Noncommutative GAGA

• Noncommutative GAGA is a framework that generalizes Serre’s GAGA by replacing classical coherent sheaves with noncommutative, Morita-theoretic analogues using stable ∞-categories.
• It employs derived Morita theory and Beauville–Laszlo gluing to establish descent, existence, and equivalence theorems for smooth–proper objects and twisted sheaves.
• The framework reveals new insights in the Brauer group by integrating non-torsion classes and proving injectivity even under minimal regularity or flatness assumptions.

Noncommutative GAGA refers to the extension of Serre’s geometric “GAGA” (an acronym for Géometrie Algébrique et Géométrie Analytique) theorems from the classical setting of coherent sheaves on algebraic schemes to the context of noncommutative algebraic geometry, formulated in terms of stable $\infty$-categories, derived Azumaya algebras, and twisted sheaves on gerbes. The framework replaces commutative geometric objects with their categorical or Morita-theoretic analogues and establishes gluing, descent, and existence theorems for these sophisticated invariants, significantly generalizing classical results and capturing new phenomena in both torsion and non-torsion Brauer classes (Binda et al., 2021).

1. Derived Morita Theory and the Derived Brauer Group

Noncommutative GAGA exploits the language of derived Morita theory, where presentable $\infty$-categories and compactly generated variants play the role of generalized module categories. For a quasi-compact, quasi-separated derived scheme $X$, $QCoh(X)$ inhabits $\Cat_\infty^{\rm cg}$ and is rigid, with perfect complexes as dualizable objects.

Given this, one defines $r_X = \Mod_{QCoh(X)}(\Cat_\infty^{\rm L})$, and similarly constructs full subcategories of $QCoh(X)$-linear, compactly generated, stable $\infty$-categories ($\_X$). These satisfy étale and Zariski descent properties on qcqs schemes.

Smooth and proper objects in $\_X$—collectively $SmPr(X)$—are dualizable in the $QCoh(X)$-linear structure, while invertible objects admit monoidal inverses. Derived Azumaya algebras are perfect $A \in \Alg(QCoh(X))$ that are étale-locally Morita-equivalent to the trivial algebra and generate $QCoh(X)$ via the equivalence $A \otimes_{QCoh(X)} A^{\rm op} \simeq \RHom_{QCoh(X)}(A,A)$. The resulting maximal $\infty$-groupoid of invertible $A$-modules defines the derived Brauer group: $dBr(X) := \pi_0\,dAz(X) \cong H^2_{\et}(X,\G_m)$ by a theorem of Toën–Lurie.

2. Beauville–Laszlo Gluing for Categories and Derived Azumaya Algebras

The noncommutative generalization of Beauville–Laszlo’s gluing theorem operates at the level of stable $\infty$-categories. Given a pullback square of affine derived schemes with $U$ the complement of an ideal in $S = \Spec A$ and $T$ inducing an isomorphism on $I$-adic completions, the restriction functor

$\Phi: \_S \longrightarrow \_T \times_{\_V} \_U$

is a symmetric-monoidal equivalence. This result is robust for smooth–proper and invertible objects, yielding

$SmPr(S) \simeq SmPr(T) \times_{SmPr(V)} SmPr(U)$

$dAz(S) \simeq dAz(T) \times_{dAz(V)} dAz(U)$

and consequently a Mayer–Vietoris long exact sequence for units, Picard, and Brauer groups. The proof involves constructing a right adjoint via fiber-product gluing and verifying dualizability, with semiorthogonal decompositions and the generators-trick ensuring preservation of compact generation.

3. Formal GAGA and Injectivity of Derived Brauer Groups

Let $R$ be a Noetherian ring complete along an ideal $I$, and $p: X \to S$ a proper derived scheme; define $X_n = X \times_S S_n$ and $\fX = \colim X_n$. The categorical properness assumption,

$Perf(X) \simeq \lim_n Perf(X_n)$

permits extension to $\Cat_\infty^{\rm cg}$: $QCoh(X) \simeq \lim_n QCoh(X_n)$ for compactly generated categories. As $SmPr(X)$ consists of dualizable objects, tensoring yields full faithfulness of the base-change functor, specifically for invertible objects,

$dAz^{\rm cat}(X) \to dAz^{\rm cat}(\fX)$

with $\pi_0$ injectivity: $dBr(X) \to dBr(\fX)$ and for cohomology,

$H^2_{\et}(X,\G_m) \to H^2_{\et}(\fX,\G_m)$

provided $\lim^1_n\Pic(X_n)=0$. In the case $R$ is Noetherian and Henselian and fibers are regular, injectivity persists absent completeness. Dimension $\le1$ ensures $\lim^1\Pic(X_n)=0$, reproducing Grothendieck’s formal-injectivity result for Brauer groups, generalized beyond torsion classes with no regularity or flatness assumptions.

4. Twisted $\infty$-Categories and Grothendieck Existence

Let $(\A,\alpha)$ be a $\G_m$-gerbe on $X$ with $\pi: \A \to X$ and canonical banding $\alpha$. Weight-$d$ subcategories $QCoh_d(\A)\subset QCoh(\A)$ are defined via homotopy idempotent projectors from the inertia action and are invertible in $\_X$. The decomposition

$\prod_{d \in \Z} QCoh_d(\A) \xrightarrow{\simeq} QCoh(\A)$

is canonical.

For geometric, categorically proper $X$, the natural functor

$Perf(\A) \longrightarrow \lim_n Perf(\A_n)$

is a symmetric-monoidal equivalence, establishing that the completion functor on twisted $Perf$-categories is fully faithful and essentially surjective. This yields a Grothendieck–existence theorem for the stable $\infty$-category of $\alpha$–twisted sheaves,

$Perf(\A) \simeq \lim_n Perf(\A_n)$

even for gerbes lacking the resolution property, i.e., not global quotients. This precedes previous results in generality.

5. Relation to Classical GAGA

Serre’s classical GAGA asserts equivalence of the analytification functor on perfect or coherent complexes for proper schemes over $\Spec \C$: $\Coh(X) \xrightarrow{\simeq} \Coh(X^{\rm an})$ with $X^{\rm an} = \colim_n X_n$ a formal colimit over thickenings when $X$ is over a complete local ring. Noncommutative GAGA recasts this, replacing $\Coh(X)$ by $SmPr(X)$ or $Perf(\A)$, and formal (analytic) completion by the system of thickenings $X_n \to X$. Categorical properness generalizes “proper and flat” assumptions, with full equivalences or injectivity analogously extended to higher-categorical invariants.

New features include:

• Consideration of stable $\infty$-categories of quasi-coherent or twisted objects, not only coherent sheaves;
• Avoidance of the resolution property or Tannakian reconstruction, allowing arbitrary gerbes (not just global quotients);
• Incorporation of non-torsion classes in the Brauer group via derived Azumaya algebras.

6. Consequences and Scope

This noncommutative GAGA framework establishes gluing and existence results for categories of smooth/proper and twisted sheaves, expanding well beyond classical GAGA and the torsion Brauer group. It provides new injectivity and descent statements for derived Brauer groups in the absence of standard regularity or flatness conditions and in settings far more general than those permitted by previous injectivity and descent theorems (Binda et al., 2021). The methodologies further facilitate comparison between classical and derived categorical approaches and open broad new vistas in noncommutative algebraic geometry.