The relative Brauer group of K (1) -local spectra

Using profinite Galois descent, we compute the Brauer group of the K (1) -local category relative to Morava E-theory. At odd primes this group is generated by a cyclic algebra formed using any primitive ( p − 1) st root of unity, but at the prime two is a group of order 32 with nontrivial extensions; we give explicit descriptions of the generators, and consider their images in the Brauer group of KO . Along the way, we compute the relative Brauer group of completed KO , using the étale locally trivial Brauer group of Antieau, Meier and Stojanoska.


Introduction
The classification of central simple algebras over a field is a classical question in number theory, and by the Wedderburn theorem any such algebra is a matrix algebra over some division algebra, determined up to isomorphism.If one identifies those algebras which arise as matrix algebras over the same division ring, then tensor product defines a group structure on the resulting set of equivalence classes.This relation is Morita equivalence, and the resulting group is the Brauer group Br(K).One formulation of class field theory is the determination of Br(K) in the case that K is a number field.
One consequence of Wedderburn's theorem is that every central simple algebra is split by some extension L/K; in fact, one can take L to be Galois.This opens the door to cohomogical descriptions of the Brauer group: by Galois descent, one obtains the first isomorphism in where PGL ∞ denotes the Galois module lim − → PGL k (K s ).The second isomorphism is [Ser79, Proposition X.9], and follows from Hilbert 90.This presents Br(K) as an étale cohomology group, and allows the use of cohomological techniques in its determination.Conversely, it gives a concrete interpretation of 2-cocycles, analogous to the relation between 1-cocycles and Picard elements.
Globalising this picture was a deep problem in algebraic geometric, initiated by the work of Azumaya, Auslander and Goldman, and the Grothendieck school.There are two ways to proceed: on one hand, one can generalise the notion of central simple algebra, obtaining that of an Azumaya algebra on X, and the group Br(X) classifying such algebras (again, up to Morita equivalence).On the other, one can use the right-hand side of (1), which globalises in an evident way; this yields the more computable group Br ′ (X) := H 2 (X, G m ) tors .A famous problem posed by Grothendieck asks if these groups agree in general.While there is always an injective map Br(X) → Br ′ (X), surjectivity is known to fail for arbitrary schemes (e.g.[Edi+01,Corollary 3.11]).The fundamental insight of Toën was that it is key to pass to derived Azumaya algebras, in which case one can represent any class in H 2 (X, G m ) as a derived Azumaya algebra, including nontorsion classes [Toë12].Toën's work shows that even the Brauer groups of classical rings are most naturally studied in the context of derived or homotopical algebraic geometry.This initiated a study of Brauer groups through the techniques of higher algebra, and in recent years they have become objects of intense study in homotopy theory.
In this context, the basic definitions appear in [BRS12]: as in the classical case, the Brauer group of a ring spectrum R classifies Morita equivalence classes of Azumaya algebras, defined by entirely analogous axioms.This gives a useful class of noncommutative algebras over an E ∞ -ring: for example, Hopkins and Lurie [HL17] computed Brauer groups as part of a program to classify E h -algebra structures on Morava K-theory, where E h denotes Lubin-Tate theory.Categorifying, the Azumaya condition on an R-algebra A is equivalent to the requirement that the ∞-category Mod A be R-linearly invertible, and so the Brauer group is also the Picard group of the E ∞ -algebra Mod R ∈ Pr L ; in this guise the Brauer space and its higher-categorical analogues are the targets for (invertible) factorisation homology, as an extended quantum field theory [Lur09; Sch14; AF17; Hau17; FH21].Computations of Brauer groups of ring spectra of particular interest appear in [AG14; GL21;AMS22].
In this document we study Brauer groups in the monochromatic setting.Recall that Hopkins and Lurie focus on the K(h)-local Brauer group of E h .Our objective is to extend this to the Brauer group of the entire K(h)-local category: explicitly, this group classifies of K(h)-local Azumaya algebras up to Morita equivalence.The computations of op.cit.show that the groups Br(E h ) are highly nontrivial, unlike the analogous Picard groups.We therefore complement this by computing the relative Brauer group, which classifies those K(h)-local Azumaya algebras which become trivial after basechange to E h .In analogy with the established notation for Picard groups (e.g.[Goe+15,p.5]),this group will be denoted Br 0 h .It fits in an exact sequence , where G h denotes the Morava stabiliser group, and this gives access (at least in theory) to the group Br h := Br(Sp K(h) ).The group Br 0 h also has a concrete interpretation in terms of chromatic homotopy theory: it classifies twists of the G h -action on the ∞-category Mod E h (Sp K(h) ).For the standard action (by basechange along the Goerss-Hopkins-Miller action on E h ), one has Sp K(h) ≃ Mod E h (Sp K(h) ) hG h (see [Mat16,Proposition 10.10] and [Mor23, Theorem A.II] for two formulations).Taking fixed points for a twisted action therefore gives a twisted version of the K(h)-local category.
Our main theorems give the computation of the relative Brauer group at height one: Theorem A (Theorem 5.13).At the prime two, (i) The Z/4-factor is mapped injectively to Br(KO 2 ), and Q ⊗2 2 ⊗ KO 2 is the image of the generator under The Z/8 factor is the relative Brauer group Br(Sp and the strict unit We have indexed generators on the filtration in which they are detected in the descent spectral sequence, which we recall later in the introduction.For the final part, note [BRS12, §4] that cyclic algebras are defined using strict units: that is, maps of spectra We give an alternative construction of cyclic algebras from strict units in Section 4, and using this we show that they are detected in the HFPSS by a symbol in the sense of [Ser79,Chapter XIV].This allows us to deduce when they give rise to nontrivial Brauer classes.
Any strict unit has an underlying unit, and we abusively denote these by the same symbol.The unit 1 + ε was shown to be strict in [CY23], which is what is what gives rise to the claimed representative for the class Q 4 at the prime two.Likewise, at odd primes the roots of unity are strict, which leads to our second main computation: Theorem B (Theorem 5.1).At odd primes, A generator is given by the cyclic algebra (KU , χ, ω), where χ : We now give an outline of the computation.Since Grothendieck, the main approach to computing Brauer groups has been étale or Galois descent, and this is the case for us too.Namely, recall that at any height h, Morava E-theory defines a K(h)-local Galois extension with profinite Galois group G h .In [Mor23], we used condensed mathematics to prove a Galois descent statement of the form and deduced from this a homotopy fixed point spectral sequence for Picard and Brauer groups, extending the Galois descent results of Mathew and Stojanoska [MS16] and Gepner and Lawson [GL21].For our purposes, the main computational upshot of that paper is: Theorem 1.1.There is a descent spectral sequence whose (−1)-stem gives an upper bound on Br 0 h .In a large range, there is an explicit comparison of differentials with the K(h)-local E h -Adams spectral sequence.
A more precise form of the theorem is recalled in Section 2. In the present paper, we determine this spectral sequence completely at height one, at least for t − s ≥ −1.Unsurprisingly, the computation looks very different in the cases p = 2 and p > 2, and the former represents the majority of our work.To prove a lower bound on Br 0 1 , we also prove a realisation result in the spirit of Toën's theorem.Namely, we show in a very general context that all classes on the E ∞ -page may be represented by Azumaya algebras: Theorem C (Corollary 3.9).Let C be a nice symmetric monoidal ∞-category and A ∈ CAlg(C) a faithful and dualisable Galois extension of the unit.Then the map sending an Azumaya algebra to its module ∞category, is an isomorphism.
See also [AG14,§6.3]and [GL21, §6.4] for related results.We refer the reader to Section 3 for the exact conditions in the theorem; most pertinently, the chromatic localisations of spectra give examples of such ∞-categories.This accounts immediately for most of the classes in Br 0 1 , by sparsity in the descent spectral sequence.At the prime two there is one final computation necessary, which is the relative Brauer group of KO 2 .Recall that the group Br(KO | KU ) was computed by Gepner and Lawson [GL21].In Section 5.2, we prove: and the basechange map from Br(KO | KU ) ∼ = Z/2 is injective.
The generator, which we denote P 2 , may be thought of as a cyclic algebra for the unit 1 + 4ζ, where ζ is a topological generator of Z 2 (see Section 4).We remark that this unit is not strict, and so the cyclic algebra cannot be constructed by hand; instead, we invoke Theorem C to prove that the possible obstructions vanish.We then show that P 2 survives the descent spectral sequence for the extension S K(1) → KO 2 , and therefore gives rise to the class Q 2 ∈ Br 0 1 .As an aside, we observe that Theorem D implies the following result, which may be of independent interest: We do not know if such an equivariant splitting exists for gl 1 KU .It seems conceivable that one would arise from a discrete model for K-theory, and this is an interesting problem.

Outline of the paper
In Section 2, we compute the E 2 -page of the descent spectral sequences and most differentials, giving an upper bound on the relative Brauer groups.The objective of the rest of the document is to show this bound is achieved.In Section 3 we prove that the property of "admitting a compact generator" satisfies Sp K(h) -linear Galois descent, which allows us to lift certain E ∞ -classes to Azumaya algebras.In Section 4 we give a construction of cyclic algebras which makes clear where they are detected in the descent spectral sequence; this allows us to assert that the cyclic algebras we form are distinct, and completes the odd-prime computation.We also give a construction of Brauer classes from 1-cocycles, which we use at the prime two.In Section 5 we compute the relative Brauer group Br(KO 2 | KU 2 ), and use this to complete the computation of Br 0 1 at the prime two.

Acknowledgements
I'm extremely grateful to my PhD supervisors, Behrang Noohi and Lennart Meier, for their support throughout the past few years; I'd also like to thank them for useful comments on an earlier draft.I also had helful conversations on this project with Zachary Halladay, Kaif Hilman, Shai Keidar and Sven van Nigtevecht.I'd especially like to thank Maxime Ramzi for suggesting the strategy of Proposition 4.9, and for comments on a previous version.This work forms part of my thesis, supported by EPSRC [grant EP/R513106/1].

Notation and conventions
• We will freely use the language of ∞-categories (modeled as quasi-categories) as pioneered by Joyal and Lurie [HTT; HA; SAG].In particular, all (co)limits are ∞-categorical.Most commonly, we will be in the context of a presentably symmetric monoidal stable ∞-category, and we use the term stable homotopy theory to mean such an object.All our computations take place internally to the K(h)-local category, and so the symbol ⊗ will generally denote the K(h)-local smash product.
• We only consider spectra with group actions, and not any more sophisticated equivariant notion.When G is a profinite group, we will write H * (G, M ) for continuous group cohomology.
• We follow the conventions of [Mor23].In particular, we direct the reader to [Mor23, §2.2] for details about the proétale site and the sheaves E and pic(E).Given a descendable G-Galois extension 1 → A in a stable homotopy theory (possibly profinite), we will implicitly use the associated (hypercomplete) sheaf A ∈ Sh(BG proét , C), writing A hG := ΓA.Using [Mor23, §3.1], we will also form the sheaf Mod A (C) ∈ Sh(BG proét , Pr L,smon ), and hence the Picard sheaf pic(A) ∈ Sh(BG proét , Sp ≥0 ).In this case the descent spectral sequence reads and the E 2 -term can often be identified with continuous cohomology of the G-module π t pic(A).
• We work at a fixed prime p and height h (mostly one).As such, p and h are often implicit in the notation: e.g.we use the symbol S for the both the sphere and its p-completion, according to context.
To avoid ambiguity, we are explicit in some cases: for example, KU will always mean integral K-theory.
• When indexing spectral sequences, we will always use s for filtration, t for internal degree, and t − s for stem.We abbreviate "homotopy fixed points spectral sequence" to HFPSS.We write "Picard spectral sequence" for the descent spectral sequence (2).
• We fix once and for all a regular cardinal κ such that (i) The Brauer space Br(Sp K(1) ) will by definition be the Picard space of Sp K(1) ∈ CAlg(Pr L κ ), which is a small space since Pr L κ is presentable.As noted in Corollary 3.9, for relative Brauer classes this is no restriction.

List of symbols
R nr Maximal unramified extension of a commutative ring spectrum R.

The descent spectral sequence
In this short section, we record the descent spectral sequence that will be the starting point for our computations.At any characteristic (p, h), this arises as the descent spectral sequence for the sheaf The main input from op. cit. is the following theorem: Theorem 2.1 ([Mor23], Theorem A and Proposition 5.11).(i) There is a strongly convergent spectral sequence (ii) Its (−1)-stem converges to π 0 (BPic(E)) hG , which contains Br 0 h as a subgroup.
(iii) Differentials on the E r page agree with those in the K-local E-Adams spectral sequence in the region t ≥ r + 1, and for classes x ∈ E r,r r we have In [Mor23, §4], we used this spectral sequence to recover the computation of Pic 1 := Pic(Sp K(1) ) (due to [HMS94]).In this case, Morava E-theory is the p-completed complex K-theory spectrum KU p , acted upon by G ∼ = Z × p via Adams operations ψ a .
Definition 2.2.We will write Br ′ (Sp K | E) := (BPic(E)) hG for the global sections of the (sheafified) proétale sheaf Br(E | E), and At height one, we will show in Section 5 that these groups agree.

Odd primes
We first consider the case p > 2. The starting page of the Picard spectral sequence is recorded below: The height one Picard spectral sequence for odd primes (implicitly at p = 3).Classes are labelled as follows: and circles denote p-power torsion (labelled by the torsion degree).Since , no differentials can hit the (−1)-stem.Differentials with source in stem t − s ≤ −2 have been omitted.
Lemma 2.3 ([Mor23], Lemma 4.15).At odd primes, the starting page of the descent spectral sequence is given by This is displayed in Fig. 1.In particular, the spectral sequence collapses for degree reasons at the E 3 -page.
Proof.The only possible differentials are d 2 -differentials on classes in the (−1)-stem; note that there are no differentials into the (−1)-stem, since every E 2 -class in the 0-stem is a permanent cycle.The generator in E 1,0 2 supports a d 2 , since this is the case for the class in E 1,0 2 of the descent spectral sequence for the C 2 -action on KU [GL21, Prop.7.15] (this is displayed in Fig. 3), and the span of Galois extensions allows us to transport this differential (see also Fig. 5).Note that the induced span on E 2 -pages is in bidegrees (s, t) = (1, 0) and (3, 1) respectively.Thus In Section 4 we will show that this bound is achieved using the cyclic algebra construction of [BRS12]; abstractly, this also follows from the detection result of Section 3.
The Picard spectral sequence for the Galois extension 1 K → E = KU2 at p = 2.We know that all remaining classes in the 0-stem survive, by comparing to the algebraic Picard group.Thus the only differentials that remain to compute are those out of the (−1)-stem; those displayed can be transported from the descent spectral sequence for Pic(KO) hC 2 -see Figs. 3 and 4. We have not displayed possible differentials out of stem ≤ −2.

The case p = 2
We now proceed with the computation of the (−1)-stem for the even prime.
• the class in bidegree (s, t) = (3, 3), which supports a d 3 in the Adams spectral sequence, is a permanent cycle.
By comparing with the Adams spectral sequence, any classes in the (−1)-stem that survive to E ∞ are in filtration at most six; on the E 2 -page, there are seven such generators.By comparing to the HFPSS for Br ′ (KO 2 | KU 2 ) = (BPic(KU 2 )) hC2 as in Section 2.1, we obtain the following differentials: • a d 2 on the class in This gives the claimed upper bound.
In Section 5 we will show that this bound is also achieved.
Remark 2.7.For later reference, we name the following generators: While q 6 survives to E ∞ by sparsity in Fig. 2, the other classes are sources of possible differentials.We will show that in fact all are permanent cycles.

Descent for compact generators
Given a Galois extension 1 → A in a stable homotopy theory C, we showed in [Mor23, §5] that the Picard spectral sequence computes the subgroup of the cohomological Brauer group.To relate this to the Brauer-Azumaya group classifying Azumaya algebras in C, we prove a descent result for compact generators valid in the K-local setting.This is entirely analogous to the theory of Example 3.2.The K-local sphere is an Sp K -compact generator of Sp K .More generally, we always have 1 ∈ C ecg since in this case (5) is the identity functor.
Our first objective is to show that Schwede-Shipley theory goes through in the presence of a C-compact generator.
Definition 3.3.Let C be a stable homotopy theory.We say C is rigidly generated if it is generated under colimits by dualisable objects.That is, the localising category generated by C dbl is C itself.
Example 3.4.(i) Sp is generated under colimits by shifts of 1, and so rigidly generated.
(ii) If C is a rigidly generated stable homotopy theory and L : C → C ′ a monoidal localisation, then C ′ is rigidly generated.Thus Sp K is rigidly generated.(i) G preserves colimits of simplicial objects: in fact G preserves all colimits.Indeed, G preserves filtered colimits since D is C-compact, and finite colimits as it is a right adjoint.
(ii) G is conservative: this is by definition of C-compact generators.
(iii) for every D ′ ∈ D and C ∈ C, the map But by (i), the functor G preserves all colimits, so by rigid generation we reduce to C dualisable.In this case, (6) is the composite equivalence To use this to produce Azumaya algebras, we first need to be able to produce C-compact generators.For this we will prove the following result: The proof relies on the following basic lemma: Lemma 3.7.Suppose C ∈ CAlg(Pr L ) and A ∈ C is dualisable, then A is faithful if and only if A ∨ is.
Proof.Assume that A is faithful, and that A ∨ ⊗ X = 0; the converse is given by taking duals.Then the identity on A ⊗ X factors as and in particular A ⊗ X = 0.By faithfulness of A, this implies X = 0.
Proof (Proposition 3.6).As in [GL21], we will make use of the functors and the adjunctions (denoted by the same symbols) between D and Mod A (D).In fact, we claim these adjunctions are C-linear.
Assuming this, we give the proof of the proposition.If D admits a C-compact generator D, it is straightforward to check that i !D ∈ Mod A (D) ecg : indeed, i !D is C-compact because D is so and i * preserves colimits, while i !D generates because D does and i * is conservative.Conversely, suppose we have D ∈ Mod A (D) ecg , and consider i * D ∈ D. By dualisability of A, the right adjoint preserves colimits, and hence the right adjoint i * : Mod A (D) → D does too.As a result, i * D is C-compact.
On the other hand, if X ∈ D and Map D (i * D, X) = 0, then Map Mod A (D) (D, i * X) = 0 and so It remains to prove that the adjunctions i !⊣ i * ⊣ i * are C-linear.For i !we observed C-linearity in the proof of Proposition 3.5.To see C-linearity for i * it is enough to prove that the canonical map is an equivalence for every C and M , and by rigid generation we reduce to C dualisable.As in [SAG, Remark D.7.4.4] one checks that Map C (C ′ , θ) is the composite equivalence for any C ′ ∈ C, which gives the claim.
Example 3.8.If A → B is an E-local Galois extension of ring spectra with stably dualisable Galois group G, then Rognes [Rog08, Proposition 6.2.1] shows that B is dualisable over A. For example, this covers the following cases: (i) E = S and G is finite or compact Lie.
(ii) E = F p and G is p-compact.
(iii) E = K and G = K(π, m) for π a finite p-group and m ≤ h.
Corollary 3.9 (Br = Br ′ ).Let 1 → A be a faithful dualisable Galois extension in a rigidly generated stable homotopy theory C, and Q ∈ π 0 BPic(A) hG a relative Brauer-Grothendieck class.Then Q is represented by some Azumaya algebra Q whose basechange to A is (Morita) trivial: that is, and Mod A⊗Q (C) ≃ Mod A (C). Thus the map is an isomorphism.
Proof.We claim that Q is κ-compactly generated, so that Q ∈ Pic(Cat C ).Given this, the result follows from Proposition 3.6: by assumption, Mod A (Q) ≃ Mod A (C), and so By descent for compact generators we obtain D ∈ Q ecg , and so Schwede-Shipley theory yields a C-linear For the claim, note that as in [Mat16, Corollary 3.42], Q is the limit of the cosimplicial diagam Remark

Explicit generators
In this section, we give some explicit constructions of Azumaya algebras from Galois extensions.Most of this section works in an arbitrary stable homotopy theory C. We will use these constructions in Section 5 to describe generators of the group Br 0 1 , and hence to solve extension problems.

Z p -extensions
We will begin with a straightforward construction for extensions with Galois group Z p , using the fact that cd(Z p ) = 1.
Remark 4.1.Let G be a profinite group, and B ∈ Sh(BG proét , S * ).Under suitable assumptions on B, décalage gives an isomorphism between the descent spectral sequence and the spectral sequence for the Čech nerve of G → * [Mor23, Appendix A].Moreover, the homotopy groups of the latter can often be identified with the complex of continuous cochains with coefficients in π t B, yielding an isomorphism on the E 2 -pages For example, this is the case for the sheaf pic(E) for any Morava E-theory E(k, Γ).
and is therefore an equivalence.
Construction 4.3.Suppose 1 → A is a descendable Galois extension in a stable homotopy theory C, with group G = Z p or Z. Then B := BPic(A) ∈ Sh(BG proét , S * ), and in good cases B satisfies the assumption that (7) be an isomorphism: for example, this is the case whenever each π t pic(A) is the limit of a tower of finite sets by [BS14, Lemma 4.3.9].Thus and so a relative cohomological Brauer class is given by the data of an A-linear equivalence In fact, if X ∈ Pic(A), then we may form the A-linear composite This gives an isomorphism Example 4.4.At the prime 2, the extension 1 K → KO 2 is a descendable Z 2 -Galois extension.As a result, Construction 4.3 applies, and we can form the Brauer class (KO 2 , X) associated to any X ∈ Pic(KO 2 ) as above.For example, since KO 2 is 8-periodic one can form the cohomological Brauer class Example 4.5.Let KO nr 2 := lim − →n (KO 2 ) W (F 2 n ) be the ind-étale KO 2 -algebra given by the maximal unramified extension of π 0 KO 2 = Z 2 ; since étale extensions are uniquely determined by their π 0 , one can also describe this as

Cyclic algebras
Suppose that C is a stable homotopy theory and 1 → A a finite Galois extension in C with group G. Suppose also given the following data: In this section, we will use this to define a relative Azumaya algebra (A, χ, u) ∈ Br(C | A).
Let us first recall the construction when C is the category of modules over a classical ring.Then we begin with a G-Galois extension R → A of rings, and define a G-action on the matrix algebra M k (A) as follows: we let σ := χ −1 (1) act as conjugation by the matrix Since u k = uI k ∈ GL k A is central, this gives a well-defined action on M k A. We can use this to twist the usual action of G on the matrix algebra; passing to fixed points, we obtain the cyclic algebra The construction for general C will be directly analogous.Moreover, when R is a field it is well-known (see for example [CS21]) that under the isomorphism the cyclic algebra (A, χ, u) maps to the cup-product β(χ) ∪ u, where β denotes the Bockstein homomorphism and we use the Z-module structure on A × .As a result, we obtain an isomorphism A, χ, u).We will prove an analogous result for cyclic algebras in arbitrary stable homotopy theories, which will allow us to detect permanent cycles in the descent spectral sequence; conversely, this will allow us to assert that the cyclic algebras we construct are nontrivial.
We now begin the construction of (A, χ, u) for arbitrary C. Definition 4.6.Let C be a stable homotopy theory.Given A ∈ CAlg(C) and k ≥ 1, we will write These are all E 1 -monoids under composition, and we have E 1 maps the first of which is by definition.The second will be defined immediately below.
Remark 4.7.If R is an E ∞ -ring, then Gepner and Lawson [GL21, Corollary 5.19] show there is a fibre sequence Here Mod cg R ⊂ ιMod R denotes the classifying space of compact generators, and Az triv R ⊂ ιAlg R the space of Morita trivial Azumaya algebras.To see this, we consider the components over Mod R ∈ Br(R) in the commutative square noting that this diagram is Cartesian [GL21, Proposition 5.17]: this follows by identifying those objects in which correspond to the R-linear equivalences.By [HL17, Corollary 2.1.3],C-linear equivalences between module categories in an arbitrary C correspond to bimodules M that are full and dualisable, and so in this context there is an analogous pullback diagram for any Azumaya algebra A. Restricting again to the unit component in Br(C), we obtain a fibre sequence In particular, taking A = M k (1) yields the fibre sequence and restricting to M = 1 ⊕k ∈ C fd gives the desired map GL k (1) → PGL k (1) after taking loops.
Remark 4.8.Suppose that C is a stable homotopy theory and u ∈ π 0 GL 1 (1) = π 1 Pic(C).We define a map û : Z/k → PGL k (1) by virtue of the following commutative diagram, whose bottom row is a shift of the fibre sequence (13): is given by the matrix in (10)-commutativity of the right-hand square above is implied by the computation In fact, when the chosen unit is strict, we will prove at the end of the section that u deloops: Remark 4.10.The choice of lift u is only determined up to homotopy in Construction 4.11.Suppose that 1 → A is a finite G-Galois extension in C, and u ∈ π 0 G m (1).We obtain a commutative square of spaces with G-action where the action on the top row is trivial.On homotopy fixed points, we obtain the square whose bottom right term is the relative Brauer space Br(1 | A).
Definition 4.12.Given a Galois extension 1 → A with group G in a stable homotopy theory C, and given χ : The final isomorphism is inverse to the map Proof.Going down and left in (16), we see that (A, χ, u) is described as follows: it is the A-semilinear action on Mod A (C) ≃ Mod M k (A) (C) given by the G-equivariant map This is the same action that Baker, Richter and Szymik define at the level of M k (A).
Theorem 4.14.Suppose given a strict unit u ̸ = 1 ∈ π 0 G m (1).Its image in π 0 GL 1 (1) is detected in the HFPSS for pic(C) ≃ pic(A) hG by a class v ∈ E s,s+1 2 , and we assume that one of the following holds: (i) v is in positive filtration; (ii) v is in filtration zero, and has nonzero image in H 0 (G, (π 0 A) × ).
Then the cyclic algebra (A, χ, u) is detected by the symbol In particular, Proof.The square of G-spaces (15) gives rise to a commutative square of HFPSS as below: Note that the maps on E 2 need not preserve filtration.By definition, (A, χ, u) is the image of χ ∈ π 0 (BZ/k) BG under the composite to π 0 (BPic(A)) hG , and is therefore detected on the E 2 -page for (BPic(A)) hG by u * β * (χ), as long as this class is nonzero.It is standard that the map β * is indeed the Bockstein, and we claim that the map induced on E 2 -pages by B 2 u : B 2 Z → BPic(A) agrees with v ∪ −.Indeed, this map can be identified with the composition and the induced map on HFPSS shows that by compatibility of the cup and composition products.The class β(χ) ∪ v is nonzero since Tate cohomology is β(χ)-periodic, which gives the result.

Proof of Proposition 4.9
We will write CMon ⊂ Fun(Fin * , S) for the ∞-category of special Γ-spaces, which admits a unique symmetric monoidal structure making the free commutative monoid functor S → CMon symmetric monoidal [GGN16]; its unit is the nerve of the category of finite pointed sets and isomorphisms, Definition 4.15.We will denote by Rig := Alg E1 (CMon) the ∞-category of associative semirings 2 .Likewise, we will write CRig := Alg E∞ (CMon) for the ∞-category of commutative semirings.Since the adjunction S ⇄ CMon is symmetric monoidal, it passes to algebras to yield monoid semiring functors, which factor the respective monoid algebra functors to spectra.
2 that is, rings without negatives Remark 4.16.The key observation for proving Proposition 4.9, pointed out to us by Maxime Ramzi, is that for R ∈ CRig and u ∈ G m (R), the map u ∈ π 0 Map(Z/k, PGL k (R gp )) exists in π 0 Map(Z/k, PGL k (R)) before group completion.This is because the matrix u in (10) is defined without subtraction, as are all its powers.This allows us to make use of the fact that the representing object F[u ±1 ] := F[Z] for G m is substantially simpler than S[u ±1 ], as the following remark shows.
Remark 4.17.Consider the square of [GGN16,§7].By Lemma 4.18 below, we have an equivalence Since CMon is preadditive and R : CMon → S preserves sifted colimits, the underlying space of F[u ±1 ] is lim − →n [−n,n] F. One therefore has a pullback square for any basepoint of F[u ±1 ].
Lemma 4.18.Let (L ⊣ R) : C ⇄ D be a symmetric monoidal adjunction between presentably symmetric monoidal ∞-categories.Let λ be an uncountable regular cardinal, and suppose that O ⊗ → Sym ⊗ is a fibration of ∞-operads compatible with colimits.Then the square Proof.In the notation of Remark 4.17, we want to show that the mate is an equivalence, and since Alg O (C) is generated under sifted colimits by free algebras (as noted in the proof of [HA, Corollary 3.2.3.3]) and U, U ′ preserve sifted colimits, it is enough to check this on free algebras F C, C ∈ C.But L ′ F ≃ F ′ L, and using the description of F as an operadic Kan extension [HA, Proposition 3.1.3.13]we factor (19) as an equivalence Every nonzero coefficient in the matrix (10) is 1, so it is a consequence of Remark 4.17 that all relevant components of F[u ±1 ] are contractible (which is certainly not true of S[Z] = S[u ±1 ]).This will allow us to prove that there is an essentially unique lift of semiring maps To match other figures, we have shifted everything in degree by one (so one may think of this as the spectral sequence for Σ −1 br ′ ).The extension in the 0-stem is 4 ∈ Ext(Z/8, Z2) ≃ Z/8, which gives π0Σ −1 br ′ (1 We will use this to compute the group Br(1 K | KU 2 ) by Galois descent along the Z 2 -Galois extension 1 K → KO 2 .Namely, we use the iterated fixed points formula to form the descent spectral sequence Since Z 2 has cohomological dimension one, there is no room for differentials and the spectral sequence collapses immediately.To determine Br ′ (Sp K | KU 2 ), what remains to compute is the following: This is achieved in the next couple of results, and the result is displayed in Fig. 6.Note that we have shifted degrees by one to match other figures, so that the relative Brauer group is still computed by the (−1)-stem.
Proposition 5.11.We have Br(KO 2 | KU 2 ) 1+4Z2 = Z/4, so the map Proposition 5.12.The relative cohomological Brauer group at the prime two is Proof.Based on Fig. 6, what remains is to compute the extension from Br We will conclude by showing that the extension is split.To prove the claim, note that both (KO 2 , ΣKO 2 ) and (KO nr 2 , Σ 2 KO nr 2 ) split over KO nr 2 , so that the inclusion of Br We will compute the relative Brauer group Br ′ (1 K | KO nr 2 ) by means of the descent spectral sequence which collapses at the E 3 page since Z 2 × Z has cohomological dimension two for profinite modules.To compute the E 2 -page, note that The action on Pic(KO nr 2 ) is trivial, while the action on π 0 GL 1 (KO nr 2 ) is trivial for the Z 2 -factor, and Frobenius for the Z-factor.In particular, note that H 0 ( Z, W × ) = Z × 2 ; using the multiplicative splitting by Hilbert 90.This implies that H 2 (Z 2 × Z, π 0 GL 1 (KO nr 2 )) = 0, since cd(Z 2 ) = cd( Z) = 1 for profinite coefficients.The (−1)-stem of the E 2 -page is hence concentrated in filtration one, and hence agrees with the E ∞ -page in this range.Thus and Br ′ (1 K | KU 2 ) → Z/8 ⊕ Z/8, which implies the claim.

Generators at the prime two.
To deduce Theorem A from Proposition 5.12 we will appeal to the results of Section 3.
Theorem 5.13.The relative Brauer group at the prime two is Proof.To lift the cohomological Brauer classes generating Br ′ (Sp K | KU 2 ) ∼ = Z/8⊕Z/4 to Azumaya algebras, it is enough by Corollary 3.9 to prove that they are trivialised in some finite extension of 1 K .Recall from the previous subsection that: , so ) ⊂ Br 0 1 .
• Under the base-change to KO 2 , the generators q 2 and q 6 map to the E 2 -classes representing P 2 , P 6 = P 2 2 ∈ Br(KO 2 | KU 2 ).The splitting in Theorem 5.13 of the surjection Br 0 1 ↠ Br(KO 2 | KU 2 ) of Proposition 5.11 gives a canonical choice of classes Q 2 , Q 6 = Q 2 2 ∈ Br 0 1 lifting these.In particular, q 2 must also be a permanent cycle.
Remark 5.15.From the form of the spectral sequence, it follows that the class in E 7,5 2 survives to E ∞ -this should have implications for the nonconnective Brauer spectrum of Sp K , as defined in [Hau17].

Figure 3 :
Figure 3: The E3-page of the Picard spectral sequence for KO, as in [GL21, Figure 7.2].

Figure 4 :
Figure 4: The Picard spectral sequence for KO2.Differentials come from the comparison with KO, and the resulting map of spectral sequences.See Section 5 for the extension in the (−1)-stem.

(1
iii) For a compact lie group G, the ∞-category S G of G-spaces is generated under colimits by orbits G/H (e.g.[MM02, Theorem 1.8]).Its stabilisation Sp G U at any G-universe U (as defined in [GM23, Corollary C.7]) is generated under colimits by shifts Σ −V Σ ∞ U G/H + as V ranges over representations in U , and if U is complete then these are invertible by virtue of the Wirthmüller isomorphism [GM95, (4.16)].Thus the ∞-category Sp G of genuine G-spectra is rigidly generated.Proposition 3.5 (Enriched Schwede-Shipley).Let C be a rigidly generated stable homotopy theory and D ∈ Cat C .Suppose that D ∈ D ecg , and write A := End D (D) ∈ Alg(C).Then there is an C-linear equivalence D ≃ LMod A (C).These might also be called enriched compact generators.Proof.The object D ∈ D determines canonically a C-linear left adjoint F : C → D, with right adjoint G := Map D (D, −).According to [HA, Proposition 4.8.5.8], it is enough to check the following:

Proposition 3. 6 (
Descent for C-compact generators).Let 1 → A be a descendable extension in a rigidly generated stable homotopy theory C, and suppose that A is dualisable.Then D ∈ Cat C admits a C-compact generator if and only if Mod A (D) := D ⊗ C Mod A (C) admits one.
Lemma 4.2.Let G = Z p or Z, and write ζ ∈ G for a topological generator.Suppose C is a stable homotopy theory and B ∈ Sh(BG proét , S * ), with B := B(G/ * ) and B hG := ΓB.If the canonical map (7) is an isomorphism, then B hG ≃ Eq (id, ζ : B ⇒ B) .(8) Proof.Write B ′ for the equaliser in (8).The G-map (id, ζ) : G → G × G gives rise to maps B ⇒ B(G × G) → B (9) factoring id, ζ : B ⇒ B, and the identification B hG ≃ lim B(G •+1 ) gives a distinguished nullhomotopy in (9) after precomposing with the coaugmentation η : B hG → B. Thus η factors through θ : B hG → B ′ .Taking fibres, the descent spectral sequence for B implies that π * θ fits in a commutative diagram
Differentials out of the (−1)-stem.Extensions in the E∞-page.

Figure 7 :
Figure 7: Detailed view of the Picard spectral sequence (Fig. 2) in low degrees.
3.10.In a previous version, we claimed that Morava E-theory is dualisable in Sp [HS99,pointed out to us by Maxime Ramzi, while E is Spanier-Whitehead self-dual [Str00; Bea+22] and hence reflexive, it is not dualisable: for example, K * E would otherwise be finite by[HS99, Theorem 8.6].We will bypass this issue at height one by showing that all generators of Br ′ (Sp K | E) are in fact trivialised in a finite Galois extension of the sphere, and hence lift to Azumaya algebras by Corollary 3.9; we do not know if Br 0 h ∼ = Br ′ (Sp K | E) at arbitrary height.