Relative GAGA: Algebraic and Analytic Comparison

• Relative GAGA theorem is a categorical principle that equates analytic and algebraic coherent complexes under properness and coherence conditions.
• It employs descent techniques, derived flatness, and spectral sequences to ensure that analytification commutes with derived direct images.
• The theorem underpins key applications such as moduli compactifications, mapping stacks, and p-adic Hodge theory, extending classical GAGA to higher and non-archimedean settings.

The relative GAGA theorem is a categorical and cohomological comparison between algebraic and analytic geometry, generalized from Serre’s foundational results to higher and derived stacks, families, non-noetherian settings, and non-archimedean analytic geometry. It asserts, under properness and coherence hypotheses, that the passage from algebraic to analytic objects—via analytification—commutes with derived direct images and cohomology, establishing equivalences on categories of coherent or (pseudo-)coherent complexes and their higher direct images.

1. Definitions and Foundational Hypotheses

Let $X$ and $S$ be derived Deligne–Mumford or Artin stacks locally almost of finite presentation over a ground field $k$ (typically $\mathbb{C}$ or a non-archimedean field). Such $X$ are Zariski-locally presented as affine derived schemes $\Spec A$ with $A$ a simplicial $k$-algebra, $\pi_0 A$ finitely presented, $\pi_i A$ coherent $\pi_0 A$-modules. Coherent complexes on $X$ form a stable $\infty$-category $\Coh(X)$; $\Coh^-(X)$ denotes those bounded above in cohomological grading. The morphism $f\colon X \to S$ is proper if its truncation is proper in the classical sense.

Analytification is effected by a left adjoint functor

$( - )^{\an}\colon \{\text{derived algebraic stacks over $k$}\} \to \{\text{derived analytic stacks}\}$

which for underived schemes coincides with classical analytification and is derived-flat: for affine $A$, the unit map $\eta_A\colon A \to (A^\an)^{\alg}$ satisfies $\pi_0(\eta_A)$ classically flat and, for $i>0$, $\pi_i(A)\otimes_{\pi_0A}\pi_0((A^\an)^{\alg}) \cong \pi_i((A^\an)^{\alg})$ (Porta, 2015).

In the stack context, $n$-geometric stacks admit smooth atlases with diagonals $(n-1)$-representable, analytification extends to these, and the notion of properness via representability and properness of the truncated morphism persists for analytic stacks (Porta et al., 2014).

2. Statement of the Relative GAGA Theorem

Given a proper morphism $f\colon X \to S$ of (derived or higher) algebraic stacks locally of finite presentation, and their analytifications $f^\an\colon X^\an \to S^\an$, the theorem states:

For every $\mathcal{F}\in \Coh^-(X)$ (or, in higher stack settings, $\mathcal{F} \in \Coh^+(X)$), the natural base-change map

$(Rf_*\mathcal{F})^\an \to Rf^\an_*(\mathcal{F}^\an)$

is an equivalence in the derived $\infty$-category of $\mathcal{O}_{S^\an}$-modules (Porta, 2015, Porta et al., 2014). On cohomology, this specializes to

$H^i(X, \mathcal{F}) \cong H^i(X^\an, \mathcal{F}^\an)$

and, for $X$ proper over $k$, analytification induces equivalences

$\Coh(X) \xrightarrow{\sim} \Coh(X^{\an})$

for coherent or perfect complexes, including extensions to pseudo-coherent and almost-perfect complexes in condensed and non-archimedean settings (Wang, 18 Jan 2026, Hall, 2018).

3. Proof Methodology and Key Structural Tools

The proof leverages three principal components:

3.1 Descent and $\infty$-Categorical Formalism

Analytic and algebraic stacks are embedded via functor-of-points into hypercomplete $\infty$-topoi on sites of affine schemes or (analytic) Stein spaces, with descent for quasi-coherent and coherent complexes effected by stack-theoretic sheafification. The lisse–étale site and $\infty$-categorical adjunctions enable formulation of derived direct and inverse image functors as adjoints in presentable stable $\infty$-categories, ensuring functoriality and commutation with colimits (Porta et al., 2014).

3.2 Flatness and Reduction to Classical GAGA

The analytification morphism, being derived-flat, ensures that base-change holds universally from the classical to the derived and higher categorical setting. Compatibility of $f_*$ with analytification for algebraic generators (i.e., underived coherent sheaves) suffices to guarantee the corresponding property for all coherent complexes, due to exactness and closure under cones and filtered colimits (Porta, 2015).

3.3 Cohomological Descent and Spectral Sequences

A proper smooth surjection $P\to X$ is chosen, and its Čech nerve is applied to both algebraic and analytic sides. Spectral sequences

$$E_1^{p,q} = R^q f_*(\sigma^p_* \mathcal{F}) \implies R^{p+q} f_*\mathcal{F}$$

and their analytic analogues are compared term-by-term, ensuring the equivalence for derived direct images and hence for derived categories. This method is robust for both classical and higher stacks (Porta et al., 2014, Hall, 2018).

Key results include Grauert’s theorem for stacks—derived pushforward of bounded-below coherent complexes preserves coherence (Porta et al., 2014), with analogous results in the pseudo-coherent and perfectoid settings by use of condensed mathematics and solid tensor product formalism (Wang, 18 Jan 2026).

4. Extensions, Generalizations, and Formal Variants

The relative GAGA principle has been further extended along several axes:

• Non-noetherian settings: The comparison theorem holds for analytic or formal completions over general base rings, provided pseudo-coherent morphisms and descent data, as established in (Hall, 2018).
• Stacks and families: Functorial equivalences are established not only for spaces but for algebraic and analytic stacks, Hilbert stacks, and moduli stacks (e.g., modular compactifications of the moduli of curves), using the machinery of pseudosheaves and devissage (Hall, 2011).
• Non-archimedean geometry: In condensed mathematics, perfectoid and Fredholm analytic rings yield precise GAGA equivalences for perfect and pseudo-coherent complexes on base-changed spaces (Wang, 18 Jan 2026).
• Relative hypercohomology: The relative hyper-GAGA theorem asserts equivalence of relative hypercohomology groups for bounded complexes with supports or complements, further generalizing Serre’s classical result (Haibara et al., 2024).

5. Applications and Corollaries

Prominent applications include:

• Mapping stacks: Relative GAGA establishes the equivalence between the analytification of algebraic mapping stacks and intrinsic analytic mapping stacks, allowing analytic moduli problems (e.g., $G$-bundles, Higgs bundles, Simpson correspondence) to be treated via algebraic models (Wang, 18 Jan 2026).
• Moduli and compactification: Analytic modular compactifications of moduli spaces of curves are shown to algebraize, with a categorical equivalence of corresponding analytic and algebraic stacks (Hall, 2011).
• Comparison results in arithmetic geometry: The theorem underpins p-adic Hodge theory, comparison between algebraic and rigid/Berkovich fibers, and compactness results for analytic moduli spaces (Porta et al., 2014, Wang, 18 Jan 2026).
• Lefschetz and vanishing theorems: Relative GAGA methods yield relative Lefschetz results, dimension-1 full faithfulness, and Kodaira-type vanishing via support-theoretic arguments in derived categories (Hall, 2018).

6. Technical Lemmas and Further Directions

Key results supporting the theorem include:

• Flatness of Analytification (Lemma): The unit $\eta_A$ of the adjunction $A \to (A^\an)^{\alg}$ is flat in the derived sense (Porta, 2015).
• Pushforward Preserves Coherence (Proposition): For proper morphisms, derived pushforward of coherent complexes remains coherent (Porta et al., 2014).
• Base-Change for Locally Closed Inclusions: Analytification commutes with $i_!$, $i_*$ for locally closed inclusions, essential for relative hypercohomology (Haibara et al., 2024).

Further research directions include extensions to mixed-characteristic settings, formal and Berkovich–adic GAGA, and usage in almost mathematics frameworks (Hall, 2018). The relative GAGA framework supplies a unifying categorical mechanism for algebraic–analytic cohomology comparison, moduli theory, and derived deformation theory, generalizing classical geometry to higher and derived settings.