∕( 2 − ).By the Hensel property for and , the ring is a product of either 1 or 2 local rings (see for example [63, Tag 04GG]) and ∧ 2 is a product of the same number by looking at fraction fields.If is a square in , then ∧ 2 is a product of 2 local rings, but then the same is true of .
Remark 3.4.The above result can also be read off from a much more sophisticated result due to Clausen, Mathew, and Morrow.They show in [20, Thm.A] that if is -torsion free, henselian along , and ∕ is perfect, then (2) and = ( 2 ), the 2-completion of .Applying the Clausen-Mathew-Morrow result to and , one obtains using that for any local ring we have an isomorphism K 1 ( ) ≅ × and that TC( )∕ ≃ TC( ∧ )∕ for any and any prime , for example by [20,Lem. 5.3].
To depict the spectral sequence (3.1), we will use symbols to denote the various sheaves and Table 1 can be used as a legend.
Table 1: An assortment of étale sheaves.Proof.Note that our spectral sequence consists on the ℰ 2 -page of quasi-coherent sheaves above the antidiagonal + = = 1.We will identify quasi-coherent sheaves on Spec 0 KO with their abelian groups of global sections.Since our spectral sequence can be seen as the sheafification of a presheaf of Picard homotopy fixed point spectral sequences, we can freely use the tools from [50].In particular, [50, Comparison Tool 5.2.4] implies that any 3differential originating from above the + = = 3 antidiagonal can be directly read off its counterpart in the homotopy fixed point spectral sequence for KU ℎ 2 ≃ KO.As in [50, Example 7.1.1],the claim follows.
Proof.The first claim follows from [50, Theorem 6.1.1],see also Example 7.1.1 in loc.cit.for the worked example in the case of the abelian group version of the spectral sequence (3.1).The map is surjective away from 2 since both sides vanish in that case.At 2, the map is surjective because ∕2 has stalks given by separably closed fields.The identification of the kernel is similar.
The ℰ 2 -page of the spectral sequence (3.1).All differentials on all pages above the anti-diagonal line + = 4 agree with their linear counterparts by [50].Not all information is shown in degrees ≤ −2.Dashed black arrows potentially differ from their linear partners, but they do not figure into the calculation of 0 KO .The dashed and dotted red arrow is non-linear and figures into the calculation of 0 KO .
Remark 3.7.By [30, Proposition 7.15], the differentials 1,0 2 , 2,0 2 and 2,1 3 are nonzero on global sections (where our spectral sequence is isomorphic, at least before differentials, to the usual Picard spectral sequence for KO).The first two differentials have ℤ∕2 as source and are thus determined by global sections: 1,0  2 is an isomorphism and 2,0 2 is the unique injection ℤ∕2 → ∕2.The differential 2,1 3 ∶ ∕2 → ∕2 is not determined by global sections, however, and thus remains unresolved.None of these differentials are needed for our computation of the Picard sheaf and hence of LBr(KO), though their result on global sections is used in the Gepner-Lawson computation of Br(KU|KO), which we will come back to in Remark 3.14..These computations determine the associated graded of 0 KO , but we can also resolve the extension problems as follows.

Proposition 3.8.
There is a filtration on 0 KO with associated graded pieces ℤ∕2, ℤ∕2, and * ℤ∕2, where is the closed inclusion Spec 2 → Spec ℤ.There is a surjective map from the constant sheaf ℤ∕8 to 0 KO , resulting in a non-trivial extension Proof.The first statement was proved in the lemmas above, namely we get a filtration on the E ∞ -page of the spectral sequence (3.1) with This filtration gives an inclusion * ℤ∕2 ≅ F 2 → 0 KO , and we need to identify the quotient with ℤ∕4.This quotient sits in an extension The filtration implies that the group of global sections H 0 (Spec ℤ, 0 KO ) is a finite group of cardinality at most 8. On the other hand, note that since H 1 (Spec ℤ, ) = Pic(ℤ) = 0, Proposition 2.25 implies that the homomorphism Pic(KO) → H 0 (Spec ℤ, 0 KO ) is an injection.Composing with the isomorphism we obtain a map of sheaves ℤ∕8 → 0 KO , which must be an isomorphism on global sections.The above also gives a map ℤ∕8 → that is the surjection ℤ∕8 → ℤ∕4 on global sections, implying that the extension (3.10) is non-trivial.But the only non-trivial extension of ℤ∕2 by ℤ∕2 on Spec ℤ, which has ℤ∕4 as global sections, is the constant sheaf ℤ∕4. 3 This identifies the quotient in (3.9), and to see that this extension is also not split, we again compare with the global sections.
Corollary 3.11.Let be an étale extension of ℤ.Then there is a short exact sequence If Spec is connected, the last term sits in an extension of the form where is the number of factors when decomposing ∕2 as a product of fields.
Proof.We first show the second part.The long exact sequence in cohomology associated to the extension in Proposition 3.8 takes the form is a surjection and thus we obtain the second claim.For the first part, we can assume that Spec is connected and thus a regular integral domain.From Proposition 2.25, we have a natural exact sequence Since Pic(KO ) maps surjectively onto ℤ∕4, the image of is the image of the restriction ′ ∶ (ℤ∕2) → Br( ).The 3 Indeed, Ext Spec ℤ (ℤ, ℤ∕2) ≅ H 1 (Spec ℤ; ℤ∕2) = 0 and thus the short exact sequence map → [ 1  2 ] induces a commutative square in which the horizontal arrows are the restricted boundaries ′ for and [ 1  2 ] respectively.The right-hand vertical map is an injection by Theorem 2.5 since Spec [ 1  2 ] ⊂ Spec is dense.Thus ′ = 0. Remark 3.12.As a consequence of the preceding corollary, we see that it is not true that for every étale extension ℤ ⊂ with Spec connected, we have Pic(KO ) ≅ Pic( ) × ℤ∕8 or Pic( ) × ℤ∕4.For example take the field = ℚ( √ 17), 2 , and set = ℤ[ ][ 1  17 ].Here we have 2 = −(1 + )(2 − ) and thus ∕2 ≅ 2 × 2 .We obtain Pic(KO ) ≅ ℤ∕8 × ℤ∕2.In the Picard spectral sequence for KO , the "exotic" elements arise as the kernel of the 3 -differential is bigger than ℤ∕2, namely (ℤ∕2) 2 in our example.How can we understand these additional classes?Let us sketch a conjectural general picture of the filtration on Pic( ) from the Picard spectral sequence for a faithful -Galois extension → .Let ∈ Pic( ).The 0-line detects the image ⊗ ∈ Pic( ).If ⊗ ≃ (and such an equivalence is chosen), the 1-line H 1 ( ; 0 ) describes how the -action on * ( ⊗ ) is twisted in comparison to that on * .Thus, the E 2 -term of the homotopy fixed point spectral sequence for ( ⊗ ) ℎ 2 ≃ is isomorphic to that for ℎ 2 ≃ if has filtration at least 2, which we will assume now.We fix such an isomorphism.We conjecture that if has filtration ≥ 2, its reduction to H ( , ( )) ≅ H ( ; −1 ) equals (1) in the homotopy fixed point spectral sequence for ( ⊗ ) ℎ ≃ .Back to our example, this means that the three non-trivial classes in Pic(KO ) of filtration 3 correspond conjecturally to invertible KO -modules such that 3 (1) is 1, and 1 + respectively.
The identification of KO allows us to compute the local Brauer group of KO.Recall in this context that Gepner and Lawson proved in [30,Proposition 7.17] that the subgroup Br(KU|KO) ⊆ Br(KO) of classes killed by the extension KO → KU is isomorphic to ℤ∕2.We will show that LBr(KO) is also ℤ∕2, and in fact it will be isomorphic to Br(KU|KO).
Proof.To use the exact sequence in Proposition 2.25, we first need to compute H 1 (Spec ℤ, 0 KO ), which we will do using Proposition 3.8.Since there is a unique ℤ∕2-Galois extension of Spec 2 , Theorem 2.9 implies Moreover H 1 (Spec ℤ, ℤ∕4) = 0 as there are no unramified ℤ∕4-Galois extensions of ℚ.Since furthermore H 0 (Spec ℤ, KO ) ≅ Pic(KO) → H 0 (Spec ℤ, ℤ∕4) is surjective, the long exact cohomology sequence associated with the short exact sequence of sheaves in (3.9) implies that H 1 ét (Spec ℤ, 0 KO ) is isomorphic to ℤ∕2.To conclude LBr(KO) ≅ ℤ∕2 using Proposition 2.25, it remains to show the vanishing of the differential We show this by comparison to KU: The map KO → KU induces a map of presheaves KO → KU on the étale site of Spec ℤ, which we identify with either of the étale sites of KO and KU using the isomorphism 0 KO ≅ ℤ ≅ 0 KU.Thus, we get an induced map of descent spectral sequences and in particular a commutative diagram where the right vertical map is an equality.Since H 1 (Spec ℤ, 0 KU ) ≅ H 1 (Spec ℤ, ℤ∕2) = 0, we see that the top differential must vanish.Therefore, LBr(KO) ≅ H 1 (Spec ℤ, 1 KO ) ≅ ℤ∕2.For the second part of our claim, note first that Br(ℤ[ 1  2 , 4 ]) and Br(ℤ[ ).This is clear in the first case as * ℤ∕2 restricted to Spec ℤ[ 1  2 ] vanishes.In the second case, we use that the extension 2 ⊂ 4 ≅ 2 [ 3 ] kills the non-trivial element of H 1 (Spec 2 , ℤ∕2).Remark 3.14.Note that since LBr(KU) = 0, functoriality of the local Brauer group implies that the non-zero class ∈ LBr(KO) ≅ ℤ∕2 is killed by the ℤ∕2-Galois extension KO → KU, i.e. lies in the relative Brauer group Br(KU|KO).By the main result of [30], LBr(KO) thus agrees with Br(KU|KO) though a priori we only get an inclusion.This gives a new proof of that the Galois-cohomological Brauer class found in [30,Proposition 7.15] is representable by an Azumaya algebra, which Gepner and Lawson prove instead with an unstable descent spectral sequence.See also Example 4.15 for another perspective.
We urge the reader to consider the analogue of the descent spectral sequence computation of Br(KU|KO) as in [30, Figure 7.2] in the case of the relative Brauer group of KO[ 1  3 , 3 ] with respect to KU[ 1  3 , 3 ].As the class in filtration six contributing to Br(KU|KO) has to die in Br(KO[ 1  3 , 3 ]), there must be a new 3 killing it.This 3 is given by the formula in [50, Theorem 6.1.1],the point being that the image of  → + 2 on ℤ[

Brauer groups of nonconnective spectral DM stacks
In this section, we turn to Brauer groups of nonconnective spectral Deligne-Mumford (DM) stacks.A significant difference will be that the Brauer group is in general no longer 0 of the global sections of the Brauer sheaf, yielding to a distinction between Brauer group and cohomological Brauer group, which we will explain below.
To fix notation, we recall the following definition from Lurie [47].
Definition 4.1.A nonconnective spectral DM stack is a spectrally ringed ∞-topos ( , ) such that there exists a covering ∐ ∈ → * of the final object where for each there is an equivalence ( ∕ , | ) ≃ Spec for some commutative ring spectrum . 4If is connective, we say that ( , ) is a connective spectral DM stack; if is discrete, we say that ( , ) is a classical DM stack.Definition 4.6.We let Br ′ ( , ) = 0 Γ( , ) = 0 ( ( )).This is the cohomological Brauer group of .Similarly, the cohomological local Brauer group of ( , ) is We call the space of global sections ( ) the Brauer space and similarly for the local Brauer space ( ) ≃ ( ).
For the following definition, recall that a quasi-coherent sheaf is perfect if it is dualizable or, equivalently, if it becomes a compact object when restricted to an affine.Definition 4.9.A quasi-coherent sheaf of -algebras on a nonconnective spectral DM stack ( , ) is an Azumaya algebra if the following equivalent conditions hold: is perfect, locally generates QCoh( , ), and the natural map op ⊗ → ( ) is an equivalence; (ii) there is an étale cover {Spec ← ← ← ← ← ← ← ← → ( , )} ∈ such that * is an Azumaya -algebra for all .
Definition 4.10.Any Azumaya algebra on ( , ) defines a point of and hence an element [ ] of Br ′ ( , ), called the class of .If is an Azumaya algebra, then so is the opposite algebra op and we have . These assertions may be verified locally using Definition 4.9(ii) and Definition 2.11(b).Let Br( , ) ⊆ Br ′ ( , ) be the subgroup consisting of the classes of Azumaya algebras.Let LBr( , ) = LBr ′ ( , ) ∩ Br( , ) inside Br ′ ( , ).We call these the Brauer and local Brauer groups of ( , ).
Proof.If is an Azumaya algebra representing , define QCoh( , ) as the limit of Mod ( ) over all étale maps Spec → ( , ); this can be identified with a full subcategory of Mod (Shv Sp ( )).We have QCoh( , ) ≃ QCoh( , ) and under this equivalence corresponds to a perfect local generator.Conversely, given a perfect local generator ℱ of QCoh( , ), the sheaf of endomorphisms (ℱ ) is an Azumaya algebra with class .
Proposition 4.14 will not be enough to show the agreement of Br ′ and Br for the derived moduli stack of elliptic curve since the moduli stack of elliptic curves does not have a finite étale cover by an affine scheme [68].This issue will be solved by Theorem 4.17 below.Before we state it, we introduce the following definition needed for its proof.Definition 4.16.Let ( , ) be a nonconnective spectral DM stack.Let ∈ Br ′ ( , ) be a Brauer class and let ℱ ∈ QCoh( , ) be a perfect local generator.We say that ℱ is a global generator if ℱ is compact and if QCoh( , ) is compactly generated by ℱ .The proof follows the work of [67] and [3] which uses older arguments of Bökstedt-Neeman [14] and Bondal-van den Bergh [15] who showed that for a quasi-compact and quasi-separated scheme , the derived category of complexes of -modules with quasi-coherent cohomology sheaves admits a single compact generator, which is global in the sense above.Other important examples of Br = Br ′ in the non-derived and derived context have been established in [28,22,35,18].

Proof. Note first that each
⊆ is relatively scalloped in the sense of [47, 2.5.4.1]. 5  We glue local perfect generators as in [3,Theorem 6.11] or [67, Proposition 5.9], taking care in each step to produce a global generator.Let be the union 1 ∪ ⋯ ∪ in .It is enough to prove that there is a global generator of QCoh( , ) for each = 1, … , and hence | is in Br( , ) for each .The base case follows from assumption (b).Suppose the conclusion holds for some 1 ≤ < .Set = ∩ +1 and consider the pullback square 5 Quasi-affine morphisms are relatively scalloped; these will be enough for our applications.
Using that the square (4.18) is a pullback, the vertical fibers are equivalent stable ∞-categories.Thus, corresponds to a compact object of QCoh( +1 , ) which vanishes on .On the other hand, by induction there is a global generator ℋ of QCoh( , ).Our goal will be to lift ℋ to +1 .The fact that QCoh( +1 , ) → QCoh( , ) is a localization and preserves compact objects implies that QCoh( , ) is generated by the image of ℱ +1 .Since the kernel is compactly generated by a compact object of QCoh( +1 , ) we are in the setting of Thomason's extension proposition [66, 5.2.2] (see [55,Corollary 0.9] for the generality needed here), which says that if ℬ → → is a Verdier sequence of idempotent complete stable ∞-categories, then an object ℳ ∈ lifts to if and only if its class [ℳ] ∈ K 0 ( ) lifts to K 0 ( ).Thus, possibly by replacing ℋ by ℋ ⊕ Σℋ (which always has vanishing class in K 0 ), we see that the restriction of ℋ to QCoh( , ) lifts to a compact object ℋ +1 of QCoh( +1 , ).Gluing ℋ and ℋ +1 via the pullback (4.18), we obtain a compact object ℰ of QCoh( , ).Let = ℰ ⊕ .We claim that is a global generator of QCoh( +1 , ).Verification is standard and left to the reader.
This corollary will be applied in Proposition 8.1 to the derived moduli stack of elliptic curves.
In classical algebraic geometry, there are few 0-affine DM stacks.If is a scheme, is 0-affine if and only if it is quasi-affine, which is to say quasi-compact and can be embedded as an open subscheme of Spec for some .In this case, one can take = H 0 ( , ).More generally, quasi-affine connective spectral DM stacks are 0-affine.
Remarkably, in the theory nonconnective spectral DM stacks, there is an additional wealth of non-classical examples, as supplied by the following theorem of [49].
Our main example will be (ℳ, ), where ℳ is the moduli stack of elliptic curve and is the weakly 2-periodic sheaf of ∞ -ring spectra defined by Goerss, Hopkins and Miller [25].Later, the nonconnective spectral DM stack (ℳ, ) was reinterpreted and reconstructed by Lurie to classify oriented spectral elliptic curves [43] and we will refer to it as the derived moduli stack of elliptic curves.
Our computation gives examples of Brauer classes on a commutative ring spectrum which are not killed by an étale cover.To see this, pick ∈ Br( ) and suppose that is killed by an étale cover → .Then, ⊗ defines a new quasi-affine connective spectral DM stack ( , ).(The underlying ∞-topos is naturally equivalent to × 4 Spec 0 .)By quasi-affineness, it follows that restricts to 0 on ( , ).However, the induced map on the Brauer group is This map is equivalent to which is injective as → is faithfully flat.Thus, = 0 and so no nonzero class in the Brauer group can be killed by an étale cover.

The Picard sheaf of TMF
To compute the local Brauer group of TMF, it is necessary to first compute the Picard sheaf of TMF, which we will attack in this section.Our key tool is a sheafy version of the Picard spectral sequence used in [50], which we will introduce next.Let (ℳ, ) be the derived moduli stack of elliptic curves, where denotes the Goerss-Hopkins-Miller-Lurie sheaf of ∞ -ring spectra.By Proposition 6.6, the descent spectral sequence identifies 0 TMF with H 0 (ℳ, 0 ) and the latter one computes to be ℤ[ ].Thus, the underlying classical morphism of (ℳ, ) → Spec TMF is the map ∶ ℳ → 1 = Spec ℤ[ ]; we will denote (ℳ, ) → Spec TMF by as well For every étale map ∶ Spec → 1 , we obtain an induced sheaf of ∞ -ring spectra on the base change ℳ = ℳ × 1 Spec .Let denote the Picard sheaf corresponding to on ℳ (with subscript left out if Spec = 1 ).We obtain a Picard spectral sequence H (ℳ ; ) ⇒ − (ℳ , ).Sheafification thus yields a spectral sequence in the abelian category of étale sheaves of abelian groups on Spec ℤ[ ] = 1 .We note that (ℳ , ) ≃ (TMF ) where TMF is the étale extension of TMF realizing .Indeed: the natural map As (6.1) arises as the sheafification of Picard spectral sequences, we can freely apply the tools from [50] for Picard spectral sequences.More precisely, these apply to the comparison to the analogous spectral sequence R * ⇒ − Spec TMF .Viewing a quasi-coherent sheaf on Spec 0 TMF ≅ Spec ℤ[ ] as a ℤ[ ]-module, this agrees with the usual descent spectral sequence for computing * TMF, but remembering the ℤ[ ]-module structure.See in particular Proposition 6.7 for a precise statement we will be using.Warning 6.3.In contrast to the descent spectral sequence for * TMF, the Picard spectral sequence will in general not be ℤ[ ]-linear even in the range where its E 2 -term agrees with a shift of the descent spectral sequence (i.e. for ≥ 2).We do, however, have ℤ[ ]-linearity in the range specified by Proposition 6.7 below.This should be seen in light of (a sheafy analogue of) [50, Corollary 5.2.3].Remark 6.4.Alternatively, the sheafy Picard spectral sequence can be constructed as the relative descent spectral sequence for * , i.e. the spectral sequence associated to applying (sheafy) * to the tower * ≤⋆ .Indeed: the presheaf of Picard spectral sequence considered above is obtained by applying presheaf homotopy groups pre * to the tower * ≤⋆ , and thus its sheafification agrees with the relative descent spectral sequence.
We will not compute the whole spectral sequence (6.1), but obtain the following result about the 0-stem, which will be crucial to our results about the local Brauer group.ℳ is a subsheaf of ∕(2, ); all other graded pieces vanish.
In fact, in the last two items we describe the graded pieces as subsheaves of what we see on the E 6 -page, but there are (at most) 2 more potential differentials originating from these spots.
The rest of this section will be devoted to the proof of the theorem.We will use Table 2 for notation for sheaves on Spec ℤ[ ] appearing in the spectral sequence (6.1).Fig. 3 on Page 24 shows the E 2 -page of the spectral sequence (6.1).The general pattern follows from the work of Mathew-Stojanoska [50] and the computations of the homotopy groups of TMF, as in Bauer [9].
To prove Theorem 6.5, we show in the next subsection that there are no contributions in filtration degrees above 7.Then, we analyze each remaining filtration in turn.

High filtrations
In this section, we use the comparison tool of [50] to narrow down the possible filtration degrees computing to 0 * ℳ .We use the following facts about the large-scale structure of the spectral sequence which can be read off from the charts in [9] for tmf or [41] for Tmf by inverting the discriminant modular form Δ (or rather Δ 24 since only this is a permanent cycle).(2) The longest differential in the additive spectral sequence is a 23 .
Here, column always refers to − = , i.e. to the column if drawn in Adams grading.We recall the following key tool from [50].
The E 3 -page of the sheafy spectral sequence (6.1) for Pic of the moduli stack.Above the + = 1 diagonal, only 2-primary torsion information is shown.
Proof.By [50,Comparison Tool 5.2.4], this is true for each term in the presheaf of Picard spectral sequence and is thus also true after sheafification.
Using these results, we can indeed show that the Picard spectral sequence eventually vanishes in high enough degrees.Proposition 6.8.Everything above row 7 in column 0 vanishes in the E ∞ -page of the Picard spectral sequence; likewise above row 30 in column −1.After inverting 2, the latter vanishing holds already above row 14.
Proof.The claim about column 0 follows from the Comparison Tool (Proposition 6.7) and the further claim that in the additive spectral sequence E , 2 = H (ℳ, ℳ ) ⇒ − TMF every spot in the (−1)-column above row 7 is killed by or supports a -differential, which is an isomorphism and satisfies ≤ for = + being the antidiagonal of origin.Indeed, the corresponding differential also has to occur in the Picard spectral sequence and the isomorphism of groups becomes an isomorphism of quasi-coherent sheaves.
By inspection the further claim is true up to row 23 (on the E 5 -page there is only one class in column −1 between row 7 and 24, namely in row 19 and this is killed by a 9 ).As noted in Proposition 6.6, the longest possible differential is a 23 and the E ∞ -term vanishes; thus everything above row 23 is killed by or supports a differential, which is at most a 23 .Moreover, by inspection, nothing in the additive spectral sequence in column 0 below row 23 supports a differential killing a class above row 23 in column −1.
The proof for column −1 of the Picard spectral sequence is analogous.
The Proposition 6.8 implies that to prove Theorem 6.5 it is enough to analyze gr 0 * ℳ for 0 ≤ ≤ 7.

Row 0
Since the geometric fibers of ∶ ℳ → Spec ℤ[ ] are connected and 0 ℳ ≅ ℤ∕2, we have R 0 * ℤ∕2 ≅ ℤ∕2.This term does not support any differentials since TMF [1] is a global section of the Picard sheaf which restricts to a generator of ℤ∕2 everywhere; this proves part (0) of Theorem 6.5.

Row 1 and the algebraic Picard sheaf
The next term to understand is R 1 * , which appears on the E 2 -page at ( , ) = (1, 1).This calculation is done on the classical moduli stack.The sheaf R 1 * is the sheafification of the presheaf that sends every étale → Spec ℤ[ ] to Pic(ℳ × Spec ℤ[ ] ).Thus our next lemma can be seen as a sheafy analogue of the classical computation that Pic(ℳ) ≅ ℤ∕12 (see [27]), where a generator is given by the Hodge bundle that arises as the pushforward of the sheaf of differentials of the universal elliptic curve.We will indeed use the stronger result from [27] that the same isomorphism holds over any reduced and normal base ring with vanishing Picard group.Moreover, we will use that Pic( 1 ) ≅ Pic( ) for any regular noetherian , where 1 = 1 × Spec ; this follows e.g. from the 1 -invariance of the divisor class group as in [36, Proposition 6.6, Corollary 6.16].Proposition 6.9.Denote by ∶ Spec ℤ → Spec ℤ[ ] the inclusion corresponding to the value of the function on Spec ℤ[ ] and by the inclusion of its complement.
Proof.We will explain first why it suffices to show the surjectivity of ℤ∕12 → R 1 * and identify its kernel.Note that there is an exact sequence for any and any étale sheaf ℱ .Thus we obtain the claimed extension from the proposition by quotienting the first two terms of the exact sequence 0 → ℤ∕3 ⊕ ℤ∕2 → ℤ∕12 → ℤ∕2 → 0 by ( 0 ) ! ℤ∕3 ⊕ ( 1728 ) ! ℤ∕2 and using the snake lemma if indeed is exact.This exactness can be checked on the level of stalks, which is the content of the rest of the argument.Let ∶ Spec → 1 be a geometric point (corresponding to some ∈ ) and ∶ Spec → ℳ its unique lift.We will show that ℤ∕12 → (R 1 * ) is surjective with the prescribed kernel.One can deduce from [1, Lemma 2.2.3] that the base change of ℳ to the étale stalk of 1 at is equivalent to the quotient stack [Spec ∕Aut( )], where is strictly Henselian with residue field and Aut( ) is acting trivially on (cf.[52,Proposition 8]).We can compute the stalk (R 1 * ) as Pic([Spec ∕Aut( )]) ≅ H 1 (Aut( ); ( )).For the values of Aut( ) we refer to [62,Section III.10].We will proceed with a case distinction based on and the characteristic of .decomposes into a ℚ-vector space and a torsion group, which maps isomorphically to ( )[ 1 ] ≅ ℚ∕ℤ[ 1 ] (cf. the proof of [52,Lemma 9]).We obtain If is of characteristic not 2 or 3, we have Aut( ) ≅ ℤ∕4 if = 1728 and Aut( ) ≅ ℤ∕6 if = 0, which implies that the corresponding stalks of R 1 * are the Pontryagin duals of ℤ∕4 and ℤ∕6, i.e. isomorphic to ℤ∕4 and ℤ∕6 as well.(Note that in these cases H 1 (Aut( ); ( )) is 12-torsion, so inverting changes nothing.) Concretely, the map ℤ∕12 ≅ Pic(ℳ) → Hom(Aut( ); ℚ∕ℤ[ 1 ]) sends a line bundle ℒ to the action of Aut( ) on ℒ by the roots of unity ℚ∕ℤ[ 1 ] ≅ ∞ ⊂ ( ).By the proof of [62,Theorem III.10.1], in our case a generator of Aut( ) acts by a fourth respectively a sixth root of unity on the invariant differential and thus on (for the standard generator of Pic(ℳ) as above).Thus summarizing, we see that the map ℤ∕12 → (R 1 * ) is surjective with the prescribed kernel unless char( ) = 2, 3 and corresponds to = 0 ≡ 1728.In particular, we see that Case 3: char( ) = 2 or 3 and = 0 = 1728: From now on let ∶ → 1 be a geometric point with char( ) = for = 2, 3 corresponding to = 0.For a base ring , denote by ℳ the base change ℳ × Spec .We will show that ∶ ℤ∕12 → (R 1 * ) is an isomorphism by comparison with the known computation of the Picard group of Pic(ℳ ) for certain .To that purpose we will use the Leray spectral sequence for the map ∶ ℳ → 1 .Let us display the part relevant for the computation of Pic.
If were not injective, there would be some [ ] ∈ ℤ∕12 (namely [4] or [6]) such that ([ ]) is zero in every étale stalk in characteristic and hence Φ would not be injective either.We see that is thus indeed injective and thus ℱ agrees with the image of .
Moreover, H 1 ( 1 ; ℤ∕2) ≅ H 1 ( 1 ; 2 ) sits in a short exact sequence with H 1 ( 1 ; )∕2 = 0 and H 2 ( 1 ; ) [2] = 0 and thus has to vanish as well.We conclude that H 1 ( 1 ; ℱ ) = 0. Summarizing, we have an exact sequence We have seen above that the natural morphisms from ℤ∕12 to the first two non-trivial groups are isomorphisms and thus the map between them is an isomorphism.Thus H 0 ( 1 ; ) vanishes.As is supported at , we see that its stalk at vanishes and thus that is also surjective.
We claim that there are no differentials out of E 1,1 2 = R 1 * . Indeed, by the preceding proposition is a surjective map of sheaves on 1 = Spec ℤ[ ].Thus, as long as the classes of ℤ∕12 lift to invertible sheaves on the derived moduli stack, surjectivity of the map means there cannot be differentials.But, this ℤ∕12 is generated by TMF [2], which gives part (1) of Theorem 6.5.

3-torsion in Row 5
For higher filtrations it is necessary to compute differentials.Differentials , in the sheafified Pic spectral sequence where ≤ − 1 (i.e., where the "length" of the differential is smaller than the coordinate = + of the antidiagonal of origin) can be directly read off the descent spectral sequence computing * TMF by Proposition 6.7.We will use this fact without further comment.
For the rest of the analysis, we will work separately with 2 and 3 inverted, to analyze the 3 and 2-torsion, respectively.The only possible contribution to 3-torsion in 0 * ℳ is the kernel of the 9 differential on E 5,5 9 .We will implicitly invert 2 throughout this section.Lemma 6.10.The differential is surjective and the kernel is * ℤ∕3, where is the closed inclusion of Spec 3 into Spec ℤ[ ] at = 3 = 0.
As Fig. 4 on Page 29 proves, the lemma shows that Part (5) of Theorem 6.5 holds with 2 inverted and all the other graded pieces vanish with 2 inverted.Thus it remains to analyze the 2-torsion and we will implicitly work 2-locally everywhere.We will compute two further differentials affecting the zeroth column of the Picard spectral sequence and then give an outlook on what remains to be done to compute all remaining differentials.

Row 3
There is a 3 -differential, This differential is of the form There will be no further differentials from this spot because all possible further targets are supported at (2, ).

Long differentials
As already established in Proposition 6.8, in column 0 everything above row 7 must be zero on the E ∞ -page.As Fig. 6 on Page 31 and the preceding discussion shows, the only remaining possible differentials are a 13 and 25 originating in row 5 and a 11 and a 23 originating in row 7. We can show the vanishing of one of these differentials.
: The E 7 -page of the sheafy spectral sequence (6.1) for Pic of the moduli stack.Above the + = 1 diagonal, only 2-primary torsion information is shown.
we obtain an exact sequence The composition is a surjection by comparison to TMF (similar to the preceding example) since H 0 ( 1 ; ) → H 0 ( 1 ; ) is an isomorphism (using a comparison on associated graded pieces and the five lemma).This implies that H 0 ( 1 ; ) ≅ ℤ∕24 and that Eq. (7.4) is short exact.
For the first claim, we recall from Proposition 2.25 the exact sequence ) by Theorem 2.5 and ℐ ( 1 ) = 0.

The local Brauer groups of TMF and (ℳ, )
The aim of this section is to show that local Brauer groups of TMF and the derived moduli stack (ℳ, ) are infinitely generated and to compute them up to finite ambiguity.First, we observe the coincidences of various Brauer groups pertinent to this example.Proof.Parts (i) and (iii) follow from Corollary 4.19.Indeed, we can use the cover with opens ℳ[ 1 2 ] = ℳ×Spec ℤ[ 1  2 ] and ℳ[ 1  3 ] = ℳ × Spec ℤ[ 1  3 ].Condition (a) of Theorem 4.17 follows because the kernels of QCoh(ℳ[ 1  2 ]) → QCoh(ℳ[ 1  6 ]) and QCoh(ℳ[ 1  3 ]) → QCoh(ℳ[ 1  6 ]) are generated by the compact objects ∕3 and ∕2, respectively.Moreover, both ℳ[ 1  2 ] and ℳ[ 1  3 ] admit a finite étale cover from an affine scheme, for example the moduli stacks ℳ(4) and ℳ(3) of elliptic curves with full level 4 and full level 3 structures, respectively.Thus, by (the proof of) Proposition 4.14, -twisted sheaves on ℳ[ 1  2 ] and ℳ[ Proof.By the previous proposition, we can apply the spectral sequence from Construction 4.8 for the computation of LBr(ℳ, ) = 0 (ℳ).Up to a onefold shift, this agrees with the Picard spectral sequence for TMF from [50].Note that H 1 (ℳ; ℤ∕2) = 0 (since ℳ has no finite covers) and H 2 (ℳ, ) = Br(ℳ) = 0 by [4].The non-sheafy version of Proposition 6.8 holds by the same arguments and thus only terms of filtration at most 30 can survive in the Picard spectral sequence in column (−1).By the results from [50], we know all differentials from the 0-column to the (−1)-column of the Picard spectral sequence: up to row 30 there are only 3 and they are 2-local (cf.especially Figure 6 to 10 in [50]).One thus observes that the -torsion is as stated for ≥ 3.In the E ∞ -term, we have 2-locally the kernel of an unknown 9 -differential from coker( 3 ∶ 2 [ ] → 2 [ ] ⊕ ℤ∕2) in row 6 to a ℤ∕2 in row 15 (which must be abstractly isomorphic to (ℤ∕2) ∞ , as in the proof of Eq. (8.5) below), and further copies of ℤ∕2 in rows 10, 18 and 30, which cannot support differentials.Here, we use [50, Comparison Tool 5.2.4], both to show that possible targets of differentials vanish and to show the vanishing of a possible 5 on the class in row 10.This implies the result.
Next, we give a similar (but less precise) computation for LBr(TMF).Later, we will compare the two calculations.
Thus, the image of H 2 ( ∩ ; ) → H 3 ( ; ) is 2-locally of the form (finite ⊕ divisible).Since the image of TMF 3 must be an 2 -vector space (as the image of an 2 -vector space), the image of TMF 3 must be finite.We deduce that the kernel of TMF consists of the finite group H 1 ( 1 ; F 5 0 * ℳ ) plus an infinite-dimensional subspace of H 1 ( 1 ; * !ℤ∕2) ≅ ∞ 2 , as claimed.It remains to show that the restrictions of TMF to and are zero.The case of is clear as * !ℤ∕2 is supported outside of .For the case of , recall from [61, Lemma 3.2] that the base change ℳ × 1 is equivalent to × B 2 , i.e. the stack quotient of by the trivial 2 -action; this yields in particular an étale map → ℳ × 1 → ℳ.We obtain a diagram / / H 3 ( ; ).
Here, ( ) refers to the boundary map in the long exact sequence from Proposition 2.25 for the ring spectrum ( → ℳ), while ( , TMF ) uses the restriction of the spectral scheme structure of Spec TMF to ; note that both affine spectral schemes here have underlying scheme .In particular, the rightmost vertical map is an isomorphism.Note further that F 3 0 = 0 since all terms in the sheafy Picard spectral sequence of filtration 3 and higher are of the form H 2 +1 ( ; 2 ) for ≥ 1, which all vanish since is an affine scheme.Thus, we see that TMF is indeed zero after restricting to .
Since Br(TMF) → Br(ℳ, ) is an isomorphism, LBr(TMF) → LBr(ℳ, ) is an injection.We want to describe how to obtain a computational handle on this injection.In conjunction with Theorem 8.2, this will also provide an alternative proof of Theorem 8.3.
Consider the sheaf * on 1 .It assigns to every étale open , the spectrum ( × 1 ℳ, ).The relative descent spectral sequence (cf.Remark 6.4) for * takes the form and provides thus a method to compute * * .But since is just a suspension of , this spectral sequence is up to a shift actually the same as the sheafy Picard spectral sequence considered in Section 6.In particular, one observes that * ≅ −1 TMF for ≥ 1, but have additionally interesting sheaves for ≤ 0, which are computed by the ( − 1)-column of the sheafy Picard spectral sequence.We obtain a descent spectral sequence Note that the map TMF → * induces a map of descent spectral sequences, which is essentially the inclusion of the top two anti-diagonals.Fig. 7 gives a schematic picture of part of this map, with the image of the descent spectral sequence of TMF colored in blue.Theorem 8.7.The injection LBr(TMF) → LBr(ℳ, ) has finite cokernel and is an isomorphism after inverting 2.
Regarding the 2-local picture, Fig. 6 shows that the only possible contributions are in Rows 6, 18 and 30 and each of them is an ∕(2, ) on the E 7 -page.The same argument as provided in the proof of Theorem 8.3 for the finiteness of H 1 ( 1 ; F 5 0 * ℳ ) (2) shows also the finiteness of H 0 ( 1 , 0 * ).This in turn implies the finiteness of the cokernel of LBr(TMF) → LBr(ℳ, ).

Fig. 1 and
Fig.1and Fig.2show the spectral sequence (3.1).Several lemmas explain the nature of the differentials and the calculation of the ℰ 4 -page.Lemma 3.5.The ℰ 4 -page is zero in column 0 above row 3.

Figure 2 :
Figure 2: A part of the ℰ 4 -page of the spectral sequence (3.1).

Example 4 . 5 . 1 0≅
In general, Brauer classes on connective commutative ring spectra are étale-locally trivial by[3, Theorem 5.11].We also have 1 ≥0 ≅ ℤ by the computation of Picard groups of connective commutative ring spectra.On the other hand, 0 and 1 are highly dependent on the nature of itself.

Figure 4 :Figure 5 :
Figure4: The E 5 -page of the sheafy spectral sequence (6.1) for Pic of the moduli stack.Above the + = 1 diagonal, only 3-primary torsion information is shown.