Derived Brauer Group in Algebraic Geometry

• Derived Brauer group is the extension of classical Brauer theory using derived Azumaya algebras, capturing twisting data beyond torsion classes.
• It employs categorical Morita theory with étale descent methods to represent cohomological classes in derived algebraic geometry.
• The framework uses Brauer spaces and descent spectral sequences to compute invariants across various settings, including derived schemes and spectral stacks.

The derived Brauer group is the Morita-theoretic extension of classical Brauer theory obtained by replacing ordinary Azumaya algebras with derived Azumaya algebras, or equivalently with invertible linear stable $\infty$-categories. In derived algebraic geometry it is used to encode twisting data beyond the strictly classical setting, including cohomological classes that need not be torsion. Its modern study combines étale descent, higher Artin geometry, and categorical Morita theory, and its behavior depends strongly on the ambient context: connective derived schemes, spectral Deligne–Mumford stacks, and strict differential graded models over a field lead to related but distinct theories (Antieau et al., 2012, Binda et al., 2021, Antieau et al., 2022, Zimmermann, 2023).

1. Definitions and scope

Following Toën’s framework as presented in later work, a derived Azumaya algebra on a derived scheme $X$ is an object $A\in \mathrm{Alg}(\mathrm{Perf}(X))$ such that the natural map

$$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$

is an equivalence and $A$ is a compact generator for $QCoh(X)$. Up to Morita equivalence, such objects define the derived Brauer group. For a quasi-compact, quasi-separated $X$, one has the homotopical description

$$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$

so every class in $H^2_{\mathrm{ét}}(X,\mathbb G_m)$, not only torsion classes, is represented by a possibly derived Azumaya algebra (Binda et al., 2021).

Antieau and Gepner use a closely related formulation for a connective $E_\infty$-ring or a qcqs derived scheme: an Azumaya $X$0-algebra is an $X$1-algebra $X$2 that is a compact generator of $X$3 and satisfies

$X$4

Its Brauer group is the set of Morita-equivalence classes of such algebras. In that setting, for qcqs derived schemes $X$5, they prove

$X$6

so every torsion cohomological Brauer class is represented by a derived Azumaya algebra (Antieau et al., 2012).

This suggests that the literature uses related but not identical conventions: some authors reserve “Brauer group” for the torsion Azumaya-representable part, while others use “derived Brauer group” for the fuller object detected by $X$7.

Notion	Defining objects	Characteristic feature
Classical Brauer group	Morita classes of Azumaya algebras	Classically tied to $X$8
Derived Brauer group $X$9	Morita classes of derived Azumaya algebras	For qcqs $A\in \mathrm{Alg}(\mathrm{Perf}(X))$0: $A\in \mathrm{Alg}(\mathrm{Perf}(X))$1
Local Brauer group $A\in \mathrm{Alg}(\mathrm{Perf}(X))$2	Étale-locally trivial spectral Azumaya classes	Proper subgroup of $A\in \mathrm{Alg}(\mathrm{Perf}(X))$3 in general spectral settings
dg Brauer group $A\in \mathrm{Alg}(\mathrm{Perf}(X))$4	Central simple dg $A\in \mathrm{Alg}(\mathrm{Perf}(X))$5-algebras modulo stabilization by dg endomorphism algebras	Over a field: $A\in \mathrm{Alg}(\mathrm{Perf}(X))$6

2. Morita theory, étale local triviality, and representability

A fundamental structural result is that derived Azumaya algebras are Morita-theoretic objects: $A\in \mathrm{Alg}(\mathrm{Perf}(X))$7 is Azumaya if and only if $A\in \mathrm{Alg}(\mathrm{Perf}(X))$8 is invertible in the symmetric monoidal $A\in \mathrm{Alg}(\mathrm{Perf}(X))$9-category of compactly generated $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$0-linear categories (Antieau et al., 2012). This recasts Brauer theory in categorical terms and makes invertibility, rather than presentation by a literal algebra object, the primary invariant.

For connective commutative ring spectra, Antieau and Gepner prove étale local triviality: if $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$1 is an Azumaya algebra over a connective commutative ring spectrum $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$2, then there exists a faithfully flat étale $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$3-algebra $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$4 such that $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$5 is Morita equivalent to $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$6. In other words, Azumaya algebras are étale locally trivial in this derived setting (Antieau et al., 2012). This is one of the main mechanisms allowing cohomological classes to be realized by geometric or categorical objects.

The proof uses geometricity statements for moduli of compact objects. If $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$7 is a stable $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$8-category of finite type, then the moduli space $$A\otimes_{\mathcal O_X}A^{\mathrm{op}}\to \operatorname{End}_{\mathcal O_X}(A)$$9 of compact objects is locally geometric, and the sheaf of Morita equivalences from $A$0 to $A$1 is smooth and surjective over $A$2 (Antieau et al., 2012). These higher-geometric properties provide étale local sections and thereby connect categorical invertibility to descent-theoretic triviality.

A second structural ingredient is the local-to-global principle for compact generators. If an $A$3-linear category with descent acquires a compact generator after an étale cover $A$4, then it already has a compact generator over $A$5 (Antieau et al., 2012). Combined with étale local triviality, this yields a derived solution to Grothendieck’s $A$6 problem for qcqs derived schemes.

3. Brauer spaces and computational machinery

The theory naturally upgrades from a set of equivalence classes to a space, and even a spectrum: the Brauer space $A$7. Antieau and Gepner describe a descent spectral sequence

$A$8

which serves as a principal computational tool (Antieau et al., 2012).

The homotopy sheaves of the Brauer space are given by

$A$9

For a connective $QCoh(X)$0-ring $QCoh(X)$1, this yields explicit formulas: $QCoh(X)$2 These formulas make the passage between étale cohomology and categorical Brauer data explicit (Antieau et al., 2012).

Several computations follow. The Brauer group of the sphere spectrum $QCoh(X)$3 vanishes, $QCoh(X)$4, using $QCoh(X)$5 and $QCoh(X)$6 (Antieau et al., 2012). The same paper also records vanishing for $QCoh(X)$7, $QCoh(X)$8, $QCoh(X)$9, and $X$0, and gives a nontrivial example

$X$1

The vanishing of $X$2 has categorical consequences: all Azumaya algebras over the sphere spectrum are Morita equivalent to $X$3, and this is used to prove uniqueness results for the stable homotopy category (Antieau et al., 2012).

4. Spectral Deligne–Mumford stacks and the local Brauer group

In spectral algebraic geometry, especially for nonconnective or periodic objects, the relation between Brauer classes and étale local triviality becomes subtler. For an $X$4-ring spectrum $X$5, the Brauer group $X$6 is defined through Morita equivalence classes of Azumaya $X$7-algebras in the language of stable $X$8-categories. Because not every spectral Azumaya algebra is étale-locally trivial, one isolates the local Brauer group $X$9, the subgroup of classes trivialized by some faithful étale extension $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$0 (Antieau et al., 2022).

For a spectral Deligne–Mumford stack $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$1, one similarly has $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$2 and a cohomological Brauer group $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$3, defined as $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$4 of the global sections of the Brauer sheaf. These need not coincide in general. Under suitable hypotheses—existence of a Zariski cover with generation conditions on overlaps and global perfect generators for the relevant twisted categories—every cohomological Brauer class is representable by an Azumaya algebra, and one has

$$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$5

(Antieau et al., 2022).

A particularly detailed computation concerns the derived moduli stack of elliptic curves $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$6 and topological modular forms $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$7. The stack $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$8 is a spectral Deligne–Mumford stack and is $$\mathrm{dBr}(X)\simeq \pi_0 \operatorname{Map}_{dSt}\!\bigl(X,(B^2\mathbb G_m)\times (B\mathbb Z)\bigr) \simeq H^2_{\mathrm{ét}}(X,\mathbb G_m)\times H^1_{\mathrm{ét}}(X,\mathbb Z),$$9-affine: $H^2_{\mathrm{ét}}(X,\mathbb G_m)$0 Consequently,

$H^2_{\mathrm{ét}}(X,\mathbb G_m)$1

(Antieau et al., 2022).

The local Brauer group is computed by a descent spectral sequence

$H^2_{\mathrm{ét}}(X,\mathbb G_m)$2

together with the exact sequence

$H^2_{\mathrm{ét}}(X,\mathbb G_m)$3

for a periodic ring spectrum of period $H^2_{\mathrm{ét}}(X,\mathbb G_m)$4 (Antieau et al., 2022).

For $H^2_{\mathrm{ét}}(X,\mathbb G_m)$5 and $H^2_{\mathrm{ét}}(X,\mathbb G_m)$6, the resulting local Brauer groups are torsion groups. Their $H^2_{\mathrm{ét}}(X,\mathbb G_m)$7-primary part is $H^2_{\mathrm{ét}}(X,\mathbb G_m)$8, they have no $H^2_{\mathrm{ét}}(X,\mathbb G_m)$9-torsion for $E_\infty$0, and their $E_\infty$1-primary part surjects onto $E_\infty$2 with kernel of order at most $E_\infty$3. Moreover, the natural map

$E_\infty$4

is injective with finite cokernel and becomes an isomorphism after inverting $E_\infty$5 (Antieau et al., 2022). The infinite $E_\infty$6-torsion is traced to kernels and cokernels of differentials in the sheafified Picard spectral sequence.

5. Gluing, completion, and formal GAGA

Derived Brauer theory has a strong formal and descent-theoretic component. A Beauville–Laszlo-type theorem for quasi-coherent sheaves of categories implies a corresponding statement for derived Azumaya algebras. Given a derived pullback square with $E_\infty$7, $E_\infty$8, $E_\infty$9, and $X$00 inducing an equivalence on $X$01-adic completions, one has

$X$02

and on invertible objects

$X$03

(Binda et al., 2021).

Restricting to units, derived Picard groups, and derived Brauer groups gives a long exact sequence of Mayer–Vietoris type: $X$04 This is a genuine gluing statement for derived twisted objects, not merely for perfect complexes (Binda et al., 2021).

Formal geometry provides another major application. If $X$05 is proper over $X$06 with $X$07 noetherian and complete along an ideal $X$08, and if $X$09 denotes the formal completion obtained from the thickenings $X$10, then the restriction on derived Azumaya objects is fully faithful, and passing to connected components yields an injective map

$X$11

This implies injectivity of

$X$12

and, under a $X$13-vanishing hypothesis on Picard groups, of

$X$14

(Binda et al., 2021). A Henselian variant gives injectivity for $X$15 under regular geometric fiber hypotheses without the same $X$16 restriction.

The same formalism yields Grothendieck existence for twisted sheaves on arbitrary $X$17-gerbes: $X$18 is an equivalence for any $X$19-gerbe $X$20 on a proper scheme over a noetherian complete base (Binda et al., 2021). This removes the classical dependence on the resolution property.

6. Limits, contrasts, and related generalizations

One recurrent misconception is that Brauer-theoretic data should be preserved by derived equivalence of ordinary algebraic varieties. For Calabi–Yau $X$21-folds this is false. There exist smooth projective derived-equivalent Calabi–Yau $X$22-folds $X$23 and $X$24 with distinct Brauer groups, $X$25 (Addington, 2013). The mechanism is topological: for any Calabi–Yau $X$26-fold,

$X$27

so derived equivalence preserves the torsion in topological $X$28-theory, not the Brauer group by itself. Consequently,

$X$29

In the Gross–Popescu/Bak–Schnell example, $X$30 is simply connected with $X$31, whereas the derived-equivalent dual fibration $X$32 has $X$33 and $X$34 (Addington, 2013). By contrast, for K3 surfaces the Brauer group is a derived invariant.

At the opposite end of the spectrum, strict dg models over a field can be too rigid to produce genuinely new Brauer data. Zimmermann defines the dg Brauer group $X$35 using central simple dg $X$36-algebras, with equivalence generated by stabilization: $X$37 for bounded complexes of finite-dimensional $X$38-vector spaces $X$39 and $X$40. The equivalence classes form an abelian group under dg tensor product, with inverse given by the opposite dg algebra, and the main theorem is

$X$41

(Zimmermann, 2023). Over a field, any dg structure on a central simple algebra is therefore trivialized up to this Morita-type equivalence. The same paper notes that homology does not descend well to $X$42, since $X$43 need not equal $X$44 (Zimmermann, 2023). This sharp contrast with the broader derived-Azumaya theory indicates that “derived Brauer group” is sensitive to the chosen model.

A separate line of generalization, related in spirit but not identical in definition, is Sakagaito’s motivic extension via Bloch’s cycle complex. For an equi-dimensional scheme $X$45, the generalized Brauer group is defined by

$X$46

recovering the usual Brauer group in regular situations and supporting Gersten-type exact sequences in several low-dimensional mixed-characteristic cases (Sakagaito, 2015). This is a motivic enlargement rather than a derived-Azumaya one, but it belongs to the same broader effort to reinterpret Brauer theory in higher and more flexible cohomological frameworks.