The derived Brauer map via twisted sheaves

Let X be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of X and which we call Br†(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{Br}^\dagger (X)$$\end{document} following Lurie. Toën introduced a map ϕ:Br†(X)→He´t2(X,Gm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi :\textsf{Br}^\dagger (X)\rightarrow H ^2_{\acute{e }t }(X,{\mathbb {G}}_{\textrm{m}})$$\end{document} which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} to a subgroup Br(X)⊂Br†(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{Br}(X)\subset \textsf{Br}^\dagger (X)$$\end{document}, which we call the derived Brauer group, on which ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} becomes an isomorphism Br(X)≃He´t2(X,Gm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{Br}(X)\simeq H ^2_{\acute{e }t }(X,{\mathbb {G}}_{\textrm{m}})$$\end{document}. This map may be interpreted as a derived version of the classical Brauer map which offers a way to “fill the gap” between the classical Brauer group and the cohomogical Brauer group. The group Br(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{Br}(X)$$\end{document} was introduced by Lurie by making use of the theory of prestable ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories between the Brauer space of invertible presentable prestable OX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}_X$$\end{document}-linear categories, and the space Map(X,K(Gm,2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Map (X,K ({\mathbb {G}}_{\textrm{m}},2))$$\end{document}. We offer an alternative proof of this equivalence of ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.


Introduction
Throughout the whole work, X will be a quasicompact quasiseparated scheme over some field k of arbitrary characteristic.The categories of sheaves and the various cohomology groups will always be understood with respect to the étale topology.
In 1966, Grothendieck [Gro66] introduced the notion of Azumaya algebra over X : this is an étale sheaf of algebras which is locally of the form End(E), the sheaf of endomorphisms of a vector bundle E over X .This is indeed a notion of "local triviality" in the sense of Morita theory: two sheaves of algebras A, A ′ are said to be Morita equivalent if the category LMod A = {F quasicoherent sheaf over X together with a left action of A} and its counterpart LMod A ′ are (abstractly) equivalent: for instance, one can prove that, for any vector bundle E over X , LMod End(E) is equivalent to LMod O X = QCoh(X ) via the functor M → E ∨ ⊗ End(E) M .
The classical Brauer group Br Az (X ) of X is the set of Azumaya algebras up to Morita equivalence, with the operation of tensor product of sheaves of algebras.Grothendieck showed that this group injects into H2 (X , m ) by using cohomological arguments: essentially, he used the fact that a vector bundle corresponds to a GL n -torsor for some n, and that there exists a short exact sequence of groups The image of Br Az (X ) → H 2 (X , m ) is contained in the torsion subgroup of H 2 (X , m ), which is often called the cohomological Brauer group of X . 1ne of the developments of Grothendieck's approach to the study of the Brauer group is due to Bertrand Toën and its use of derived algebraic geometry in [Toë10].There, he introduced the notion of derived Azumaya algebra as a natural generalization of the usual notion of Azumaya algebra.Derived Azumaya algebras over X form a dg-category Deraz X .There is a functor Deraz X → g c (X ), this latter being (in Toën's notation) the dg-category of presentable stable O X -linear dg-categories 2 which are compactly generated, with, as morphisms, functors preserving all colimits.The functor is defined by A → LMod A = {quasicoherent sheaves on X with a left action of A} where all terms have now to be understood in a derived sense.One can prove that this functor sends the tensor product of sheaves of algebras to the tensor product of presentable O X -linear dg-categories, whose unit is QCoh(X ).Building on classical Morita theory, Toën defined two derived Azumaya algebras to be Morita equivalent if the dg-categories of left modules are (abstractly) equivalent.This agrees with the fact mentioned above that LMod End(E) ≃ QCoh(X ) for any E ∈ Vect(X ).
In [Toë10, Proposition 1.5], Toën characterized the objects in the essential image of the functor A → LMod A as the compactly generated presentable O X -linear dg-categories which are invertible with respect to the tensor product, i.e. those M for which there exists another presentable compactly generated O X -linear dg-category M ∨ and equivalences 1 ∼ − → M ⊗ M ∨ and M ∨ ⊗ M ∼ − → 1.The proof of this characterization goes roughly as follows: given a compactly generated invertible dg-category M , one can always suppose that M is generated by some single compact generator E M .Now, compactly generated presentable O X -linear dgcategories satisfy an important étale descent property [Toë10, Theorem 3.7]; from this and from [Toë10, Proposition 3.6] one can deduce that the End M (E M ) has a natural structure of a quasicoherent sheaf of O X -algebras A, and one can prove that M ≃ LMod A as O X -linear ∞-categories.
Both Antieau-Gepner [AG14] and Lurie [Lur18,Chapter 11] resumed Toën's work, using the language of ∞-categories in replacement of that of dg-categories.Lurie also generalized the notion of Azumaya algebra and Brauer group to spectral algebraic spaces, see [Lur18, Section 11.5.3]3 .He considers the ∞-groupoid of compactly generated presentable O X -linear ∞-categories which are invertible with respect to the Lurie tensor product ⊗, and calls it the extended Brauer space Br † X .This terminology is motivated by the fact that the set π 0 Br † X has a natural abelian group structure, and by [Toë10, Corollary 2.12] is isomorphic to H 2 ét (X , m ) × H 1 ét (X , ): in particular, it contains the cohomological Brauer group of X .At the categorical level, Lurie proves that there is an equivalence of ∞-groupoids between

INTRODUCTION
Br † (X ) and Map Stk k (X , B 2 m × B ) (Stk k is the ∞-category of stacks over the base field k).In particular, Br † (X ) is 2-truncated.
We can summarize the situation in the following chain of functors: where the left term is the maximal ∞-groupoid in the localization of the ∞-category of derived Azumaya algebras to Morita equivalences.At the level of dg-categories, this chain of equivalences is proven in [Toë10, Corollary 3.8].At the level of ∞-categories, this is the combination of [Lur18, Proposition 11.5.3.10] and [Lur18, 11.5.5.4]Note that, while in the classical case we had an injection Br Az (X ) → H 2 (X , m ), in the derived setting one has a surjection Br † (X ) := π 0 Br † (X ) ։ H 2 (X , m ).If H 1 (X , ) = 0 (e.g. when X is a normal scheme), then the surjection becomes an isomorphism of abelian groups.
While the first equivalence in (1.1) is completely explicit in the works of Toën and Lurie, the second one leaves a couple of questions open: • since the space Map(X , B 2 m × B ) is the space of pairs (G, P ), where G is a m -gerbe over X and P is a -torsor over X , it is natural to ask what are the gerbe and the torsor naturally associated to an element of Br † (X ) according to the above equivalence.This is not explicit in the proofs of Toën and Lurie, which never mention the words "gerbe" and "torsor", but rather computes the homotopy group sheaves of a sheaf of spaces Br † X over X whose global sections are Br † (X ).
• conversely: given a pair (G, P ), what is the ∞-category associated to it along the above equivalence?
The goal of the present paper is to give a partial answer to the two questions above.The reason for the word "partial" is that we will neglect the part of the discussion regarding torsors, postponing it to a forthcoming work, and focus only on the relationship between linear ∞categories/derived Azumaya algebras and m -gerbes.
In fact, we will work with an intermediate group Br(X ), which we call the derived Brauer group of X (see Definition 2.26), sitting in a chain of injective maps The derived Brauer group is always isomorphic to H 2 ét (X , m ) (Theorem 2.27) and the first injection is (up to that identification) exactly the classical Brauer map.
The following is our main theorem.The next two subsections will provide the necessary vocabulary to understand its statement.Section 2.3 provides a sketch of the main geometric ideas behind the proof.
Theorem (Theorem 2.34).Let X be a qcqs scheme over a field k.Then there is a symmetric monoidal equivalence of ∞-groupoids4 Φ : Br(X ) ⊗ ←→ Ger m (X ) ⋆ : Ψ where We will define all the necessary vocabulary in Section 2. Informally, Triv ≥0 (M ) is a stack over X which, étale-locally in X , parametrizes equivalences between M and QCoh(X ) ≥0 , the unit object in Br(X ); QCoh id (G) ≥0 is the ∞-category of quasicoherent sheaves over G which are connective, and homogeneous with respect to the identity character of m in an appropriate sense (Remark 2.9): without the connectivity assumption, these are known in the literature as twisted sheaves.
Remark 1.1.By looking at Definition 2.26, one sees that by taking the π 0 of the equivalence in Theorem 2.34 one obtains an isomorphism of abelian groups This isomorphism also appears in [Lur18, Example 11.5.7.15], but as mentioned before, it is a consequence of the equivalence of ∞-categories whose proof does not use the interpretation of the right-hand-side as the space of m -gerbes over X .
Remark 1.2.Proposition 3.2 and Proposition 3.4, which constitute the two steps of the proof that Ψ carries a symmetric monoidal structure, provide a proof of [BP21, Conjecture 5.27].

Construction of the correspondence
In the present section, we construct the two functors Ψ (Definition 2.31) and Φ (Definition 2.33) appearing in Theorem 2.34.).Let X be a quasicompact quasiseparated (qcqs) scheme over a field k.A m -gerbe over X is the datum of: • a stack in groupoids G with a map α : G → X which is, étale-locally in X , -nonempty (it has a section) connected (every two sections are isomorphic).
• an isomorphism of group stacks over which we call banding.
A morphism of m -gerbes over X is a morphism of stacks whose induced morphisms at the level of inertia groups commutes with the bandings.
Definition 2.3.Let X be a qcqs scheme.We denote by Ger m (X ) the (2, 0)-category whose objects are m -gerbes over X , whose 1-morphisms are morphisms of m -gerbes and whose 2-morphisms are natural transformations of maps of stacks.
The following theorem can be proven by using the same arguments of [Ols16, Theorem 12.2.8],but recording the identifications used in the proof as 1-and 2-morphisms.
We will not use this theorem in the proof of Theorem 2.34, but we will use it to justify some comments and comparisons with perspectives used by other Authors.Construction 2.6.Let G be a m -gerbe over X .The derived ∞-category of quasicoherent sheaves on G is denoted by QCoh(G) and it is a presentable stable compactly generated O Xlinear category.This follows from the combination of the following facts: • it is true for B m ([HR17, Example 8.6] or [BP21, Remark 5.8]); • every m -gerbe G over X is étale-locally equivalent to B m ×X (see [SPA, Tag 06QG]); • presentable compactly generated categories satisfy étale descent ([Lur18, Theorem D.5.3.1]. We now recall the definition of G-twisted sheaves on X .This notion dates back to Giraud [Gir71] and later to Max Lieblich's thesis [Lie08], and has been developed in the derived setting by Bergh and Schnürer [BS19] (using the language of triangulated categories) and Binda and Porta [BP21] (using the language of stable ∞-categories).
Let now F ∈ QCoh(G).We have that F is endowed with a canonical right action by I G , called the inertial action (see e.g.[BS19, Section 3]).Proposition 2.7.Let G be a m -gerbe over a qcqs scheme X .The pullback functor QCoh(X ) → QCoh(G) establishes an equivalence between QCoh(X ) and the full subcategory of QCoh(G) spanned by those sheaves on which the inertial action is trivial.
Note that the banding m × G → I G induces a right action ρ of m on any sheaf F ∈ QCoh(G), by composing the banding with the inertial action.On the other hand, for any character χ : m → m , m acts on F on the left by scalar multiplication precomposed with χ .Let us call this latter action σ χ .
Let G be a m -gerbe over a qcqs scheme X , and χ a character of m .We define the category of χ -homogeneous sheaves over G, informally, as the full subcategory QCoh χ (G) of QCoh(G) spanned by those sheaves on which ρ = σ χ .
Remark 2.9.The above definition is a little imprecise, in that it does not specify the equiv- There, the Authors define an idempotent functor (−) χ : QCoh(G) → QCoh(G), taking the "χ -homogeneous component".This functor is t-exact ([BP21, Proof of Lemma 5.17]), and comes with canonical maps i χ ,F : F χ → F. The category QCoh χ (G) is defined as the full subcategory of QCoh(G) spanned by those F such that i χ (F) is an equivalence.To fix the notations, let us make these constructions explicit.If X is a scheme and G a m -gerbe, we have the following diagram where p, q are the projections, u is the atlas of the trivial gerbe and act α is the morphism induced by the banding α of G. See [BP21, Section 5] for more specific description of these maps.We denote by L χ the line bundle over B m associated to the character χ , while and χ a character of m , we define (following [BP21, Construction 5.17 and proof of Lemma 5.20]) Intuitively, act * α F gains a decomposition into m -equivariant summands; tensoring with L ∨ χ shifts the weights of this decomposition by χ −1 , making the χ -equivariant summand into the 1-equivariant (i.e.invariant) summand.Pushing forward along p selects this invariant component as a quasicoherent sheaf over G.
Note that (F) χ can be also seen as (notations as in the diagram above) By pulling back along u the counit of the adjunction p * ⊣ p * we obtain a canonical morphism i χ (F) : (F) χ → F.
[BP21] also prove that there is a decomposition building on the fact that F ≃ χ ∈Hom( m , m ) F χ .The same result was previously obtained by Lieblich in the setting of abelian categories and by Bergh-Schnürer in the setting of triangulated categories.
Definition 2.10.In the case when χ is the identity character id : m → m , QCoh χ (G) is usually called the category of G-twisted sheaves on X .
The relationship between categories of twisted sheaves and Azumaya algebras has been intensively studied, see [DJ04], [DJ], [Lie08], [HR17], [BS19], [BP21].Given a m -gerbe G over X , its category of twisted sheaves QCoh id (G) admits a compact generator, whose algebra of endomorphisms is a derived Azumaya algebra A G .In contrast, if we restrict ourselves to the setting of abelian categories and consider abelian categories of twisted sheaves, this reconstruction mechanism does not work anymore.This is one of the reasons of the success of Toën's derived approach.
Remark 2.11.Let G, G ′ be m -gerbes over X , and χ , χ ′ two characters of m .A straightforward generalization of the above definitions allows to define the sub-∞-category spanned by "(χ , χ ′ )-homogeneous sheaves": it suffices to replace tha line bundle L χ by the external product Remark 2.12.Let C be a presentable ∞-category, p : C → C be an endofunctor and η : p ⇒ id C be a natural transformation.We have a diagram of ∞-categories.Let C 0 be the subcategory of C spanned by the elements X of C such that η(X ) is an equivalence.Now consider (C 1 , p 1 , η 1 ) and (C 2 , p 2 , η 2 ) triples as the above one, and let ρ : C 1 → C 2 be a functor and α : ) is a morphism between the two diagrams and therefore there exists a unique morphism C 0 1 → C 0 2 compatible with all the data.If G is a m -gerbe and χ is a character of m , the endofunctors p = (−) χ , q = id of QCoh(G) and the natural transformation η = i χ introduced in Remark 2.9 form a diagram as above.In this situation, QCoh χ (G) is our C 0 .Therefore, a morphism G → G ′ induces a morphism QCoh χ (G) → QCoh χ (G ′ ).Along the same lines, one proves that the association yields a well-defined functor Ger m (X ) → LinCat St (X ).We end this subsection by describing the symmetric monoidal structure on the category of m -gerbes.
Let X be a scheme and G 1 and G 2 be two m -gerbes on X .One can construct the product G 1 ⋆ G 2 , which is a m -gerbe such that its class in cohomology is the product of the classes of G 1 and G 2 (see [BP21, Conjecture 5.23]).Clearly, this is not enough to define a symmetric monoidal structure on the category of m -gerbes.The idea is to prove that this ⋆ product has a universal property in the ∞-categorical setting, which allows us to define the symmetric monoidal structure on the category of m -gerbes using the theory of simplicial colored operads and ∞-operads (see Chapter 2 of [Lur17]).Construction 2.13.Let AbGer(X ) be the (2, 1)-category of abelian gerbes over X and AbGr(X ) the (1, 1)-category of sheaves of abelian groups over X .We have the so-called banding functor ).It is easy to prove that Band is symmetric monoidal with respect to the two Cartesian symmetric monoidal structures of the source and target, that is it extends to a symmetric monoidal functor Band : AbGer(X ) × → AbGr(X ) × of colored simplicial operads (and therefore also of ∞-operads).
Recall that, given a morphism of sheaf of groups φ : µ → µ ′ and a µ-gerbe G, we can construct a µ ′ -gerbe, denoted by φ * G, and a morphism ρ φ : G → φ * G whose image through the banding functor is exactly φ.This pushforward construction is essentially unique and verifies weak funtoriality.This follows from the following result: if G is a gerbe banded by µ, then the induced banding functor Band G/ : AbGer(X ) G/ −→ AbGr(X ) µ/ is an equivalence and the pushforward construction is an inverse (see [BS19, Proposition 3.9]).This also implies that Band is a coCartesian fibration.
Let Fin * be the category of pointed finite sets.Consider now the morphism Fin * → AbGr(X ) × induced by the algebra object m,X in AbGr(X ) × . 5We consider the following pullback diagram where B is again a coCartesian fibration.This implies that G is a symmetric monoidal structure over the fiber category G 〈1〉 := B −1 (〈1〉) which is exactly the category of m -gerbes.The operadic nerve of G will be the symmetric monoidal ∞-category of m -gerbes (see Because of this, one can also prove that under the identification Ger m (X ) ≃ Map(X , B 2 m ) from Theorem 2.5 the ⋆-product coincides also with the product induced by the multiplication B Definition 2.15.We denote by Ger m (X ) ⋆ the symmetric monoidal ∞-groupoid of mgerbes over X , with the symmetric monoidal structure given by Construction 2.13.

Reminders on stable and prestable linear categories
In this subsection, we recall some definitions and needed properties in the context of linear prestable ∞-categories.Almost all notations, and all the results, are taken from [Lur18]: the purpose of this summary is only to gather all the needed vocabulary in a few pages, for the reader's convenience.Definition 2.16.Let C be an ∞-category.We will say that C is prestable if the following conditions are satisfied: • The ∞-category C is pointed and admits finite colimits.
• The suspension functor Σ : C → C is fully faithful.
• For every morphism f : Y → ΣZ in C, there exists a pullback and pushout square Proposition 2.17 ([Lur18, Proposition C.1.2.9]).Let C be an ∞-category.Then the following conditions are equivalent: • C is prestable and has finite limits.
• There exists a stable ∞-category D equipped with a t-structure (D ≥0 , D ≤0 ) and an equivalence C ≃ D ≥0 .
A stable presentable O X -linear ∞-category is an object of Mod QCoh(X ) (Pr L,⊗ St ).
Remark 2.22.The category Mod QCoh(X ) ≥0 (Groth ⊗ ∞ ) has a tensor product − ⊗ QCoh(X ) ≥0 − (which we will abbreviate by ⊗) induced by the Lurie tensor product of presentable ∞categories.See [Lur18, Theorem C.4.2.1] and [Lur18, Section 10.1.6]for more details.The same is true for Mod QCoh(X ) (Pr L,⊗ St ).There is a naturally induced stabilization functor which is symmetric monoidal with respect to these structures (again by an argument analogous to the one sketched in the nonlinear setting).
Theorem 2.24.Let X be a qcqs scheme over k.Then the global sections functors are symmetric monoidal equivalences.This theorem means that, if X is a qcqs scheme over k, every (Grothendieck pre)stable presentable O X -linear ∞-category C has an associated sheaf of ∞-categories on X having C as category of global sections.We will make substantial use of this fact in the present work.The main reason why we will always assume our base scheme X to be qcqs is because it makes this theorem hold.Definition 2.25.We denote the two right-hand sides of the equivalences in Theorem 2.24 respectively by LinCat St (X ) and LinCat PSt (X ).
Definition 2.26.We denote by Br(X ) the maximal ∞-groupoid contained in LinCat PSt (X ) and generated by ⊗ QCoh(X ) ≥0 -invertible objects which are compactly generated categories, and equivalences between them.We denote by Br † (X ) the maximal ∞-groupoid contained in LinCat St (X ) generated by ⊗ QCoh(X ) -invertible objects which are compactly generated categories, and equivalences between them.We call Br(X ) := π 0 Br(X ) the derived Brauer group of X and Br † (X ) := π 0 (Br † (X )) the extended derived Brauer group of X . 6By [Lur18, Theorem 10.3.2.1], to be compactly generated is a property which satisfies descent.The same is true for invertibility, since the global sections functor is a symmetric monoidal equivalence by Theorem 2.24.Theorem 2.27 ([Lur18, Example 11.5.7.15 and Example 11.5.5.5]).Let X be a qcqs scheme.There are equivalences of spaces Br(X ) ≃ Map(X , K( m , 2)) and Br † (X ) ≃ Map(X , K( m , 2) × K( , 1)) 6 Our notation here slightly differs from the one used in [Lur18, Definition 11.5.2.1, Definition 11.5.7.1], in that we write Br(X ), Br † (X ) in place of Br(X ), Br † (X ) in order to avoid confusion with the classical Brauer group.For the same reason, we introduce the terminology "derived Brauer group" (resp."extended derived Brauer group") which interpolates between the one used by Lurie and the one appearing in [Toë10, Definition 2.14].Indeed, our extended derived Brauer group Br † (X ) is the same as what Toën in loc.cit.calls dBr cat (X ), the derived categorical Brauer group of X , whereas there is no notation corresponding to Br(X ) in Toën's paper.

and therefore bijections
which can be promoted to isomorphisms of abelian groups.
Remark 2.28.By See [Lur18, Remark 11.5.7.3], the stabilization functor mentioned in Remark 2.22 restricts to a functor Br(X ) → Br † (X ), whose homotopy fiber at any object C ∈ Br † (X ) is discrete and can be identified with the collection of all t-structures (C ≥0 , C ≤0 ) on C satisfying the following conditions: • The t-structure (C ≥0 , C ≤0 ) is right complete and compatible with filtered colimits.
However, the stabilization functor is faithful ([Lur18, Proposition C.3.1.1]),and the induced map of groups is injective, in that it corresponds via Theorem 2.27 to the injection induced by the zero element of H 1 ét (X , ).We do not prove this last assertion, which however follows easily from the results in [Lur18, Section 11.5], since the relationship between the Brauer space and the extended Brauer space from the point of view of gerbes and torsors will be analyzed in a future work.We will never make use of this result in our proofs.
Remark 2.29.We now describe the symmetric monoidal structure of LinCat St (X ) using the language of simplicial colored operads.The same description will apply to the prestable case.This will come out useful in the rest of the paper.
The simplicial colored operad LinCat St (X ) can be described as follows: Remark is the smallest full subcategory of C ⊗ D containg C ≥0 ⊗ D ≥0 .The analogue statement in the O X -linear setting implies that, for G, G ′ → X m -gerbes over X ,

The correspondence
We now explain in detail the statement of Theorem 2.34 and offer a sketch of its proof, which will be carried out in Section 3. The construction of the ∞-category of twisted sheaves (see Definition 2.10) gives rise to a functor Ger m (X ) (the fact that it takes values in Br † (X ) is proven both in Theorem 3.This section is neither fully faithful nor essentially surjective.However, one can observe that QCoh id (G) is not just an O X -linear stable ∞-category, but also carries a t-structure which is compatible with filtered colimits, since as recalled in Remark 2.9 the functor (−) id : QCoh(G) → QCoh(G) is t-exact.This additional datum allows to "correct" the fact that QCoh id (−) is not an equivalence.More precisely, the fact that QCoh(G) has a functorial t-structure follows from [Lur18, Example 10.1.6.2, 10.1.7.2].One defines where the intersection is understood via the fully faithful functor QCoh(G) ≥0 → QCoh(G).Functoriality of this last operation follows from the functorialities made explicit in Remark 2.12.Definition 2.31.We denote by the functor assigning G → QCoh id (G) ≥0 .
In analogy to the stable setting, we will prove in Theorem 3.1 that Ψ factors through the (not full) subcategory Br(X ) → LinCat PSt (X ).
Remark 2.32.Given a morphism of gerbes f : G → G ′ , there are induced functors and QCoh(G) We will always use the covariant functoriality.We omit the proof of the fact that f * sends twisted sheaves to twisted sheaves, which follows from a straightforward computation.
Although there are no real issues with the contravariant notation, there are some technical complications when one wants to prove that a contravariant functor is symmetric monoidal.8Note that, if f is any morphism of gerbes (that commutes with the banding), then f is an isomorphism, therefore f * = ( f * ) −1 .
The inverse to Ψ will be described as follows.By Theorem 2.24, if X is a quasicompact quasiseparated scheme, and M a prestable presentable ∞-category equipped with an action of QCoh(X ) ≥0 in Pr L , then M is the category of global sections over X of a unique sheaf of prestable presentable QCoh(X ) ≥0 -linear categories M. Definition 2.33.Let X be a quasicompact quasiseparated scheme, and M be an element of Br(X ).Then we define the Triv ≥0 (M ) as the stack where the right-hand side is the space of equivalences of ∞ categories between QCoh(S) ≥0 and M(S).This is a stack because both M and QCoh(−) ≥0 are, and by Theorem 2.24.The functor will be denoted by Φ.
We will prove in Proposition 3.9 that Φ factors through the "forgetful" functor Ger m (X ) → Stk /X (we will abuse notation and denote the resulting functor again by Φ).
Note that both Br(X ) and Ger m (X ) have symmetric monoidal structures: on the first one, we have the tensor product of linear ∞-categories − ⊗ QCoh(X ) ≥0 − (Remark 2.22), and on the second one we have the rigidified product ⋆ of m -gerbes (Construction 2.13).
As anticipated, our main result is the following: Theorem 2.34.Let X be a qcqs scheme over a field k.Then there is a symmetric monoidal equivalence of ∞-groupoids9 As a corollary of the proof, we also have that Corollary 2.35 (Remark 3.6).Conjecture 5.27 in [BP21] is true.
We emphasize that symmetric monoidality is somehow a special feature of the prestable setting.Indeed, the analogous equivalence Br † (X ) ≃ Map(X , B 2 m × B ) is not symmetric monoidal, although it becomes an isomorphism of abelian groups after passing to π 0 ([Lur18, Remark 11.5.5.4,11.5.5.5]).Moreover, we use the symmetric monoidal structures on the two functors in a substantial way in order to prove other properties.
Section 3 is devoted to the proof of Theorem 2.34.Here is a sketch of our arguments.First, we establish the sought symmetric monoidal structure on Ψ (Theorem 3.1).We do this by considering the ∞-category of bi-homogeneous sheaves over G × X G ′ in the sense of Remark 2.11 and proving that the external tensor product establishes an equivalence between QCoh . This is done by reducing to the case of trivial gerbes (i.e.gerbes of the form B m × X ) and by using the fact that QCoh(B m ) is compactly generated.Then we prove that, in the special case of χ = χ ′ = id, the universal ("rigidification") map ρ : G× X G ′ → G⋆G ′ induces an equivalence QCoh (id,id) (G × X G ′ ) ≃ QCoh id (G ⋆ G ′ ) (Proposition 3.4): the fact that ρ * sends twisted sheaves to (id, id)-homogeneous sheaves is a direct computation using the universal property of the ⋆-product and the behaviour of the line bundles L id , L (id,id) , whereas the fact that it is an equivalence is checked étale-locally.
The above equivalences are both t-exact, because ρ : G × X G ′ → G ⋆ G ′ is representable and flat and thus one can apply t-exactness of ρ * on quasicoherent sheaves.Therefore, all results can be rephrased in the connective setting, i.e. for QCoh id (G) ≥0 .In particular, since any m -gerbe G is ⋆-invertible, the existence of a symmetric monoidal structure on Ψ implies that QCoh id (G) ≥0 belongs to Br(X ).
We then pass to proving that, for any M ∈ Br(X ), Φ(M ) is indeed a m -gerbe (Proposition 3.9), which follows essentially by working étale-locally on X and inspecting QCoh(X ) ≥0linear automorphisms of QCoh(X ) ≥0 itself.We also establish a symmetric monoidal structure on Φ (Lemma 3.11), which in this case follows directly from the universal properties defining ⊗ and ⋆.In Section 3.3, we conclude the proof of the main theorem.We start by proving that Φ is fully faithful (Proposition 3.13), which amounts again to a computation of automorphisms and uses the symmetric monoidal structure on Φ to easily reduce to working with the unit of Br(X ), i.e.QCoh(X ) ≥0 .Finally, we prove that there is a natural equivalence Id Ger m (X ) ⇒ ΦΨ.
The existence of a natural transformation is a consequence of the definitions, while the fact that it is an equivalence follows again by étale-local arguments.
3 Study of the derived Brauer map

Derived categories of twisted sheaves
Our aim in this subsection is to prove the following statement: Theorem 3.1.Let X be a qcqs scheme.The functors and QCoh id (−) ≥0 : Ger m (X ) ⋆ → LinCat PSt (X ) ⊗ carry a symmetric monoidal structure with respect to the ⋆-symmetric monoidal structure on the left hand side and to the ⊗-symmetric monoidal structures on the right hand sides.In particular, since every m -gerbe is ⋆-invertible and Ger m (X ) is a 2-groupoid, QCoh id (−) takes values in Br † (X ) and QCoh id (−) ≥0 takes values in Br(X ).
We will prove Theorem 3.1 in two different steps, Proposition 3.2 and Proposition 3.4, which also prove [BP21, Conjecture 5.27].
Proposition 3.2.Let X be a quasicompact quasiseparated scheme over a field k.Let G, G ′ → X be two m -gerbes over X , and χ , χ ′ two characters of m .The external tensor product establishes a t-exact equivalence where G × X G ′ is seen as a m × m -gerbe on X .
Proof.First we prove that the external tensor product induces a t-exact equivalence (3.1)By Theorem 2.24, every side of the sought equivalence is the ∞-category of global sections of a sheaf in categories over X , which means that the equivalence is étale-local on X .Therefore, by choosing a covering U → X which trivializes both G and G ′ , we can reduce to the case for every F ⊠ F ′ .With the notation of Remark 2.9, we can do this using the following chain of equivalence: A straightforward computation shows that α is in fact a natural transformation and verifies the condition described in Remark Remark 2.12.It remains to prove that the restricted functor induces an equivalence, which will automatically be t-exact.But again, it suffices to show this locally, and in the local case this just reduces to the fact that Proof.The proof is a direct application of the universal property of the ⋆-product, together with the fact that the structure map ρ is flat ([AOV08, Theorem A.1]).
Note first that the character m : m × m → m given by the multiplication induces a line bundle L id,id on B m × B m .One can prove easily that this line bundle coincides with the external product of two copies of L id , the universal line bundle on B m .This means that what we denote by L (id,id) ∈ QCoh(G × X G ′ × B m × B m ) has the form L id ⊠ L id (again with the usual notations of Remark 2.9).
We need to prove that ρ * sends id-twisted sheaves in (id, id)-twisted sheaves.To do this, we will construct an equivalence α(F) : (ρ * F) (id,id) ≃ ρ * (F id ) for every F in QCoh(G ⋆ G ′ ) and apply Remark 2.12.An easy computation shows that α is in fact a natural transformation and it verifies the condition described in Remark 2.12.
By construction of the ⋆ product, one can prove that the following diagram is commutative, where act (α,α ′ ) is the action map of G × X G ′ defined by the product banding, act αα ′ is the action map of G ⋆ G ′ and Bm is the multiplication map m : m × m → m at the level of classfying stacks.This implies that act * (α,α ′ ) (ρ * F) = (ρ, Bm) * act * αα ′ (F).
Consider now the following diagram: where q(α,α ′ ) , q (α,α ′ ) and q αα ′ are the projections.This is a commutative diagram and the square is a pullback.Finally, we can compute the (id, id)-twisted part of ρ * F: notice that since Bm * L id = L id,id we have (ρ, Bm) * L id = L id,id .Furthermore, an easy computation shows that the unit id → (id, Bm) * (id, Bm) * is in fact an isomorphism, because of the explicit description of the decomposition of the stable ∞-category of quasi-coherent sheaves over B m .These two facts together give us that (ρ * F) id,id =q (α,α ′ ), * (id, Bm) * (id, Bm) * (ρ, id) * (act To finish the proof, we need to verify that ρ * restricted to twisted sheaves is an equivalence with the category of (id, id)-homogeneous sheaves.Again, this can be checked étale locally, therefore we can reduce to the case G ≃ G ′ ≃ X × B m where the morphism ρ can be identified with (id X , Bm).We know that for the trivial gerbe we have that QCoh(X ) ≃ QCoh id (X × B m ) where the map is described by F → π * F ⊗ L id , π being is the structural morphism of the (trivial) gerbe.A straightforward computation shows that, using the identification above, the morphism ρ * is the identity of QCoh(X ).property, we can reduce to the trivial case, for which both sides of the banding are isomorphic to (X × B m ) × m while the morphism is just the identity.The second part of the statement follows from a straightforward computation.More precisely, we need to prove that the following diagram is commutative, where I Triv ≥0 ( f ) is the morphism induced by Triv ≥0 ( f ) on the inertia stacks.This follows from the trivial equality id f * id φ = id f •φ .
Remark 3.10.The following lemma is an example of a result which is more natural to prove in the context of ∞-categories, or in particular ∞-operads.In the proof of Theorem 3.1, we essentially proved that there exists an isomorphism and then proved that it is compatible with the higher structures.This follows essentially from the close relation between the ∞-categories of twisted sheaves on G × X G ′ and on G ⋆G ′ (see Proposition 3.4).
As far as the functor Triv ≥0 (−) is concerned, the same cannot be done so easily.One would like to use the morphism of gerbes but there is no natural way to construct a morphism as in the proof of Theorem 3.1.However, the two symmetric monoidal structures are constructed using the (cartesian) product structure.Therefore, the idea is to prove that the ⊗structure naturally transforms into the ⋆-structure through Triv ≥0 as ∞-operad.It is important to remark that Triv ≥0 ( M i ) is not isomorphic to Triv ≥0 (M i ).Nevertheless, we just need to universal property of the product to prove Lemma 3.11.
Lemma 3.11.The functor Triv ≥0 (−) is symmetric monoidal.Proof.To upgrade Triv ≥0 to a symmetric monoidal functor, we need to define its action on multilinear maps.Let {M i } i ∈I be a sequence of invertible prestable O X -linear ∞-categories indexed by a finite set I and N be another invertible prestable O X -linear category.Then we define Triv ≥0 (−) : in the following way: if f : i ∈I M i → N is a morphism in LinCat PSt (X ) which preserves small colimits separately in each variable, we have to define the image Triv ≥0 ( f ) as a functor such that Band(Triv ≥0 ( f )) is the n-fold multiplication of m , where n is the cardinality of I .We define it on objects in the following way: if {φ i } is an object of i ∈I (Triv ≥0 (M i )), then we set where φ i is just the morphism induced by the universal property of the product.
First of all we need to prove that f • φ i is still an equivalence.Let S ∈ Sch X and f : i ∈I M i → N morphism in LinCat PSt (X ), we can consider the following diagram: where the tensors are in fact relative tensors over QCoh(S).The diagonal map u is the morphism universal between all the QCoh(X )-linear morphisms from i ∈I M i (S) which preserve small colimits separately in each variable.Equivalently, one can say that it is a coCartesian edge in the ∞-operad LinCat PSt (S) ⊗ .
Thus, it is enough to prove that both ⊗φ i and f are equivalences.Because the source of the functor Triv ≥0 (−) is the ∞-groupoid Br(S), the morphism f (S) is an equivalence.Furthermore, the morphisms φ i are equivalences, therefore it follows from the functoriality of the relative tensor product that ⊗φ i is an equivalence.A straightforward computation shows that it is defined also at the level of 1-morphisms and it is in fact a functor.
It remains to prove that Band Triv ≥0 ( f ) is the n-fold multiplication of m , where n is the cardinality of I .It is equivalent to prove the commutativity of the following diagram: where m n is the n-fold multiplication of m .The notation follows the one in the proof of Proposition 3.9.Using diagram 3.2 again, we can reduce to the following straightforward statement: let λ 1 , . . ., λ n be QCoh(S)-linear automorphisms of id QCoh(S) , which can be identified with elements of m (S); then the tensor of the natural transformations coincide with the product as elements of m , i.e λ 1 ⊗ • • • ⊗ λ n = m n (λ 1 . . .λ n ).
Finally, because all maps are maps of gerbes, the condition of being strictly monoidal is automatically satisfied once the lax monoidal structure is given.

Proof of the main theorem
The goal of this subsection is to prove that the constructions M → Triv ≥0 (M ) and G → QCoh id (G) ≥0 establish a (symmetric monoidal) categorical equivalence between the ∞-groupoids Br(X ) and Ger m (X ), thus proving Theorem 2.34.Remark 3.12.Note that both sides of Theorem 2.34 are in fact 2-groupoids, the left-handside by [Lur18, Construction 11.5.7.13] and the right-hand-side by definition.However, we do not use this in the proof.

PROOF OF THE MAIN THEOREM
Proposition 3.14.Let X be a qcqs scheme.Then there is a natural equivalence of functors Id Ger m (X ) ⇒ Triv ≥0 • QCoh id (−) ≥0 .
Proof.Let G be a m -gerbe over X .Let us observe that for any S → X we have G(S) ≃ Equiv Ger m (S) (S × B m , G S ).
Indeed, there is a map of stacks over X where u S : S → S ×B m is induced by the canonical atlas of B m .Up to passing to a suitable étale covering of X , the map becomes an equivalence: in fact the choice of an equivalence φ : S × B m → S × B m of gerbes over S amounts to the choice of a map S → B m , because φ must be a map over S (hence pr B m •φ = pr B m ) and it must respect the banding.Therefore, F is an equivalence over X , and this endows Equiv Ger m (X ×B m , G) with a natural structure of a m -gerbe over X .The construction φ → φ * thus provides a morphism of stacks G → Equiv QCoh(X ) QCoh id (X × B m ), QCoh id (G) .Now since the pushforward φ * of an equivalence φ of stacks is t-exact, the construction φ → φ * | QCoh id (X ×B m ) ≥0 yields a map G → Equiv QCoh(X ) ≥0 QCoh id (X × B m ) ≥0 , QCoh id (G) ≥0 .and now the right-hand-side is in turn equivalent to G → Equiv QCoh(X ) ≥0 QCoh(X ) ≥0 , QCoh id (G) ≥0 = Triv ≥0 (QCoh id (G) ≥0 ).
But now, Triv ≥0 (QCoh id (G)) is a m -gerbe over X (note that this was not true before passing to the connective setting), and therefore to prove that the map is an equivalence it suffices to prove that it agrees with the bandings, i.e. that it is a map of m -gerbes.This follows from unwinding the definitions.
We are now ready to prove our main result.
Proof of Theorem 2.34: Section 3.1 and Section 3.2 tell us that the two functors are symmetric monoidal and take values in the sought ∞-categories.The fact that they form an equivalence follows from Proposition 3.13 and Proposition 3.14.