Cohomological Brauer Group Overview

• Cohomological Brauer Group is a torsion subgroup of H²ₑₜ(X, Gₘ) that captures obstructions to forming Azumaya algebras and projective bundles.
• It is computed using spectral sequences, Kummer theory, and derived algebraic geometry techniques, enabling explicit classifications in various settings.
• Its applications span arithmetic, moduli spaces, and topology, offering concrete insights for resolving obstructions in geometric and stack-theoretic contexts.

The cohomological Brauer group is a fundamental invariant in algebraic geometry, topology, and related areas, encoding subtle obstructions to the existence of Azumaya algebras, projective bundles, or central simple algebras, and interfacing with both arithmetic and geometric structure. It provides a torsion class in étale (or analytic, or topological) cohomology which governs the classification of certain types of bundles, gerbes, and derived categories. The equality of the "classical" Brauer group (Azumaya algebras up to Morita equivalence) with the cohomological Brauer group (torsion in degree-2 étale cohomology with coefficients in the sheaf of invertible functions) is an intricate problem whose solution structure depends notably on the ambient category (schemes, stacks, topological spaces, and derived or spectral settings).

1. Fundamental Definition and Context

Let $X$ be a locally ringed space (typically a scheme, Deligne–Mumford stack, or topological space). The cohomological Brauer group $\mathrm{Br}'(X)$ is defined to be the torsion subgroup of the second étale cohomology group with coefficients in the sheaf of units: $$\mathrm{Br}'(X) := H^2_{\mathrm{\acute{e}t}}(X, \mathbb{G}_m)_{\mathrm{tors}}$$ For smooth quasi-projective varieties over an algebraically closed field, $\mathrm{Br}'(X)$ agrees with the group of Morita-equivalence classes of Azumaya algebras (the "classical Brauer group" $\mathrm{Br}(X)$) via Gabber's theorem, but in the presence of singularities, stacks, or in the topological or derived settings, this relationship must be clarified using finer tools (Antieau et al., 2012, Chough, 2020). In the topological setting, an analogous definition is made using sheaf cohomology; for a CW-complex $X$, $\mathrm{Br}'(X) = \mathrm{Tors}(H^3(X,\mathbb{Z}))$ (Hornbostel et al., 2013).

2. Cohomological Brauer Group: Spectral and Derived Settings

In derived and spectral algebraic geometry (where the building blocks are connective $E_\infty$-rings and spectral stacks), the cohomological Brauer group is generalized to the setting where $X$ is a quasi-geometric spectral algebraic stack. The torsion in $H^2_{\mathrm{\acute{e}t}}(X,\mathbb{G}_m)$, computed in the fpqc or étale topology, is shown to classify Morita-classes of Azumaya algebras in these rich contexts. Explicitly, if $X$ admits a quasi-finite presentation and has quasi-affine diagonal, then the natural map

$$\mathrm{Az}(X)/\!\!\sim \longrightarrow \mathrm{Br}'(X)$$

is always an isomorphism, and any cohomological class is realized by a spectral Azumaya algebra (Chough, 2020, Antieau et al., 2012). This extends the classical result to spectral and derived stacks and underpins many modern results in higher and derived algebraic geometry.

3. Cohomological Invariants, Root Stacks, and Moduli

For algebraic stacks and moduli spaces, the cohomological Brauer group encodes rich arithmetic information and tracks invariants under stacky modifications. If $X$ is a smooth scheme over a base field $k$ and $V\subset X$ a Cartier divisor, the $r$-th root stack $X(V,r)$ produces new cohomological invariants governed by the ramification along $V$. There exists a short exact sequence (Lorenzo et al., 2020): $$0 \to \operatorname{Inv}^{*}(X,M) \to \operatorname{Inv}^{*}(X(V,r), M) \to \ker(\partial : A^*(V,M) \to A^{*+1}(X,M))[r] \to 0$$ where $M$ is a cycle module of exponent dividing $r$, $A^*$ denotes Chow groups with coefficients, and $\partial$ is the localization boundary. For $\ell$-torsion coefficient modules, this identifies $\operatorname{Br}'(X(V,r))[\ell^\infty]$ with a controlled enlargement of $\operatorname{Br}'(X)[\ell^\infty]$. Applications include explicit calculations for moduli stacks of admissible double covers (e.g., $$\mathrm{Br}(\mathcal{H}_g^{\mathrm{adm}}) \cong \mathrm{Br}(\mathcal{H}_g) \cong \mathrm{Br}(k) \oplus H^1(k, \mathbb{Z}/2) \oplus \mathbb{Z}/2$$ in characteristic $0$) and the tracking of invariants under stacky blow-ups (Lorenzo et al., 2020).

4. Moduli and Homogeneous Spaces: Finiteness, Vanishing, and Explicit Calculations

In the context of moduli spaces (curves, bundles, parabolic structures, etc.), the structure of $\mathrm{Br}'(X)$ ranges from explicit cyclic groups to full vanishing, depending on the detailed geometry:

• For moduli spaces of stable parabolic bundles on curves, $\mathrm{Br}'(\mathcal{P}\mathcal{M}_s^\alpha)$ is cyclic of order $m = \gcd(d, n, \{r_{i,j}\})$, generated by the universal projective bundle; a universal vector bundle exists if and only if $m=1$ (Biswas et al., 2010).
• For Hilbert schemes and punctual Quot schemes on smooth surfaces, formation of these moduli spaces typically does not create new Brauer classes: $\mathrm{Br}'(X^{[d]}) \cong \mathrm{Br}'(X^{(d)})$ and for $Q(r,d)\to X^{[d]}$, the pullback on Brauer groups is an isomorphism (Parameswaran et al., 2019).
• For moduli of stable curves, recent results have established that $\mathrm{Br}'(\overline{M}_{g,n})$ vanishes for $g\geq 2$ (full range), and for $g=1,1\leq n\leq 6$, and all $g\geq 4, n\geq 0$ (Bartling et al., 2024). Finiteness is established for proper, smooth Deligne–Mumford stacks over $\operatorname{Spec}\mathbb{Z}$.

In the setting of homogeneous spaces $V=G\backslash G'$, the unramified algebraic Brauer group is computed in terms of Galois cohomology of a finite abelian group associated to the stabilizer and the surjectivity of restriction maps. Explicit duality and norm conditions identify elements of $\mathrm{Br}_{\mathrm{nr},\mathrm{alg}}(V)$, reducing the calculation to group-theoretic input (Arteche, 2013).

5. Generalizations, Gersten Conjecture, and Motivic Interpretation

Recent work has reframed the cohomological Brauer group as a special case of étale motivic cohomology, particularly: $$\mathrm{Br}(X) = H^2_{\mathrm{\acute{e}t}}(X,\mathbb{G}_m) = H^{d+1}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}(d))_{\mathrm{tors}}$$ for smooth $d$-dimensional schemes over a Dedekind base, via Bloch's cycle complex (Sakagaito, 2015, Sakagaito, 2017). This perspective enables:

• A generalized Brauer group $$\mathrm{H}_B^n(X) := H^0_{\mathrm{Zar}}(X, R^{n+1}\epsilon_* \mathbb{Z}(n-1)_{\mathrm{\acute{e}t}})$$, which recovers classical and higher analogues and aligns with Gersten-type conjectures, purity, and rigidity in mixed characteristic contexts.
• Explicit isomorphisms $$\mathrm{H}_B^n(R) \cong H^{n+1}_{\mathrm{\acute{e}t}}(R, \mathbb{Z}(n-1))$$ for local rings $R$ of smooth $A$-algebras over a mixed characteristic Dedekind domain $A$ (Sakagaito, 2017).
• Purity and Gersten-exact sequences for $\mathrm{Br}'(X)$ in regular and henselian contexts.

6. The Cohomological Brauer Group in Topology

In the topological setting, the Brauer group may be defined for topological spaces $X$ in terms of Morita-equivalence classes of Azumaya algebras (or numerable projective bundle classes), with the cohomological Brauer group $\mathrm{Br}'(X)$ given by torsion in $H^3(X,\mathbb{Z})$ (Hornbostel et al., 2013). For finite or compact CW-complexes, $\mathrm{Br}(X) = \mathrm{Br}'(X)$ by Serre’s theorem; however, there exist spaces (e.g., $X=K(\mathbb{Z}/n,2)$) for which $\mathrm{Br}'(X)$ is nonzero but $\mathrm{Br}(X) = 0$, emphasizing that not every cohomological class admits an Azumaya algebra representing it in the topological context.

7. Computational Techniques and Exact Sequences

The computation of $\mathrm{Br}'(X)$ leverages spectral sequences, Kummer theory, and hypercohomological tools:

• Hochschild–Serre spectral sequence: For $X/k$, $$E_2^{p,q} = H^p(G_k, H^q_{\mathrm{\acute{e}t}}(X_{\bar{k}},\mathbb{G}_m)) \Rightarrow H^{p+q}_{\mathrm{\acute{e}t}}(X,\mathbb{G}_m)$$ allows reduction to geometric invariants and Galois cohomology (Gvirtz-Chen et al., 2019).
• Kummer exact sequences: For $n$ invertible on $X$, the étale sequence $1\to\mu_n\to \mathbb{G}_m\to \mathbb{G}_m\to 1$ gives a handle on $\ell$-primary torsion and the relationship between Néron–Severi and Brauer groups (Parameswaran et al., 2019).
• Brauer group of a product: The relative units–Picard complex enables a five-term exact sequence relating $\mathrm{Br}'(X)$, $\mathrm{Br}'(Y)$, $\mathrm{Br}'(X\times_S Y)$ and relative invariants, enabling explicit calculations in geometric situations with controlled Picard and Néron–Severi behavior (Gonzalez-Aviles, 2016).
• Leray spectral sequence for gerbes: For a $\mathbb{G}_m$-gerbe $\mathcal{G}\to S$ classified by a torsion class in $\mathrm{Br}'(S)$, there is a canonical isomorphism $$\mathrm{Br}'(\mathcal{G})\cong \mathrm{Br}'(S)/\langle [\mathcal{G}] \rangle$$ (Shin, 2018).

In the topological and equivariant analytic setting, dimensionally reduced Čech cohomology and Gysin-type sequences provide tools to classify equivariant Brauer classes and capture their invariants (Bouwknegt et al., 2011).

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In summary, the cohomological Brauer group is a deep and unifying object, controlling the classification of Azumaya algebras, measuring ramification, tracking stacky modifications, and encoding both arithmetic and geometric obstructions. Its equality with the algebraic Brauer group is broadly settled in regular, quasi-compact, both ordinary and higher-categorical contexts, but topology and stack-theoretic settings pose exceptional behaviors. Through a confluence of spectral sequence arguments, cohomological methods, and motivic interpretations, explicit calculations are possible for a wide diversity of moduli and geometric objects, as reflected in the broad array of recent and classical results referenced here.