Azumaya objects in triangulated bicategories
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Abstract
We introduce the notion of Azumaya object in general homotopytheoretic settings. We give a selfcontained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg–Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutative ring. We also discuss tilting theory as an application of invertibility in triangulated bicategories.
Keywords
Brauer group Ring spectrum Homotopical algebraMathematics Subject Classification (2000)
55U99 18D35 16K50 14F221 Introduction
The notion of Azumaya algebra over a commutative ring \(k\) generalizes the notion of a central simple algebra over a field. This was introduced by Azumaya [4] in the case that \(k\) is local and by Auslander and Goldman [1] for general \(k\). The main classification theorem of Azumaya algebras says that the set of Moritaequivalenceclasses of Azumaya \(k\)algebras is the maximal subgroup of the monoid formed by Moritaequivalenceclasses of \(k\)algebras under \(\otimes _k\). This is the Brauer group of \(k\), an invariant which carries interesting algebraic and geometric information.
1.1 Foundations of homotopical Brauer theory
This work aims to develop foundations for Brauer theory in homotopical settings. We consider Azumaya objects in closed autonomous symmetric monoidal bicategories, and in particular focus on the triangulated bicategories arising as homotopy bicategories of rings and ring spectra. (The term “closed” refers to the existence of internal homs, and “autonomous” refers to opposites such as opposite algebras (see Sects. 3.4, 3.3).
For the presentation here, we have three audiences in mind: the algebraic audience, for whom Azumaya algebras and Brauer groups of commutative rings are quite familiar; the categorical audience, for whom bicategories and invertibility therein are quite familiar; and the topological audience for whom structured ring spectra and the homotopical algebra thereof are quite familiar. We expect few readers to be in the intersection of these three audiences, and thus have attempted to write for the union of their complements.
In Sects. 1 and 2 we introduce and motivate our results in the discrete and homotopical cases with a minimum of bicategorical language. This includes a review of classical Brauer groups for commutative rings as a special case of our general approach. A survey of the relevant bicategorical background is then presented in Sect. 3, with the general characterization of Azumaya objects in Sect. 3.6. We develop the notion of Azumaya object for triangulated bicategories in Sect. 4, and discuss homotopical (derived) applications in Sect. 5.
Now we state the main definitions and results.
Definition 1.1
(Eilenberg–Watts equivalence) Let \(A\) and \(B\) be a 0cells of a bicategory \({\fancyscript{B}}\). We say \(A\) is Eilenberg–Watts equivalent to \(B\) if there exists an invertible 1cell Open image in new window . In bicategorical literature, this is known simply as equivalence of 0cells, but we prefer the more expressive term for this work.
Definition 1.2
(Eilenberg–Watts bicategory) We say that a bicategory is Eilenberg–Watts if it is closed (i.e., has internal homs); has a symmetric monoidal product; and has an autonomous structure (i.e., a notion of “opposite” objects). Each of these bicategorical structures is described in Sect. 3.
Definition 1.3
(Brauer group, Azumaya objects) Let \({\fancyscript{B}}\) be a monoidal bicategory with unit 0cell \(k\). The Brauer group of \({\fancyscript{B}}\), denoted \(Br({\fancyscript{B}})\), is the group of 1cell equivalence classes (Eilenberg–Watts equivalence classes) of 0cells \(A\) for which there exists a 0cell \(B\) such that \(A \otimes _k B\) is Eilenberg–Watts equivalent to \(k\). Such 0cells are called Azumaya objects.
Remark 1.4
What we have termed “Eilenberg–Watts equivalence” is sometimes called “Morita equivalence”, or sometimes “standard Morita equivalence” [24]. The term “Morita equivalence” is also used frequently for monoid objects to mean that their categories of modules are equivalent. The fundamental theorem of Morita theory states that these notions coincide for algebras over a commutative ring and their module categories. Moreover, Rickard [29] has shown that these notions coincide for algebras over a commutative ring and their derived categories of modules. However it is known that this coincidence does not generalize to derived categories of differential graded algebras or ring spectra [30]. Work of Dugger and Dugger–Shipley analyzes the situation for differential graded algebras and ring spectra (see [11, 12] and their related articles). The survey [21] compares the two notions abstractly for closed bicategories.
We have elected to introduce new terminology in hopes of clarifying the distinction between the two notions. See [20] for a discussion of Eilenberg–Watts theorems in various settings.
The classical treatment of Azumaya algebras depends heavily on localization arguments and reducing to the case of algebras over fields; one feature of recent work in this area is a treatment that is independent of ideal theory: by introducing a bicategorical context we are able to generalize and unify the basic theory of Azumaya algebras and Brauer groups, giving elementary formal proofs of the main classification theorems. Similar ideas have appeared in algebraic and categorical literature—notably [7, 28, 38, 39]. We extend this approach to homotopical settings, giving a classification of Azumaya objects via invertibility in triangulated bicategories in Sect. 4.2. During preparation of this article, related work has appeared in [2, 6, 13, 37].
Notation 1.5
Throughout, \(k\) will denote the unit of a symmetric monoidal bicategory. In our applications, \(k\) will often be a commutative (\(E_\infty \)) d.g. algebra or ring spectrum. We let \({\fancyscript{C}}_k\) denote the category of \(k\)modules, noting that this is a symmetric monoidal category. We let \({\fancyscript{M}}_k\) denote the bicategory of \(k\)algebras and bimodules.
When \({\fancyscript{C}}_k\) has a suitable model structure (described in Sect. 4.1, following [33]), then there is an induced model structure on \({\fancyscript{M}}_k(A,B)\) for \(k\)algebras \(A\) and \(B\). We let \({\fancyscript{D}}_k(A,B)\) denote the corresponding homotopy category.
Taking as 0cells the collection of \(k\)algebras which are cofibrant as \(k\)modules, we give one set of conditions in Proposition 4.14 which guarantee that \({\fancyscript{D}}_k\) forms a triangulated bicategory with the additional Eilenberg–Watts structure necessary for the homotopical Brauer theory of Sects. 3.6 and 4.2.
Although the bicategorical definitions of Azumaya object and Brauer group require only a monoidal structure, these are somewhat too general to be of practical interest. In an Eilenberg–Watts bicategory, we have characterization theorems for Azumaya objects in more concrete terms. In Definitions 3.10 and 3.11 we generalize the classical notions of central, (faithfully) separable, and faithfully projective \(k\)algebras to Eilenberg–Watts bicategories. We also introduce the term “endomorphic”, formally dual to the notion of centrality. We then show that these are the correct generalizations by proving the following theorem in Sect. 3.6:
Theorem 1.6
 i.
\(\lfloor A \rfloor \) is an invertible 1cell.
 ii.
\(A\) is central and faithfully separable over \(k\).
 iii.
\(A\) is endomorphic and faithfully projective over \(k\).
 iv.
\(A^e\) is Eilenberg–Watts equivalent to \(k\).
 v.
There is a 0cell \(B\) such that \(A \otimes B\) is Eilenberg–Watts equivalent to \(k\).
Remark 1.7
The equivalence of the first three conditions is formal and straightforward. The argument proceeds by explaining that each of the second two conditions is an alternate description of invertibility. It is clear that the first condition implies the last two; one goal of this work is to develop a setting in which the reverse implications are also straightforward.
Another goal of this work is to prove a further characterization of Azumaya objects when (as in the derived and homotopical cases) the ambient bicategory carries a triangulated structure. In this case, the notion of invertibility admits alternate descriptions in terms of localization. This has been used in the topological case by Baker and Lazarev [5] for \(THH\) calculations. In Sect. 4 we introduce triangulated bicategories and prove a further characterization of Azumaya objects in that context. As with the rest of this theory, the notion of localization has both left (source) and right (target) versions, but we leave these definitions until Sect. 4 where we will develop more of the underlying bicategorical structure. The following is immediate from Proposition 4.21:
Theorem 1.8
 i.
\(A\) is an Azumaya object.
 ii.
\(A\) is dualizable over \(A^e\) and central over \(k\), and \(A^e\) is strongly (target\()A\)local.
 iii.
\(A\) is dualizable over \(k\) and endomorphic over \(k\), and \(k\) is strongly (source\()A\)local.
Using these general characterizations of Azumaya objects, we show in Sect. 5 that the homotopical Brauer groups defined here agree with those of [37] for derived categories of rings and of [6] for commutative \(S\)algebras.
1.2 Brauer groups of Eilenberg–Mac Lane spectra
One application of the foundations developed here is an immediate computation of the Brauer group of an Eilenberg–Mac Lane spectrum. We include the proof because it is short and because it illustrates the utility of the setting we develop: that which should be formal (because the homotopical algebra of Eilenberg–Mac Lane spectra is equivalent to the ordinary algebra of their underlying rings) has become straightforward (because the Eilenberg–Mac Lane functor induces a local equivalence of triangulated bicategories).
Theorem 1.9
Proof
Remark 1.10
As an immediate corollary, we have a computation of the Brauer group for an Eilenberg–Mac Lane spectrum of a field in terms of the classical Brauer group:
Corollary 1.11
If \(k\) is a field, then \(Br({\fancyscript{D}}_{Hk}) \cong Br(k)\). In particular, \(Br({\fancyscript{D}}_{Hk}) = 0\) if \(k\) is finite or algebraically closed.
Proof
Proposition 5.3 says that \(Br({\fancyscript{D}}_k) \cong Br(k)\). The result then follows from Theorem 1.9. \(\square \)
In the case that \(k\) is an algebraically closed field, this was established by [6] using different methods, but also relying on [37] for the comparison with the derived Brauer group of a field.
1.3 Invertibility
Sections 3.5 and 4.2 give a general study of invertibility in triangulated bicategories, and this forms the foundation of the characterization theorems. Proposition 4.21 gives a general characterization of invertible objects in triangulated bicategories. As further applications, we recover results from the derived Morita (Eilenberg–Watts) theory of Rickard [29] and Schwede and Shipley [35]. The object \(T\) here is called a tilting object.
Proposition 1.12
 i.
\(T\) is a rightdualizable \(B\)module.
 ii.
\(T\) generates the triangulated category \({\fancyscript{D}}_k(B).\)
Note Since \(T\) is rightdualizable, \(T\) is a bounded complex of finitelygenerated and projective \(B\)modules. Therefore the derived and underived endomorphism algebras are equal.
Proposition 1.13
 i.
\(T\) is rightdualizable as a \(B\)module.
 ii.
\(T\) generates the triangulated category \({\fancyscript{D}}_k(B).\)
2 Motivation
To help motivate the generality of Sects. 3 and 4 needed for Theorems 1.6 and 1.8, we give parallel introductions of the classification theorems for classical algebra and for homotopical settings (Sects. 2.1 and 2.2, respectively). Both are special cases of the general theory. In Example 2.7 we sketch a number of additional examples to which the theory applies.
2.1 Brauer groups of discrete rings
In this section we review the classical theory of Brauer groups for commutative (discrete) rings. Let \(k\) be a commutative ring and let \({\fancyscript{B}}= {\fancyscript{M}}_k\) be the bicategory of \(k\)algebras and their bimodules. A 0cell \(A\) of this bicategory is a \(k\)algebra, and \(A^e\) is the enveloping \(k\)algebra \(A \otimes _k A^\mathsf{op }\). Note that throughout this section we regard \(A\) as a right module over \(A^e\) and a left module over \(k\). We recall the classical characterization/definition of Azumaya algebras after recalling basic terminology and a wellknown lemma. These are special cases of the more general terms introduced in Definitions 3.10 and 3.11.
Definition 2.1
 \(A\) is called separable over \(k\) if \(A\) is projective as a module over \(A^e\). Since \(A\) is always finitely generated over \(A^e\), this is equivalent to the condition that \(A\) be a dualizable module over \(A^e\). By the dual basis lemma, this is equivalent to the condition that the coevaluationbe an isomorphism. To motivate this term, it should be noted that when \(k\) is a field, \(A\) being separable over \(k\) implies that \(A\) is semisimple over \(k\) and remains semisimple upon extension of scalars over any field extension of \(k\). When \(A\) is also a field, this implies that \(A\) is a separable extension of \(k\) in the usual sense for fields [8].$$\begin{aligned} A \otimes _{A^e} {{\mathrm{Hom}}}_{A^e}(A,A^e) \rightarrow {{\mathrm{Hom}}}_{A^e}(A,A) \end{aligned}$$
 \(A\) is called central over \(k\) if the center of \(A\) is precisely \(k\); this occurs if and only if the unitis an isomorphism.$$\begin{aligned} k \rightarrow {{\mathrm{Hom}}}_{A^e}(A,A) \end{aligned}$$
 \(A\) is called faithfully projective over \(k\) if both the coevaluationand the evaluation$$\begin{aligned} {{\mathrm{Hom}}}_k(A,k) \otimes A \rightarrow {{\mathrm{Hom}}}_k(A,A) \end{aligned}$$are isomorphisms. Note, again by the dual basis lemma, that the coevaluation being an isomorphism is equivalent to \(A\) being finitelygenerated and projective as a \(k\)module. The evaluation map being an isomorphism implies that \( \otimes _{k} A\) is object faithful, meaning that \(M \otimes _{k} A = 0\) implies \(M = 0\).$$\begin{aligned} A \otimes _{A^e} {{\mathrm{Hom}}}_k(A,k) \rightarrow k \end{aligned}$$
Lemma 2.2
Here is the classical theorem, which follows directly from Theorem 1.6. It provides a link between Eilenberg–Watts equivalences and the more calculable conditions of central, separable and faithfully projective.
Theorem
 i.
\(A\) is invertible as a \((k, A^e)\)bimodule, thus providing an Eilenberg–Watts equivalence between \(A^e\) and \(k\).
 ii.
\(A\) is central and separable over \(k\).
 iii.\(A\) is faithfully projective over \(k\) and the unit mapis an isomorphism.$$\begin{aligned} A^e \rightarrow {{\mathrm{Hom}}}_{k} (A, A) \end{aligned}$$
 iv.
There is a \(k\)algebra \(B\) such that \(A \otimes _k B\) is Eilenberg–Watts equivalent to \(k\).
Remark 2.3
Note that the last condition of item iii means that every \(k\)linear endomorphism of \(A\) is given by left and right multiplication in \(A\). We call such an algebra “endomorphic” in Definition 2.4.
2.2 Homotopical Brauer groups
In Sect. 4.1 we describe a bicategorical structure on the collection of categories \({\fancyscript{D}}_k(A,B)\), where \(k\) is a commutative (\(E_\infty \)) d.g. algebra or ring spectrum, cofibrant as a module over itself, and \(A\) and \(B\) are cofibrant as \(k\)modules (see Proposition 4.15). This consists mainly of assembling the necessary results from the literature on monoidal model categories, but we have presented a careful treatment of the Eilenberg–Watts structure.
In this section we continue to motivate the general theory by describing the definitions and Azumaya characterization theorem in this special case. We use \(\wedge \) to denote the derived tensor product, and \(F\) to denote the derived hom. The isomorphisms in the homotopy category are the weak equivalences, and so we switch to this terminology. Let \(k\) be a commutative ring, and let \(A\) be a differentialgraded algebra over \(k\). For the topologically inclined reader, \(k\) and \(A\) can be taken to be structured ring spectra—the modern foundations of spectra such as [14, 25] allow us to treat the algebraic and topological cases with the same categorical arguments. For either the algebraic or topological cases, we assume throughout that \(k\) and \(A\) are cofibrant with respect to a given model structure on \({\fancyscript{C}}_k\).
To begin, we make the following definitions inspired by the classical case. They are again special cases of the general bicategorical definitions in 3.10 and 3.11.
Definition 2.4
Definition 2.5
Definition 2.6
We call \(A\) an Azumaya object if any of the equivalent conditions below hold.
Theorem
 i.
\(A\) is an invertible bimodule.
 ii.
\(A\) is central and faithfully separable over \(k\).
 iii.
\(A\) is endomorphic and faithfully projective over \(k\).
 iv.
\(A^e\) is Eilenberg–Watts equivalent to \(k\).
 v.
There exists a \(k\)algebra \(B\) such that \(A \wedge _k B\) is Eilenberg–Watts equivalent to \(k\).
2.3 Additional examples
For concreteness, we focus our main applications on the Eilenberg–Watts bicategories coming from differential graded algebras or from ring spectra. But the bicategorical setting is quite general and here we sketch several additional examples to which the theory applies.
Example 2.7
(Symmetric monoidal categories) The example of rings and modules motivates a standard construction for a bicategory \({\fancyscript{M}}_{{\fancyscript{C}}}\) from any complete and cocomplete closed symmetric monoidal category \({\fancyscript{C}}\). The 0cells are taken to be the monoids of \({\fancyscript{C}}\), and 1cells are taken to be bimodules. Bimodule maps are defined just as in the case of bimodules over rings, and these constitute the 2cells of \({\fancyscript{M}}_{{\fancyscript{C}}}\). Colimits from \({\fancyscript{C}}\) are used to construct the horizontal composition in \({\fancyscript{M}}_{{\fancyscript{C}}}\), in the same way that coequalizers give the tensor product of two modules over a ring. Limits give \({\fancyscript{M}}_{{\fancyscript{C}}}\) a closed structure in the same way that equalizers give the homomorphisms of two modules over a ring.
Remembering the underlying symmetric monoidal product of \({\fancyscript{C}}\) gives \({\fancyscript{M}}_{{\fancyscript{C}}}\) an autonomous symmetric monoidal structure, and \({\fancyscript{M}}_{{\fancyscript{C}}}\) is therefore Eilenberg–Watts. Theorem 1.6 thus characterizes the Azumaya monoids of \({\fancyscript{C}}\), and defines a Brauer group of \({\fancyscript{C}}\) which reduces to the Brauer group of \(k\) when \({\fancyscript{C}}\) is the symmetric monoidal category of modules over a commutative ring \(k\).
The Brauer group of a symmetric monoidal category has been defined directly by Pareigis [28] and, for a braided monoidal category, by Van Oystaeyen and Zhang [39]. These definitions (in the symmetric monoidal case) are equivalent to ours, but there the emphasis is on other aspects of the classical theory (e.g. separability and Casimir elements); their work does not give the full characterization of Azumaya algebras presented in Theorem 1.6. Likewise, we do not discuss the variants on Brauer theory described by Pareigis or Van Oystaeyen–Zhang. Vitale [38] gives a thorough treatment of Brauer groups for symmetric monoidal categories, and a pleasant summary of categorical approaches to this case.
Example 2.8
Example 2.9
(Rings with many objects) Let \(\mathcal{A }\) be a small category enriched in Open image in new window , the category of abelian groups. A left \(\mathcal{A }\)module is a functor Open image in new window , and a right module is a functor \(\mathcal{A }^\mathsf{op } \rightarrow \); morphisms of \(\mathcal{A }\)modules are enriched natural transformations. Now let \({\fancyscript{R}}\textit{ingoids}\) be the bicategory whose 0cells are small Open image in new window categories, 1cells are bimodules, and 2cells are natural transformations. The tensor product and hom for bimodules induce horizontal composition with a closed structure, and the tensor product in Open image in new window induces a symmetric monoidal product on \({\fancyscript{R}}\textit{ingoids}\) making it an Eilenberg–Watts bicategory. The unit object is the oneobject category whose endomorphism ring is the ring of integers.
Clearly this construction may be generalized by replacing Open image in new window with a bicomplete closed symmetric monoidal category \(\mathbf V \). We denote the bicategory so constructed by \(\mathbf V \) \({\fancyscript{R}}\textit{ingoids}\). This too is an Eilenberg–Watts bicategory, and thus Theorem 1.6 gives a characterization of Azumaya objects.
Example 2.10
(Sheaves) The Brauer group of a sheaf \({\fancyscript{O}}\) of commutative rings was introduced by Auslander in [3] and by Grothendieck in [17]. By Theorem 1.6 this is the same as the Brauer group of the symmetric monoidal category of \({\fancyscript{O}}\)modules: a map of sheaves is an isomorphism if and only if it is so locally, and thus an Azumaya object in the bicategory of \({\fancyscript{O}}\)algebras and their bimodules is precisely the same as a sheaf of Azumaya algebras. This perspective was initiated by Auslander’s work, and has been discussed by Van Oystaeyen–Zhang [39, 3.5(2); 3.7].
3 Eilenberg–Watts bicategories
Recall that we use the term “Eilenberg–Watts bicategory” for a closed autonomous symmetric monoidal bicategory. We describe these terms in Sects. 3.1 to 3.4. Section 3.5 reviews duality and invertibility in bicategories. In Sect. 3.6 we prove the characterization theorem for Azumaya objects in Eilenberg–Watts bicategories.
Notation 3.1
3.1 Monoidal bicategories
A trivial but slightly technical lemma about monoidal structure will be useful later:
Lemma 3.2
Proof
3.2 Symmetric monoidal bicategories
3.3 Autonomous structure
3.4 Closed structure
3.5 Duality and invertibility in bicategories
For general discussion about duality, we consider fixed 1cells Open image in new window in a closed bicategory \({\fancyscript{B}}\).
Definition 3.3
Lemma 3.4
Note. When \((X,Y)\) is a dual pair, we will occasionally refer to this by saying that \(X\) is dualizable over \(B\), since the unit condition often amounts to a finiteness of \(X\) over \(B\). When \((Y,X)\) is a dual pair, we will say that \(X\) is dualizable over \(A\). With this convention, the phrase “dualizable over” always references the target of the evaluation map, and uniquely determines whether we mean left or rightdualizable.
Definition 3.5
(Invertible pair) A dual pair \((X,Y)\) is called invertible if the maps \(\eta \) and \(\varepsilon \) are isomorphisms. Equivalently, the adjoint pairs described in Proposition 3.7 are adjoint equivalences.
Duality for monoidal categories has been studied at length; one reference for duality in a bicategorical context is [26, Sect. 16.4], and there are surely others in the categorical literature. The definition of duality does not require \({\fancyscript{B}}\) to be closed, but we will make use of the following basic facts about duality, some of which do require a closed structure on \({\fancyscript{B}}\). The following two results can be found in [26, Sect. 16.4].
Proposition 3.6
Proposition 3.7
 i.For any 0cell \(C\), we have two adjoint pairs of functors, with left adjoints written on top: The structure maps for the dual pair give the triangle identities necessary to show that the displayed functors are adjoint pairs.
 ii.
The right dual, \(Y,\) is canonically isomorphic to \({{\mathrm{\textit{t}Hom}}}(X,B)\), and for any 1cell Open image in new window , the coevaluation map \(W \odot {{\mathrm{\textit{t}Hom}}}(X,B) \rightarrow {{\mathrm{\textit{t}Hom}}}(X,W)\) is an isomorphism.
 iii.
The left dual, \(X,\) is canonically isomorphic to \({{\mathrm{\textit{s}Hom}}}(Y,B)\), and for any 1cell Open image in new window , the coevaluation map \({{\mathrm{\textit{s}Hom}}}(Y,A) \odot U \rightarrow {{\mathrm{\textit{s}Hom}}}(Y,U)\) is an isomorphism.
Lemma 3.8
Let Open image in new window . If \(X\) is rightdualizable and the unit \(A \rightarrow {{\mathrm{\textit{t}Hom}}}(X,X)\) is an isomorphism, then the evaluation \(X \odot \,\, {{\mathrm{\textit{s}Hom}}}(X,A) \rightarrow A\) is an isomorphism. Likewise, if \(X\) is leftdualizable and the unit \(B \rightarrow {{\mathrm{\textit{s}Hom}}}(X,X)\) is an isomorphism, then the evaluation \({{\mathrm{\textit{t}Hom}}}(X,B) \odot X \rightarrow B\) is an isomorphism.
Proof
3.6 Characterization of Azumaya objects
In this section we give a proof of Theorem 1.6 in two stages. Let \(A\) be a fixed 0cell of an Eilenberg–Watts bicategory \({\fancyscript{B}}\).
Notation 3.9
We now give the general characterization of Azumaya objects; Theorem 1.6 follows directly from this. First, we give natural generalizations of the classical terminology:
Definition 3.10
Definition 3.11
We are now ready to prove the characterization theorem for Azumaya objects in Eilenberg–Watts bicategories.
Proof of Theorem 1.6
In general, a pair of 1cells \((X,Y)\) is invertible if and only if \(X\) is rightdualizable, \(Y\) is isomorphic to the canonical right dual of \(X\), and both the unit and counit of the duality are isomorphisms. This gives the equivalence of i and iii. Likewise, \((Y,X)\) is invertible if and only if \(X\) is leftdualizable, \(Y\) is isomorphic to the canonical left dual of \(X\), and both the unit and counit of the duality are isomorphisms. This gives the equivalence of i and iv. Since \((X,Y)\) is an invertible pair if and only if \((Y,X)\) is such, the first three conditions are seen to be equivalent.
4 Triangulated bicategories
We recall first the definitions of localizing subcategory and generator for a triangulated category, and then give a definition (4.4) of triangulated bicategory suitable for our purposes. In particular, under this definition \({\fancyscript{D}}_k\) is a triangulated bicategory when \(k\) is a commutative d.g. ring or ring spectrum.
Definition 4.1
(Localizing subcategory) If \({\fancyscript{T}}\) is a triangulated category with infinite coproducts, a localizing subcategory, \({\fancyscript{S}}\), is a full triangulated subcategory of \({\fancyscript{T}}\) which is closed under coproducts from \({\fancyscript{T}}\).
Remark 4.2
This is equivalent to the definition for arbitrary triangulated categories of [19], (which requires that a localizing subcategory be thick) because a triangulated subcategory automatically satisfies the 2outof3 property and because in any triangulated category with countable coproducts, idempotents have splittings. See [27, 1.5.2, 1.6.8, and 3.2.7] for details.
Definition 4.3
(Triangulated generator) A set, \({\fancyscript{P}}\), of objects in \({\fancyscript{T}}\) (triangulated category with infinite coproducts, as above) is a set of triangulated generators (or simply generators) if the only localizing subcategory containing \({\fancyscript{P}}\) is \({\fancyscript{T}}\) itself.
Definition 4.4
 (TC0)The local suspension functorsassemble as the components of a strong transformation$$\begin{aligned} \Sigma _{A,B} : {\fancyscript{B}}(A,B) \xrightarrow {\simeq } {\fancyscript{B}}(A,B) \end{aligned}$$This implies, in particular, that for a 1cell Open image in new window there are natural structure isomorphisms$$\begin{aligned} \Sigma : {\fancyscript{B}}(,) \rightarrow {\fancyscript{B}}(,). \end{aligned}$$
 (TC1)For a 0cell \(A\), the following composite of structure isomorphisms interchanging suspension coordinates is multiplication by 1:$$\begin{aligned} \Sigma ^2 A \xrightarrow {\Sigma \beta _A^{1}} \Sigma (A \odot \Sigma A) \xrightarrow {\alpha _{\Sigma A}^{1}} \Sigma A \odot \Sigma A \xrightarrow {\beta _{\Sigma A}} \Sigma (\Sigma A \odot A) \xrightarrow {\Sigma \alpha _A} \Sigma ^2 A. \end{aligned}$$
 (TC2)
Each of \( \odot \), \({{\mathrm{\textit{s}Hom}}}(,)\), and \({{\mathrm{\textit{t}Hom}}}(,)\) is exact in both variables.
The axioms above generalize those of a triangulated monoidal category although, because we do not need it, we do not include a version of the braid axiom (TC3).
Remark 4.5
For a commutative d.g. algebra or ring spectrum \(k\), each local category \({\fancyscript{D}}_k(A,B)\) is triangulated, with the suspension functor given by horizontal composition \(\Sigma A \odot \). The monoidal category \({\fancyscript{D}}_k(k,k)\) is wellknown to satisfy the compatibility conditions, and the same reasoning shows that \({\fancyscript{D}}_k\) satisfies these conditions in general.
If \({\fancyscript{B}}\) is a triangulated bicategory and \(P\), \(Q\) are 1cells in \({\fancyscript{B}}(A,B)\), we emphasize that \({\fancyscript{B}}\) is triangulated by writing the abelian group of 2cells \(P \rightarrow Q\) as \({\fancyscript{B}}[P,Q]\) and by writing the graded abelian group obtained by taking shifts of \(Q\) as \({\fancyscript{B}}[P,Q]_*\). To emphasize the source and target of \(P\) and \(Q\), we may also write \({\fancyscript{B}}(A,B)[P,Q]_*\).
Definition 4.6
(\(\odot \)faithful 1cells) In any locally additive bicategory, \({\fancyscript{B}}\), a 1cell Open image in new window is called sourcefaithful if triviality for any 1cell Open image in new window is detected by triviality of the composite \(Z \odot W\). That is, Open image in new window is zero if and only if \(Z \odot W = 0\). A collection of 1cells, \({\fancyscript{E}}\), in \({\fancyscript{B}}(A,B)\) is called jointly sourcefaithful if the objects have this property jointly; that is, \(Z = 0\) if and only if \(Z \odot W = 0\) for all \(W \in {\fancyscript{E}}\). The term targetfaithful is defined similarly, considering \(W \odot \) instead of \( \odot W\).
Remark 4.7
If \({\fancyscript{C}}\) is an additive monoidal category with monoidal product \(\odot \), the unit object is both source and targetfaithful. In an arbitrary locally additive bicategory \({\fancyscript{B}}\), if \(A \ne B\) then \({\fancyscript{B}}(A,B)\) may not have a single object with this property. In relevant examples, however, the collection of all 1cells, ob\({\fancyscript{B}}(A,B)\), does have this property jointly.
Lemma 4.8
Let \({\fancyscript{B}}\) be a triangulated bicategory, and let Open image in new window be a generator for \({\fancyscript{B}}(A,B)\). If the collection of all 1cells, \({\fancyscript{B}}(A,B)\), is jointly sourcefaithful (resp. targetfaithful), then \(P\) is sourcefaithful (resp. targetfaithful).
Proof
Remark 4.9
Since the functors \(P \odot \) are exact, the property of \(P \odot \) detecting trivial objects is equivalent to \(P \odot  \) detecting isomorphisms (meaning that a 2cell \(f\) is an isomorphism if and only if \(P \odot f\) is so).
4.1 Homotopy bicategories
In this section, let \({\fancyscript{C}}\) be a complete and cocomplete closed symmetric monoidal model category. Let \({\fancyscript{M}}\) be the closed bicategory formed by \({\fancyscript{C}}\)monoids and their bimodules as in Example 2.7. We say that \({\fancyscript{M}}\) has a local model structure if each category of 1 and 2cells \({\fancyscript{M}}(A,B)\) is a model category.
We now describe further conditions for the model structure on \({\fancyscript{C}}\) which ensure that \({\fancyscript{M}}\) has a local model structure and that the collection of homotopy categories assembles to an Eilenberg–Watts bicategory. The first of these, pushout products, implies that the horizontal composition of 1cells descends to a homotopy bicategory. The second, unit replacement, implies that horizontal composition of 1cells on the level of homotopy is unital.
Definition 4.10
Definition 4.11
Next we recall one result which implies that \({\fancyscript{M}}\) has a local model structure, and then give another showing that this local model structure indeed descends to form a homotopy bicategory.
Proposition 4.12
[33, 4.1] Let \({\fancyscript{C}}= ({\fancyscript{C}}, \wedge , k)\) be a cofibrantly generated closed monoidal model category which satisfies the monoid axiom and in which each object is small with respect to the whole category. Let \({\fancyscript{M}}_{\fancyscript{C}}\) be the bicategory formed by \({\fancyscript{C}}\)monoids which are cofibrant in \({\fancyscript{C}}\) and their bimodules. Then \({\fancyscript{M}}_{\fancyscript{C}}\) has a local model structure, and has pushout products. Moreover, the category of \({\fancyscript{C}}\)monoids \({{\mathrm{Mon}}}({\fancyscript{C}})\) is a cofibrantly generated model category, and if \(k\) is cofibrant then every cofibration in \({{\mathrm{Mon}}}({\fancyscript{C}})\) whose source is cofibrant is also a cofibration upon forgetting to \({\fancyscript{C}}\).
This proposition applies, for example, to simplicial sets, \(\Gamma \)spaces, symmetric spectra, simplicial abelian groups, chain complexes, and \(S\)modules. Details for these and other examples are given in [33, Sect. 5].
Proposition 4.13

The unit, \(k\), is cofibrant in \({\fancyscript{C}}\).

The unit replacement condition holds for each monoid in \({\fancyscript{C}}\) which is cofibrant in \({\fancyscript{C}}\).

For each monoid \(A\) and cofibrant \(A\)module \(N\), \( \wedge _A N\) takes weak equivalences of \(A\)modules to weak equivalences in \({\fancyscript{C}}\).
Proof
Proposition 4.14
Let \({\fancyscript{C}}\) be as in Proposition 4.13. Then the derived monoidal product on \({\fancyscript{C}}\) descends to an autonomous symmetric monoidal structure on \(h{\fancyscript{M}}\), making \(h{\fancyscript{M}}\) Eilenberg–Watts.
Proof
We now give our main application, in which \(k\) is taken to be a commutative d.g. algebra or commutative ring spectrum and \({\fancyscript{C}}\!=\! {\fancyscript{C}}_k\) is taken to be the category of \(k\)modules. One may work with any monoidal model category of spectra, but we must assume that \(k\) is cofibrant in \({\fancyscript{C}}_k\) and thus the result applies only to those categories in which the unit is cofibrant (e.g., symmetric spectra or orthogonal spectra). Likewise, if \(k\) is a d.g. algebra, one must choose a model structure on \({\fancyscript{C}}_k\) for which the unit is cofibrant (e.g., the injective model structure).
Proposition 4.15
Let \(k\) be a commutative d.g. algebra or commutative ring spectrum, and let \({\fancyscript{C}}_k\) be the symmetric monoidal category of \(k\)modules, with a model structure for which \(k\) is cofibrant. Then we have Eilenberg–Watts bicategories \({\fancyscript{M}}_k = {\fancyscript{M}}_{{\fancyscript{C}}_k}\) and \({\fancyscript{D}}_k = h{\fancyscript{M}}_{{\fancyscript{C}}_k}\).
Proof
4.2 Invertibility in triangulated bicategories
Definition 4.16
Let Open image in new window be a 1cell in \({\fancyscript{D}}(A,B)\).
Baker and Lazarev describe the following in the context of spectra, but their methods generalize to our setting. The key observation is that for any 1cell \(P\) whose source is \(A\), \({{\mathrm{\textit{s}Hom}}}(T,P)\) is target\(T\)local. Likewise, if \(P^{\prime }\) is any 1cell whose target is \(B\), \({{\mathrm{\textit{t}Hom}}}(T,P^{\prime })\) is source\(T\)local.
Proposition 4.17
Proposition 4.18
Proof
Corollary 4.19
Let Open image in new window be as in Proposition 4.18. Then \(T\) is targetfaithful (Definition 4.6) if and only if localization induces an equivalence between the category of 1cells \({\fancyscript{B}}(B,B)\) and the target\(T\)local subcategory. In this case each of the three adjoint pairs of Proposition 4.17 (at left) is an equivalence. We have a corresponding statement for the sourcefaithful case.
Proof
Definition 4.20
Combining the previous results yields a characterization of invertible objects in triangulated Eilenberg–Watts bicategories. Applying this to the case \(T = \lfloor A \rfloor \) immediately gives the characterization of Azumaya objects in Theorem 1.8.
Proposition 4.21
 i.
\(T\) is invertible.
 ii.
 a)
\(T\) is rightdualizable.
 b)
The unit induces \(A \cong {{\mathrm{\textit{t}Hom}}}(T,T)\).
 c)
\(B\) is strongly target\(T\)local.
 a)
 iii.
 a)
\(T\) is leftdualizable.
 b)
The unit induces \(B \cong {{\mathrm{\textit{s}Hom}}}(T,T)\).
 c)
\(A\) is strongly source\(T\)local.
 a)
Remark 4.22
4.3 Application to tilting theory
The work in this section allows us to give a unified proof of results from the tilting theory of [29] and [35]:
Proof of 1.12 and 1.13
Let \(\widetilde{T}\) denote \(T\) regarded as a bimodule over \(A = {{\mathrm{\textit{t}Hom}}}(T,T)\). Since \(T\) is rightdualizable, \(\widetilde{T}\) is rightdualizable in \({\fancyscript{D}}_k(A,B).\) Moreover, \(A \simeq {{\mathrm{\textit{t}Hom}}}(\widetilde{T},\widetilde{T}).\)
By Remark 4.5 and Proposition 4.15, \({\fancyscript{D}}_k\) is a triangulated Eilenberg–Watts bicategory. Since \(k\) is the unit of \({\fancyscript{D}}_k\), the 1cells of \({\fancyscript{D}}_k(k,B)\) are jointly targetfaithful (Definition 4.6). Since \(T\) generates \({\fancyscript{D}}_k(k,B),\) Lemma 4.8 shows that \(T\) is targetfaithful and this means that \(\widetilde{T}\) is also targetfaithful. The result then follows from Corollary 4.19 and Proposition 4.21. \(\square \)
5 Homotopical Brauer groups
This section describes homotopical Brauer groups for rings and ring spectra; these constitute our main applications of the preceding theory. We also give explicit comparisons between the Brauer group as characterized by Theorems 1.6 and 1.8 and as it appears in related work on Brauer groups in homotopical settings. We begin with the derived Brauer group of a ring and then address the Brauer groups of ring spectra.
5.1 The derived Brauer group of a ring
Definition 5.1
We refer to \(Br({\fancyscript{D}}_k)\) as the derived Brauer group of \(k\), to distinguish it from the generally different classical Brauer group, \(Br(k)\).
Toën [37] introduces the notion of derived Azumaya algebras (the two conditions appearing in Proposition 5.2), and describes a Brauer group formed by Eilenberg–Watts equivalence classes of such. We show that the notion of derived Azumaya algebra is equivalent to the notion of Azumaya object in the derived category, and therefore the resulting Brauer groups are isomorphic.
Proposition 5.2
 i.
The underlying \(k\)module of \(A\) is a compact generator of the triangulated category \({\fancyscript{D}}_k(k,k)\).
 ii.
The map \(\mu :A^e \rightarrow F_k(A,A)\) is an equivalence.
Proof
Toën [37, 2.8] shows that the conditions above are equivalent to the condition that \(A^e\) be Eilenberg–Watts equivalent to \(k\) in the bicategory \({\fancyscript{D}}_k\), and this is one of the conditions appearing in the characterization Theorem 1.6. \(\square \)
Combining this with [37, 2.12] we have the following:
Proposition 5.3
Let \(k\) be a field. Then \(Br({\fancyscript{D}}_k)\) is isomorphic to the classical Brauer group \(Br(k)\).
5.2 The Brauer group of a ring spectrum
When \(k\) is a commutative ring spectrum Baker, Richter and Szymik have introduced and studied the notion of topological Azumaya \(k\)algebra [6]. Their definition (the three conditions of Proposition 5.4 below) makes sense in any modern monoidal model category of spectra, and we show it is equivalent to the definition of Azumaya used here. Since the various modern categories of spectra are all strong monoidal Quillen equivalent [34], the results of homotopical Brauer theory transfer between any of them.
Recall that Open image in new window denotes \(A\) regarded as a right module over \(A^e\), and \({_k}A_k = \lfloor A \rfloor _k\) is the underlying \((k,k)\)module of \(A\).
Proposition 5.4
 i.
\(A\) is dualizable in the homotopy category of \(k\)modules.
 ii.
The map \(A^e \rightarrow F_k(A,A)\) is a weak equivalence of bimodules over \(A^e\).
 iii.
\(A\) is faithful as a \(k\)module.
Proof
Baker, Richter, and Szymik show that these conditions imply that \(\lfloor A \rfloor \) is central and separable over \(k\) [6, 1.3, 1.4]. They are working with \(S\)algebras, but those results apply equally well in any modern category of spectra. Now if \(\lfloor A \rfloor _k = \lfloor A \rfloor \odot A_k\) is faithful, then \(\lfloor A \rfloor \) must be targetfaithful (Definition 4.6). Remark 4.22 explains that under these circumstances targetfaithfulness of \(\lfloor A \rfloor \) is equivalent to \(A^e\) being strongly target\(\lfloor A \rfloor \)local, and thus \(A\) is Azumaya by Proposition 4.21.
For the converse, we again refer to Proposition 4.21. Lemma 3.4 shows that \(\lfloor A \rfloor \) being leftdualizable implies \(\lfloor A \rfloor _k\) is leftdualizable, and under these circumstances \(k\) being strongly source\(\lfloor A \rfloor \)local is equivalent to \(\lfloor A \rfloor \) being sourcefaithful (Remark 4.22). Since restriction of scalars along the unit map \(k \rightarrow A^e\) is faithful, the composite \(\lfloor A \rfloor _k\) must also be faithful. \(\square \)
One minor subtlety remains in the choice of equivalence relation with which we form a Brauer group. There are two standard choices, Eilenberg–Watts equivalence and Brauer equivalence. We finish by showing that the two resulting groups are isomorphic. This is a classical fact for discrete rings which holds for formal reasons.
Definition 5.5
Proposition 5.6
[6, 2.4] The collection of Azumaya \(k\)algebras modulo Brauer equivalence forms a group.
Lemma 5.7
The group of Brauer equivalence classes of Azumaya \(k\)algebras is isomorphic to the group of Eilenberg–Watts equivalence classes of such algebras.
Proof
For a dualizable \(k\)module \(M\), \(F_k(M,M)\) is always Eilenberg–Watts equivalent to \(k\). So if \(A_1\) is Brauer equivalent to \(A_2\) then it is Eilenberg–Watts equivalent to \(A_2\). Therefore Brauer equivalence implies Eilenberg–Watts equivalence and there is a surjective group homomorphism from Azumaya algebras modulo Brauer equivalence to Azumaya algebras modulo Eilenberg–Watts equivalence. But if \(A\) is Eilenberg–Watts equivalent to the ground ring, \(k\), then there is an invertible bimodule \(M\) giving the equivalence, and \(A \simeq F_k(M,M)\), so \(A\) is also Brauer equivalent to \(k\). Thus this homomorphism is an isomorphism. \(\square \)
Notes
Acknowledgments
This work developed from a portion of the author’s Ph.D. thesis. The author thanks his advisor, Peter May, for reading it many times in draft form and offering a number of suggestions both mathematical and stylistic. The author also thanks Michael Ching for his significant encouragement. The exposition has benefited from the helpful suggestions of anonymous referees, and from the audiences who have listened to talks and read drafts of this article. The terminology “Eilenberg–Watts equivalence” was suggested by Justin Noel during one of many helpful conversations.
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