Normalized Formal Blow-Up in Adic Geometry

• Normalized formal blow-up is a formal-geometric modification that performs an admissible blow-up followed by normalization to preserve integral closure in the generic fiber.
• It addresses limitations of classical blow-ups in non-Noetherian or non-finite-type settings by ensuring compatibility between formal models and their analytic counterparts.
• This construction underpins a Raynaud-style theory by providing a universal factorization that uniquely identifies integrally closed formal models over a Tate affinoid base.

Searching arXiv for papers on normalized formal blow-ups and related blow-up/normalization contexts. Normalized formal blow-up is a formal-geometric modification in which an admissible formal blow-up is followed by normalization in the generic fiber. In the setting of uniform qcqs adic spaces over a Tate affinoid base, this construction is introduced as the replacement for classical admissible formal blow-ups when finite-type hypotheses are absent (Dine, 15 Jul 2025). The central idea is that an ordinary formal blow-up may fail to preserve integral closure in the generic fiber, whereas the normalized variant restores that property and thereby becomes suitable for a Raynaud-style theory of formal models beyond the Noetherian finite-type regime (Dine, 15 Jul 2025). The expression also sits near two adjacent but distinct usages of blow-up in the literature: blow-up of conductor ideals as a mechanism for recovering normalization in algebraic geometry (Birghila et al., 2016), and formal geometric blow-up as a desingularizing coordinate method for dynamic bifurcations in PDEs (Jelbart et al., 2023). These uses share the language of blow-up and normalization but address different categories, objects, and universal properties.

1. Formal definition in the adic setting

The modern formal-model notion is formulated over a complete adic ring $R$ whose ideal of definition is principal, generated by a non-zero-divisor $\varpi\in R$. Writing $\overline R$ for the integral closure of $R$ in $R[\varpi^{-1}]$, one sets

$S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$

One then considers adic spaces $X\to S$ which are quasi-compact, quasi-separated, and uniform; a formal $R$-model of $X$ is an adic formal $R$-scheme $\varpi\in R$0 whose adic analytic generic fiber $\varpi\in R$1 recovers $\varpi\in R$2 (Dine, 15 Jul 2025).

Within this framework, if $\varpi\in R$3 is a locally rig-sheafy, $\varpi\in R$4-torsion-free, qcqs adic formal $\varpi\in R$5-scheme whose generic fiber $\varpi\in R$6 is uniform, a morphism

$\varpi\in R$7

is called a normalized formal blow-up if it factors as

$\varpi\in R$8

where $\varpi\in R$9 is a $\overline R$0-torsion-free admissible formal blow-up and the first arrow is the normalization of $\overline R$1 in its generic fiber (Dine, 15 Jul 2025). Equivalently, there exist

$\overline R$2

with $\overline R$3 the normalization in the generic fiber (Dine, 15 Jul 2025).

This definition is designed to take over the role played by admissible formal blow-ups in the classical Raynaud theory, but now in a context where integral closure in the generic fiber must be enforced explicitly rather than assumed automatically (Dine, 15 Jul 2025).

2. Admissible blow-up and normalization as component operations

The first ingredient is the classical admissible formal blow-up. If $\overline R$4 is an adic formal $\overline R$5-scheme of finite ideal type and $\overline R$6 is an admissible ideal, meaning locally containing an ideal of definition and of finite type, one forms

$\overline R$7

by completing, on each affine chart $\overline R$8, the usual algebraic blow-up $\overline R$9 (Dine, 15 Jul 2025).

The second ingredient is normalization inside the generic fiber. For a locally rig-sheafy, $R$0-torsion-free, qcqs adic formal $R$1-scheme $R$2 with generic fiber $R$3, pushing forward the sheaf of integral functions on $R$4 yields an adically quasi-coherent $R$5-algebra

$R$6

The resulting $R$7 is again $R$8-torsion-free and integrally closed in its generic fiber; this morphism is called the normalization of $R$9 in $R[\varpi^{-1}]$0 (Dine, 15 Jul 2025).

The normalized formal blow-up is therefore a two-step procedure: first perform the admissible modification prescribed by an ideal, then replace the resulting formal model by its integral closure in the generic fiber. The paper stresses that the normalization step is essential in general when $R[\varpi^{-1}]$1 is non-Noetherian or when $R[\varpi^{-1}]$2 is not of finite type but merely uniform (Dine, 15 Jul 2025). This suggests that “normalized” is not a cosmetic modifier but the condition ensuring compatibility between formal models and the integral structure carried by the generic fiber.

3. Universal property and coordinate construction

A central structural fact is the universal property of normalization. If

$R[\varpi^{-1}]$3

are affine morphisms of $R[\varpi^{-1}]$4-torsion-free formal $R[\varpi^{-1}]$5-schemes, both integrally closed in their common generic fiber $R[\varpi^{-1}]$6, with $R[\varpi^{-1}]$7, then there is a unique isomorphism $R[\varpi^{-1}]$8 over $R[\varpi^{-1}]$9 (Dine, 15 Jul 2025). In particular, the factorization of a normalized formal blow-up is characterized universally among formal modifications dominating the same admissible blow-up (Dine, 15 Jul 2025).

On an affine chart $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$0, if $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$1 is an open ideal of definition, the admissible blow-up is covered by affines

$S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$2

where

$S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$3

The normalization step then replaces each completed blow-up algebra by its integral closure in its generic fiber (Dine, 15 Jul 2025). A global description is also available as the relative $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$4 of the pushed-forward integral ring of power-bounded functions on the generic fiber (Dine, 15 Jul 2025).

Two consequences follow directly from the source material. First, the normalized blow-up is determined simultaneously by a birational modification on the formal side and by an integrality condition on the analytic generic fiber. Second, because the final object is integrally closed in its generic fiber, it occupies the correct place in the category of formal models used to recover uniform qcqs adic spaces (Dine, 15 Jul 2025).

4. Relation to classical admissible formal blow-ups

The relation to classical admissible blow-ups is explicit. If the original ideal $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$5 is already normal, meaning $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$6 for all $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$7, then the normalization step does nothing, and a normalized blow-up agrees with the classical admissible blow-up (Dine, 15 Jul 2025). In that case the older and newer notions coincide.

In general, however, the extra normalization is what ensures that all formal models produced remain integrally closed in their generic fibers (Dine, 15 Jul 2025). This distinction is decisive outside the classical Noetherian finite-type regime. The finite-type case behaves more rigidly: the paper states that when $S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$8 is topologically of finite type, any normalized formal blow-up is dominated by a single admissible blow-up (Dine, 15 Jul 2025). It also notes that, in the finite-type case, the resulting equivalence recovers the classical Raynaud–Bosch–Lütkebohmert equivalence once one observes that normalization is automatic in the Noetherian setting (Dine, 15 Jul 2025).

A compact comparison is useful.

Notion	Defining step	When it suffices
Admissible formal blow-up	Complete the algebraic blow-up along an admissible ideal	Classical finite ideal type setting (Dine, 15 Jul 2025)
Normalized formal blow-up	Admissible formal blow-up followed by normalization in the generic fiber	Uniform qcqs, non-finite-type, integrally closed formal-model setting (Dine, 15 Jul 2025)

The broader significance is methodological. The normalized variant is not merely a refinement of the admissible blow-up; it is the morphism class by which the category of integrally closed formal models must be localized in order to match the category of uniform qcqs adic spaces (Dine, 15 Jul 2025).

5. Structural properties and role in Raynaud-style theory

Several stability properties are recorded. If

$S \;=\;\bigl(R[\varpi^{-1}],\,\overline R\bigr).$9

are normalized formal blow-ups, then the composite is again a normalized formal blow-up (Dine, 15 Jul 2025). Formation of admissible blow-ups and of the subsequent normalization commutes with base change of divided power-free formal schemes (Dine, 15 Jul 2025). On each closed fiber $X\to S$0, the normalized blow-up is proper and generically an isomorphism, hence finite over a dense open (Dine, 15 Jul 2025).

These properties are tied to the categorical role of the construction. The paper states an analog of Raynaud theory: the category of qcqs, stably uniform, integrally closed formal $X\to S$1-schemes, localized by normalized blow-ups, is equivalent to the category of uniform qcqs adic spaces over $X\to S$2 (Dine, 15 Jul 2025). Normalized blow-ups therefore supply the “birational” moves needed for descent, comparison of special and generic fibers, and the study of coherent sheaves on non-Noetherian adic spaces, including perfectoid spaces (Dine, 15 Jul 2025).

This placement in the formal-model equivalence clarifies why normalized formal blow-ups are introduced at all. Classical admissible blow-ups suffice in the conventional Noetherian finite-type environment, but once one passes to uniform qcqs adic spaces without finite-type assumptions, admissible blow-ups alone no longer generate the correct localization unless integral closure in the generic fiber is restored (Dine, 15 Jul 2025).

6. Examples and adjacent notions of normalization by blow-up

Three examples in the formal-model setting illustrate the construction. For the open unit disc

$X\to S$3

blowing up $X\to S$4 gives charts

$X\to S$5

and normalizing restores the usual two-chart description of the unit disc after removing the origin (Dine, 15 Jul 2025). Over a perfectoid base, starting from

$X\to S$6

a similar blow-up in $X\to S$7 yields the perfectoid open disc, and the normalization step reflects passage to the integral perfectoid algebra $X\to S$8 (Dine, 15 Jul 2025). For the Fargues–Fontaine curve, one forms the formal model $X\to S$9 and then inverts the divisor $R$0 by blow-up followed by normalization; this recovers the subspace $R$1 whose quotient by Frobenius is the Fargues–Fontaine curve (Dine, 15 Jul 2025).

A related but algebraically different phenomenon appears in the study of conductor ideals. For a reduced Nagata scheme $R$2 with finite normalization $R$3, the blow-up of the conductor $R$4 recovers the normalization precisely when the pulled-back conductor $R$5 is an invertible fractional ideal on $R$6 (Birghila et al., 2016). In dimension one, $R$7 always, recovering Wilson’s statement for curves (Birghila et al., 2016). In the reduced Gorenstein Nagata case with Cohen–Macaulay normalization, one has

$R$8

(Birghila et al., 2016).

The formal-model normalized blow-up and the conductor blow-up theorem both intertwine blow-up with normalization, but they do so in distinct categories. The former normalizes a formal blow-up in its generic fiber; the latter characterizes when a blow-up of a specific ideal already equals the normalization of a scheme (Dine, 15 Jul 2025, Birghila et al., 2016). A plausible implication is that both bodies of work fit a broader pattern in which blow-ups of carefully chosen ideals become effective only after one imposes an invertibility or integral-closure condition on the transformed object.

7. Terminological boundaries and common confusions

The phrase “formal blow-up” also occurs in a separate analytic-dynamical literature where it denotes a geometric desingularization method for bifurcation problems in PDEs rather than a formal-model construction. In the Swift–Hohenberg equation with slow parameter drift,

$R$9

the blow-up introduces

$X$0

together with the normalization condition $X$1 (Jelbart et al., 2023). The origin $X$2 is replaced by a blow-up cylinder $X$3, $X$4, and overlapping charts yield modulation equations of real Ginzburg–Landau type (Jelbart et al., 2023).

Despite the shared vocabulary of blow-up and normalization, this PDE usage is conceptually different from normalized formal blow-up in adic geometry. Here “normalization” is a gauge condition removing homogeneity scaling in $X$5, not normalization in the sense of integral closure (Jelbart et al., 2023). Confusing these notions would conflate a coordinate desingularization in infinite-dimensional dynamics with a birational-integral modification of formal schemes.

For that reason, precise usage is important. In contemporary arithmetic and adic geometry, “normalized formal blow-up” refers to the morphism obtained by taking an admissible formal blow-up and then normalizing it in the generic fiber, with universal and categorical properties tailored to the theory of formal models of uniform qcqs adic spaces (Dine, 15 Jul 2025). In adjacent literatures, similar words may encode fundamentally different constructions.