Raynaud's Theory of Formal Models

• Raynaud's Theory of Formal Models is a framework in nonarchimedean analytic geometry that identifies analytic spaces with the generic fibers of formal schemes.
• It explains how admissible and normalized formal blow-ups preserve analytic equivalence and integral structures in both classical and relative settings.
• The theory extends to uniform qcqs adic spaces over Tate affinoid bases, offering a robust method for constructing analytic spaces from formal data.

Searching arXiv for relevant papers on Raynaud's theory of formal models and related extensions. Raynaud’s theory of formal models is the framework in nonarchimedean analytic geometry that identifies analytic spaces with generic fibers of formal schemes and interprets analytic equivalence as birational equivalence on the formal side. In its classical form, it concerns quasi-compact, quasi-separated rigid-analytic spaces over a nonarchimedean field and admissible formal schemes over the valuation ring; the analytic category is obtained by localizing the formal category at admissible formal blow-ups, which do not change the generic fiber. In the relative and non-Noetherian setting, this principle has been extended from rigid spaces over a field to uniform qcqs adic spaces over an arbitrary Tate affinoid base by replacing admissible formal blow-ups with normalized formal blow-ups, whose role is to preserve integrality and uniformity in the absence of finite-type hypotheses (Dine, 15 Jul 2025).

1. Classical Raynaud localization

Raynaud’s original insight is that rigid-analytic geometry over a nonarchimedean field $K$ can be reconstructed from a birational category of formal schemes over the valuation ring $\mathcal{O}_K$. The formal objects are admissible formal $\mathcal{O}_K$-schemes, meaning adic formal schemes locally of the form $\operatorname{Spf}(A)$ where $A$ is a $\pi$-adically complete $\mathcal{O}_K$-algebra topologically of finite type and $\pi$ generates an ideal of definition. Their generic fibers are rigid spaces $\mathfrak{X}_\eta$, with affine generic fiber given by the Huber pair $(A[\pi^{-1}], A^+)$, where $\mathcal{O}_K$0 is taken as the integral closure of $\mathcal{O}_K$1 in $\mathcal{O}_K$2 for precise functoriality and sheafiness (Dine, 15 Jul 2025).

In this setting, admissible formal blow-ups are blow-ups along admissible ideals followed by completion along the special fiber. If $\mathcal{O}_K$3 is generated by $\mathcal{O}_K$4, the blow-up is obtained from

$\mathcal{O}_K$5

with local charts

$\mathcal{O}_K$6

The key property is that these blow-ups are invisible to the generic fiber: $\mathcal{O}_K$7 This is the mechanism behind Raynaud’s localization theorem (Dine, 15 Jul 2025).

The classical theorem states an equivalence

$\mathcal{O}_K$8

with generic fiber as the localization functor. The theorem is fully faithful and essentially surjective: rigid spaces admit formal models, morphisms descend after admissible blow-up of the source, and any two formal models become isomorphic after admissible blow-ups (Dine, 15 Jul 2025).

A closely related exposition appears in the study of universal compactifications of rigid spaces, where Raynaud’s theory is used as the formal-geometric input for constructing Huber’s compactification by inverse limits of compactified special fibers (Kobak, 2023). That work stays in the classical tft world but illustrates how Raynaud’s framework serves as a bridge between rigid spaces, formal models, and valuation-theoretic compactification.

2. Formal schemes, generic fibers, and specialization

The generic fiber functor is the central construction of the theory. For an affine formal model $\mathcal{O}_K$9, the generic fiber passes from the $\mathcal{O}_K$0-adic ring $\mathcal{O}_K$1 to the analytic ring $\mathcal{O}_K$2, retaining an integral structure through $\mathcal{O}_K$3. This produces the rigid or adic analytic space associated with the formal model (Dine, 15 Jul 2025).

In the more general relative framework over a base $\mathcal{O}_K$4, one considers adic formal $\mathcal{O}_K$5-schemes that are locally rig-sheafy. For affine $\mathcal{O}_K$6, after choosing generators $\mathcal{O}_K$7 of an ideal of definition $\mathcal{O}_K$8, one defines

$\mathcal{O}_K$9

and rational opens

$\operatorname{Spf}(A)$0

The analytic generic fiber is then

$\operatorname{Spf}(A)$1

This glues on affine opens and is functorial in $\operatorname{Spf}(A)$2 (Dine, 15 Jul 2025).

Associated to the generic fiber is the specialization map

$\operatorname{Spf}(A)$3

characterized on affine opens by

$\operatorname{Spf}(A)$4

It is a morphism of locally $\operatorname{Spf}(A)$5-ringed spaces compatible with valuations on stalks, and it has strong topological properties: continuity, surjectivity under mild hypotheses, closedness, and spectral and generalizing behavior (Dine, 15 Jul 2025).

The same specialization paradigm underlies compactification theory. In the compactification construction for qc separated rigid spaces, a point of the universal compactification is analyzed via its minimal vertical generalization in the interior and via the valuation ring on its residue field; this valuation determines compatible centers on compactified special fibers of formal models (Kobak, 2023). This shows that Raynaud’s specialization formalism is not only local-model theoretic but also global and valuative.

3. Adic spaces and the relative extension

Huber’s adic spaces enlarge the rigid category to include non-Noetherian and infinite-dimensional analytic phenomena. In the relative theory, one fixes a complete adic ring $\operatorname{Spf}(A)$6 with ideal of definition generated by a single non-zero-divisor, non-unit $\operatorname{Spf}(A)$7, assumes $\operatorname{Spf}(A)$8 is sheafy, and sets

$\operatorname{Spf}(A)$9

where $A$0 is the integral closure of $A$1 in $A$2. One then studies analytic adic spaces over $A$3 (Dine, 15 Jul 2025).

Uniformity is fundamental in this setting. A Tate ring $A$4 is uniform if $A$5 is bounded, and stably uniform if every rational localization is uniform. On the analytic side, the relevant class is uniform qcqs adic spaces; on the formal side, the corresponding condition is local stable uniformity. This is the correct substitute for finite type when working with adic spaces far from the Noetherian regime (Dine, 15 Jul 2025).

The extension of Raynaud’s theory to this context is not a formal translation of the classical theorem. Admissible blow-ups by themselves no longer suffice, because the absence of Noetherian hypotheses means that integrally closed rings of definition can be lost after formal blow-up. The relative theory therefore has to reformulate the relation between the formal category and the analytic category so that $A$6, integrality, and boundedness remain well behaved (Dine, 15 Jul 2025).

This broader perspective aligns with other attempts to push Raynaud’s framework beyond the tft situation. For example, non-Noetherian perfectoid-type formal schemes built from fractional-power restricted series were proposed as a rational-degree extension of the classical framework, with admissible blow-ups and a generic fiber functor adapted to non-Noetherian rings (Bedi, 2019). That work suggests a parallel route beyond classical rigid geometry, though it does not provide the same relative adic localization theorem as the uniform qcqs theory.

4. Normalization and normalized formal blow-ups

The decisive innovation in the modern extension is the normalized formal blow-up. In the classical finite-type setting, admissible formal blow-ups preserve the information needed on the generic fiber, because integrality behaves well enough under blow-up. In the general uniform qcqs adic setting, this fails: starting from $A$7, blow-up along a finitely generated open ideal $A$8 that is not normal can produce local rings that are not integrally closed in the generic fiber. This obstructs the use of integrally closed rings of definition and thereby obstructs uniformity (Dine, 15 Jul 2025).

To repair this, one normalizes the formal model inside its generic fiber. If $A$9 is a locally rig-sheafy, $\pi$0-torsion-free, qcqs adic formal $\pi$1-scheme with generic fiber $\pi$2, then the direct image

$\pi$3

is an adically quasi-coherent $\pi$4-algebra, and its relative formal spectrum

$\pi$5

comes with a canonical affine morphism

$\pi$6

with source $\pi$7-torsion-free and integrally closed in $\pi$8 (Dine, 15 Jul 2025).

A normalized formal blow-up is then defined by first taking an admissible formal blow-up $\pi$9 along an admissible ideal $\mathcal{O}_K$0, then normalizing inside the generic fiber: $\mathcal{O}_K$1 By construction,

$\mathcal{O}_K$2

Thus normalized blow-ups replace admissible blow-ups as the morphisms to be inverted in the non-finite-type localization theorem (Dine, 15 Jul 2025).

The conceptual content is that normalization is performed not abstractly on the formal scheme alone but inside the analytic generic fiber. This makes $\mathcal{O}_K$3 the controlling object. A plausible implication is that the theory is best understood not merely as a birational theory of formal schemes, but as a birational-integral theory in which the formal category must remember the integral structure induced by powerbounded elements.

5. Main equivalences and structural consequences

The principal theorem in the relative theory states that, under the hypotheses on $\mathcal{O}_K$4 and $\mathcal{O}_K$5, the generic fiber functor induces an equivalence

$\mathcal{O}_K$6

The conclusions parallel the classical theorem: faithfulness, fullness after normalized blow-up of the source, and essential surjectivity via existence of integrally closed formal models (Dine, 15 Jul 2025).

A second theorem treats the finite-type relative setting. When one restricts to formal $\mathcal{O}_K$7-schemes topologically of finite type over $\mathcal{O}_K$8 and adic spaces of finite type over $\mathcal{O}_K$9, admissible blow-ups again suffice: $\pi$0 Thus the normalized theory strictly extends classical Raynaud localization rather than replacing it universally (Dine, 15 Jul 2025).

Several structural consequences mirror the classical package. Every uniform qcqs adic space $\pi$1 over $\pi$2 admits a $\pi$3-torsion-free, locally stably uniform formal $\pi$4-model integrally closed in its generic fiber; for affinoid $\pi$5, the canonical model is $\pi$6 when $\pi$7 is taken integrally closed in $\pi$8. Any two such models become isomorphic after normalized formal blow-ups. Morphisms between analytic spaces lift to formal morphisms after normalized blow-up of the source (Dine, 15 Jul 2025).

The sheaf $\pi$9 is the algebraic backbone of these results. For a formal model $\mathfrak{X}_\eta$0, the direct image $\mathfrak{X}_\eta$1 yields the normalization, and globally one has a Bhatt-type limit theorem: $\mathfrak{X}_\eta$2 This recovers the analytic space and its integral structure as an inverse limit over formal modifications (Dine, 15 Jul 2025).

6. Applications, variants, and related developments

Raynaud’s theory is not only a classification theorem; it is a technique for constructing analytic spaces from formal or algebraic data. In the construction of Huber’s universal compactification for a qc separated rigid space over a non-archimedean field, one starts from the cofiltered system of compactified special fibers $\mathfrak{X}_\eta$3 of formal models $\mathfrak{X}_\eta$4, takes the inverse limit of the compactified special fibers, and proves that it is canonically homeomorphic to the universal compactification $\mathfrak{X}_\eta$5. The stalk comparison is expressed in terms of valuation rings in residue fields, again showing the centrality of formal models and their specialization maps (Kobak, 2023).

Another direction concerns algebraizable formal models. For a variety $\mathfrak{X}_\eta$6 over a complete nontrivially valued field that admits a closed embedding into a toric variety, one can construct an algebraizable formal model of $\mathfrak{X}_\eta$7 by taking the closure of $\mathfrak{X}_\eta$8 inside a suitable toric $\mathfrak{X}_\eta$9-model of the ambient space and completing along the special fiber. The generic fiber is then identified with $(A[\pi^{-1}], A^+)$0 using tropicalization and polyhedral combinatorics (Coles et al., 2023). This is a classical Raynaud-style construction in spirit, but it emphasizes algebraizability and toric/tropical control rather than localization by blow-ups.

A third line of development attempts to extend the formal-model viewpoint to non-Noetherian perfectoid-type rings by replacing ordinary topological finite presentation with “topologically finite eka presentation” and using fractional-power restricted series such as $(A[\pi^{-1}], A^+)$1. In that setting one can still formulate admissible blow-ups, generic fibers, and flatness statements via Gabber-type lemmas (Bedi, 2019). This suggests that the pressure to move beyond Noetherian hypotheses is longstanding, though the normalized formal blow-up of the uniform qcqs adic theory provides a different and more categorical resolution of the integrality problem.

The following table summarizes the main regimes.

Setting	Formal side	Morphisms inverted
Classical rigid geometry	admissible formal schemes of finite type over $(A[\pi^{-1}], A^+)$2	admissible formal blow-ups
Relative finite-type adic setting	locally rig-sheafy, $(A[\pi^{-1}], A^+)$3-torsion-free formal $(A[\pi^{-1}], A^+)$4-schemes of tft	admissible formal blow-ups
Relative uniform qcqs adic setting	locally stably uniform, integrally closed, $(A[\pi^{-1}], A^+)$5-torsion-free qcqs formal $(A[\pi^{-1}], A^+)$6-schemes	normalized formal blow-ups

These variants should not be conflated. The finite-type theorem is a relative non-Noetherian generalization of the classical situation, whereas the general uniform qcqs theorem depends essentially on normalization inside the generic fiber (Dine, 15 Jul 2025).

7. Significance, limitations, and misconceptions

The significance of Raynaud’s theory lies in its translation of analytic geometry into formal birational geometry. Analytic spaces become generic fibers of formal models, and analytic equivalence becomes formal equivalence after localizing by blow-ups. In the modern relative theory, this principle survives in a broader analytic world that includes perfectoid spaces, infinite-level towers, and other non-Noetherian phenomena, but only after strengthening blow-ups by normalization (Dine, 15 Jul 2025).

A common misconception is that admissible blow-ups are always sufficient. They are sufficient in the classical finite-type setting and in the relative finite-type theorem, but not for general uniform qcqs adic spaces. The obstruction is not merely technical; it is the failure of integrality of rings of definition after blow-up, which can destroy uniformity and the good behavior of $(A[\pi^{-1}], A^+)$7 (Dine, 15 Jul 2025).

Another misconception is that formal models are unique in any literal sense. What the theory provides is uniqueness only after passing to the localized category. Two models of the same analytic space need not be isomorphic as formal schemes, but they become canonically comparable after admissible or normalized formal blow-ups, depending on the regime (Dine, 15 Jul 2025).

The hypotheses in the relative theory are essential. One requires $(A[\pi^{-1}], A^+)$8 to be $(A[\pi^{-1}], A^+)$9-adically complete, $\mathcal{O}_K$00 a non-zero-divisor generating an ideal of definition, $\mathcal{O}_K$01 sheafy, uniformity on the analytic side, and local stable uniformity plus integrality in the generic fiber on the formal side. These conditions are precisely what make $\mathcal{O}_K$02 behave as a ring of definition compatible with rational localization (Dine, 15 Jul 2025).

Taken together, these results show that Raynaud’s theory has evolved from a Noetherian theorem about rigid spaces over a field into a general localization principle for relative analytic adic geometry. The classical category of admissible formal models localized at admissible formal blow-ups remains the prototype, but in the non-finite-type uniform setting the normalized formal blow-up becomes the indispensable replacement that preserves the defining integral structure of the analytic generic fiber (Dine, 15 Jul 2025).