Analytic Prismatization: Concepts & Applications

• Analytic prismatization consists of extending theories of prismatic geometry to various non-archimedean settings, including $p$-adic formal schemes, perfectoid rings, and $B_{\mathrm{dR}^+}$-thickenings, where structured thickenings are equipped with distinguished Cartier divisors.
• The stack-theoretic formulation of analytic prismatization is a conceptual transformation that operates using Cartier–Witt stacks to encode prismatic geometry, demonstrating funct inferior relations to Nygaard data, Crystals, and Frobenius-compatible topics.
• Constructs, including rational prismatization and Robba stackification, illustrate topological elements of spatial frameworks, locating significance for overall cohomological insights derived from analytic prismatization.

Analytic prismatization is a family of constructions extending prismatic geometry from $p$-adic formal schemes to rigid, adic, perfectoid, condensed, or related non-archimedean settings. The term has no single universally accepted meaning: it may refer to a rigid-analytic $B_{\mathrm{dR}^+}$-prismatic site, an analytic realization of perfect-prismatic $F$-crystals as $Y$-bundles or shtukas, a formal or stack-theoretic construction whose generic fibres are analytic, or a rational moduli stack of degree-one divisors on thickened Fargues–Fontaine spaces. The common principle is to replace a geometric object by a site or moduli stack of structured thickenings equipped with a distinguished Cartier divisor, and to recover cohomological, Frobenius, Hodge–Tate, or étale information from the resulting structure sheaf and its descent data.

1. Prismatic origins and stack-theoretic formulation

Classical prismatic geometry begins with a prism $(A,I)$, where $A$ is a $\delta$-ring, $I$ is an invertible ideal defining a Cartier divisor, $A$ is derived $(p,I)$-complete, and $B_{\mathrm{dR}^+}$0. The associated Frobenius lift is

$B_{\mathrm{dR}^+}$1

A prism is perfect when Frobenius is an isomorphism, bounded when $B_{\mathrm{dR}^+}$2 has bounded $B_{\mathrm{dR}^+}$3-torsion, orientable when $B_{\mathrm{dR}^+}$4 is principal, and crystalline when $B_{\mathrm{dR}^+}$5. Perfect prisms are equivalent to perfectoid rings through

$B_{\mathrm{dR}^+}$6

General prisms can consequently be regarded as deperfections of perfectoid rings: they retain a Frobenius lift without requiring Frobenius to be invertible (Bhatt et al., 2019).

For a smooth $B_{\mathrm{dR}^+}$7-adic formal scheme $B_{\mathrm{dR}^+}$8 over $B_{\mathrm{dR}^+}$9, the prismatic site $F$0 consists of bounded prisms $F$1 over $F$2 together with maps

$F$3

Its structure sheaves are

$F$4

The associated cohomology

$F$5

is a derived $F$6-complete commutative algebra over $F$7 with Frobenius. Its reduction modulo $F$8 satisfies the Hodge–Tate comparison

$F$9

and its crystalline, de Rham, étale, $Y$0-de Rham, and perfectoid specializations recover several established cohomology theories (Bhatt et al., 2019).

The stack-theoretic reformulation replaces the prismatic site by a Cartier–Witt stack. For a $Y$1-nilpotent ring $Y$2, a Cartier–Witt divisor is a generalized Cartier divisor

$Y$3

on the Witt-vector ring. For a bounded $Y$4-adic formal scheme $Y$5, the Cartier–Witt stack $Y$6 classifies such divisors together with a map from the corresponding quotient to $Y$7. This stack is the prismatization of $Y$8: $Y$9 It carries a Frobenius endomorphism and is functorial in $(A,I)$0. Every object of the prismatic site gives a map

$(A,I)$1

so the stack is assembled from the affine prism charts of the site (Bhatt et al., 2022).

For suitable $(A,I)$2-quasisyntomic or locally complete-intersection formal schemes, quasicoherent complexes on $(A,I)$3 recover prismatic crystals: $(A,I)$4 Likewise,

$(A,I)$5

The Hodge–Tate locus of $(A,I)$6 is a gerbe banded by a twisted tangent group, and Hodge–Tate crystals become Higgs-type objects under a splitting of that gerbe (Bhatt et al., 2022).

The prismatization of $(A,I)$7 admits three related formal-stack versions: $(A,I)$8 The first uses invertible Witt modules and primitive maps; the second allows admissible, possibly noninvertible, Witt-module extensions and is the filtered prismatization; the third glues two copies of $(A,I)$9 inside $A$0 and is the syntomification. Their universal ring stacks encode crystals, Nygaard data, and Frobenius-compatible coefficient theories (Drinfeld, 2020).

2. Formal, algebraic, and infinitesimal realizations

Before the recent use of “analytic prismatization,” several constructions supplied its formal and algebraic infrastructure. The integral prismatization over $A$1 constructs an algebraic base stack $A$2 whose fibers are

$A$3

and

$A$4

Thus its rational fiber is a formal Cartier or filtered de Rham parameter space, whereas its $A$5-adic fiber is the usual prism stack. For a scheme $A$6, the algebraic prismatization $A$7 specializes to the filtered de Rham stack over $A$8 and to the $A$9-adic prismatization over $\delta$0 (Gurney, 2023).

The rational fiber is governed by the Rees construction. If $\delta$1 is a line bundle with section, the formal Cartier stack $\delta$2 parametrizes filtered objects. Its universal ring stack is modeled by

$\delta$3

For a smooth scheme $\delta$4, the resulting de Rham prismatization has cohomology corresponding under Rees equivalence to the Hodge-filtered de Rham complex

$\delta$5

This construction is formal and algebraic rather than rigid-analytic (Gurney, 2023).

A related filtered object is the Nygaard prismatization. For a perfect ring $\delta$6 of characteristic $\delta$7 and an animated $\delta$8-algebra $\delta$9, its base is

$I$0

The loci $I$1, $I$2, $I$3, and $I$4 recover ordinary prismatic, crystalline, Hodge-filtered de Rham, and conjugate-filtered Hodge–Tate theories. The principal theorem identifies the Nygaard-filtered prismatization with the relative spectrum of the Rees algebra of Nygaard-filtered prismatic cohomology: $I$5 This result is called de Rham affineness: the stack-valued functor is represented, relative to its filtered base, by a functorially constructed cohomological algebra (Sahai, 22 Dec 2025).

The $I$6-truncated prismatizations

$I$7

are nilpotent neighborhoods of the Hodge–Tate locus. The stack $I$8 imposes the condition that the image of the $I$9-fold tensor power of the Cartier divisor vanishes in the test ring. For $A$0, it has a description as a truncated symmetric algebra over the Hodge–Tate locus. In the case $A$1, the truncations are related to Sen operators and to complexes of modules over slightly enlarged Breuil–Kisin rings. The genuine integral classification has the bound

$A$2

where $A$3 is the ramification index; the unramified case gives $A$4. The slightly shrunk construction obtained by adjoining $A$5 works for arbitrary $A$6 (Liu, 2024).

For cyclotomic bases, $A$7-prism charts provide explicit local models. Let

$A$8

with $A$9 mapping to a suitable $(p,I)$0-power root of unity. The resulting chart supports a twisted derivation

$(p,I)$1

Pullbacks of quasicoherent complexes acquire operators satisfying a twisted Leibniz rule and become modules over an Ore extension

$(p,I)$2

Relative geometric directions yield commuting $(p,I)$3-Higgs operators and a $(p,I)$4-de Rham complex. Quasicoherent complexes on the prismatization are classified by complete modules with locally nilpotent reductions of these operators (Liu, 9 Apr 2025).

3. Rigid-analytic $(p,I)$5-prismatization

A direct rigid-analytic form of prismatization is obtained by replacing integral $(p,I)$6-rings with $(p,I)$7-adic thickenings over Fontaine’s de Rham period ring. Let $(p,I)$8 be a complete algebraically closed $(p,I)$9-adic field, let

$B_{\mathrm{dR}^+}$00

be Fontaine’s canonical surjection, and choose

$B_{\mathrm{dR}^+}$01

A $B_{\mathrm{dR}^+}$02-prism over an affinoid rigid space $B_{\mathrm{dR}^+}$03 is a diagram

$B_{\mathrm{dR}^+}$04

such that $B_{\mathrm{dR}^+}$05 is noetherian, $B_{\mathrm{dR}^+}$06-torsionfree, $B_{\mathrm{dR}^+}$07-adically complete, finite type modulo powers of $B_{\mathrm{dR}^+}$08 over Tate algebras, and equipped with the indicated map to $B_{\mathrm{dR}^+}$09. These objects have no $B_{\mathrm{dR}^+}$10-structure or Frobenius lift; the role of the prismatic ideal is played by $B_{\mathrm{dR}^+}$11.

The analytic prismatic site $B_{\mathrm{dR}^+}$12 is the opposite category of such prisms with the indiscrete topology. Its structure sheaves are

$B_{\mathrm{dR}^+}$13

The corresponding cohomology theories are

$B_{\mathrm{dR}^+}$14

and

$B_{\mathrm{dR}^+}$15

The former is $B_{\mathrm{dR}^+}$16-linear and the latter is $B_{\mathrm{dR}^+}$17-linear.

For a surjection $B_{\mathrm{dR}^+}$18 with kernel $B_{\mathrm{dR}^+}$19, the analytic prismatic envelope is

$B_{\mathrm{dR}^+}$20

The variables $B_{\mathrm{dR}^+}$21 represent $B_{\mathrm{dR}^+}$22. For an l.c.i. presentation

$B_{\mathrm{dR}^+}$23

the envelope

$B_{\mathrm{dR}^+}$24

provides a covering object whose Čech–Alexander complex computes analytic prismatic cohomology.

Coherent prismatic crystals on this site correspond to finite $B_{\mathrm{dR}^+}$25-modules with integrable, quasi-nilpotent logarithmic connections

$B_{\mathrm{dR}^+}$26

The connection is logarithmic because it may have a pole along $B_{\mathrm{dR}^+}$27; crystals arising from infinitesimal geometry have integral connections landing in the unlocalized differential module (Guo, 2021).

The relationship with derived de Rham theory is expressed by a Simpson-type transform. For a filtered $B_{\mathrm{dR}^+}$28-complex $B_{\mathrm{dR}^+}$29, set

$B_{\mathrm{dR}^+}$30

For an affinoid l.c.i. space,

$B_{\mathrm{dR}^+}$31

In the smooth case this yields the $B_{\mathrm{dR}^+}$32-divided de Rham complex

$B_{\mathrm{dR}^+}$33

The infinitesimal-to-prismatic map factors through

$B_{\mathrm{dR}^+}$34

For smooth affinoids the first map is an equivalence onto $B_{\mathrm{dR}^+}$35, while equality with the full prismatic complex is generally false globally.

The reduced theory has a Hodge–Tate filtration whose graded pieces are

$B_{\mathrm{dR}^+}$36

For smooth $B_{\mathrm{dR}^+}$37, this becomes

$B_{\mathrm{dR}^+}$38

For smooth rigid spaces, the theory agrees with pro-étale $B_{\mathrm{dR}^+}$39-cohomology: $B_{\mathrm{dR}^+}$40 It therefore connects infinitesimal, Hodge–Tate, de Rham-period, and pro-étale cohomology (Guo, 2021).

4. Analytic prismatic $B_{\mathrm{dR}^+}$41-crystals, local systems, and shtukas

A different analytic realization retains the analytic locus of a prism rather than replacing the prism by a $B_{\mathrm{dR}^+}$42-thickening. For a prism $B_{\mathrm{dR}^+}$43, define

$B_{\mathrm{dR}^+}$44

An analytic prismatic crystal is a compatible system of vector bundles on these opens: $B_{\mathrm{dR}^+}$45 An analytic prismatic $B_{\mathrm{dR}^+}$46-crystal adds a Frobenius isomorphism after inverting $B_{\mathrm{dR}^+}$47: $B_{\mathrm{dR}^+}$48 The ordinary prismatic category embeds fully faithfully into the analytic category, but generally not essentially surjectively: an analytic object need not extend across $B_{\mathrm{dR}^+}$49 to a vector bundle on all of $B_{\mathrm{dR}^+}$50 (Guo et al., 2022).

For a smooth $B_{\mathrm{dR}^+}$51-adic formal scheme $B_{\mathrm{dR}^+}$52 over $B_{\mathrm{dR}^+}$53, with rigid generic fiber $B_{\mathrm{dR}^+}$54, étale realization gives

$B_{\mathrm{dR}^+}$55

The main comparison identifies the source with crystalline $B_{\mathrm{dR}^+}$56-local systems: $B_{\mathrm{dR}^+}$57 The realization is given locally by

$B_{\mathrm{dR}^+}$58

and its inverse is described by Frobenius-fixed points. This equivalence holds for general crystalline coefficients and ramified base fields (Guo et al., 2022).

The distinction between analytic and ordinary prismatic objects is essential. A crystalline local system is prismatically good reduction when its analytic prismatic inverse image extends across the prismatic divisor to an ordinary prismatic $B_{\mathrm{dR}^+}$59-crystal. For a point $B_{\mathrm{dR}^+}$60 every crystalline lattice has this property, whereas in positive relative dimension it can fail. Consequently,

$B_{\mathrm{dR}^+}$61

where the right side is the full tensor subcategory of crystalline local systems with prismatically good reduction (Imai et al., 2024).

Perfect-prismatic $B_{\mathrm{dR}^+}$62-crystals admit a more explicitly adic realization. Restricting to perfect prisms identifies them with finite projective crystals over integral perfectoid rings. Evaluating on perfectoid test objects and restricting from Witt-vector spaces to relative $B_{\mathrm{dR}^+}$63-spaces produces integral $B_{\mathrm{dR}^+}$64-bundles. For a complete topological ring $B_{\mathrm{dR}^+}$65,

$B_{\mathrm{dR}^+}$66

The equivalence is exact, tensor-compatible, and extends to affine flat group schemes. Adding Frobenius gives

$B_{\mathrm{dR}^+}$67

The analytic object is a family of vector bundles on

$B_{\mathrm{dR}^+}$68

with a meromorphic Frobenius away from the untilt divisor (Güthge, 2023).

The same framework has a Tannakian form. For a reductive group scheme $B_{\mathrm{dR}^+}$69, analytic and ordinary prismatic $B_{\mathrm{dR}^+}$70-crystals with $B_{\mathrm{dR}^+}$71-structure are exact tensor functors

$B_{\mathrm{dR}^+}$72

or into the analytic variant. The resulting category of ordinary $B_{\mathrm{dR}^+}$73-objects is equivalent to $B_{\mathrm{dR}^+}$74-local systems with prismatically good reduction: $B_{\mathrm{dR}^+}$75 A compatible shtuka realization sends analytic prismatic $B_{\mathrm{dR}^+}$76-objects to $B_{\mathrm{dR}^+}$77-shtukas and agrees with the shtuka realization of the corresponding de Rham local system (Imai et al., 2024).

5. Rational analytic prismatization and Fargues–Fontaine geometry

A genuinely analytic rational construction is formulated using Gelfand rings and Gelfand stacks. These provide a category containing Banach $B_{\mathrm{dR}^+}$78-algebras, Berkovich spaces, dagger-rigid spaces, derived Berkovich spaces, perfectoid objects, and nilperfectoid objects. A nilperfectoid ring is an overconvergent thickening whose dagger reduction is a perfectoid uniform completion.

For a perfectoid space $B_{\mathrm{dR}^+}$79 in characteristic $B_{\mathrm{dR}^+}$80, let $B_{\mathrm{dR}^+}$81 be the relative Fargues–Fontaine disc and

$B_{\mathrm{dR}^+}$82

its punctured disc. A rational $B_{\mathrm{dR}^+}$83-space is a Berkovich or Gelfand-space thickening $B_{\mathrm{dR}^+}$84 of $B_{\mathrm{dR}^+}$85 equipped with Frobenius and an identification on diamonds

$B_{\mathrm{dR}^+}$86

A rational analytic prism is a rational $B_{\mathrm{dR}^+}$87-space together with a degree-one Cartier divisor $B_{\mathrm{dR}^+}$88. Degree one means that the divisor corresponds, on the Fargues–Fontaine side, to an untilt.

For a Gelfand stack $B_{\mathrm{dR}^+}$89, the analytic prismatic site consists of tuples

$B_{\mathrm{dR}^+}$90

Its moduli stack is

$B_{\mathrm{dR}^+}$91

The rational analytic prismatization is the realization

$B_{\mathrm{dR}^+}$92

Its points consist of degree-one divisors in analytic $B_{\mathrm{dR}^+}$93-spaces together with maps from those divisors to $B_{\mathrm{dR}^+}$94 (Anschütz et al., 20 Oct 2026).

Perfection is obtained by iterating Frobenius: $B_{\mathrm{dR}^+}$95 The quotient by Frobenius,

$B_{\mathrm{dR}^+}$96

is the $B_{\mathrm{dR}^+}$97-adic twistor stack. For a perfectoid base, the ordinary relative Fargues–Fontaine curve is

$B_{\mathrm{dR}^+}$98

The analytic twistor stack retains overconvergent information that is invisible on the underlying diamond and maps naturally to a de Rham or Hyodo–Kato stack: $B_{\mathrm{dR}^+}$99

The Hodge–Tate locus is obtained by pulling back along the Hodge–Tate point: $F$00 For smooth analytic $F$01, the map

$F$02

is controlled by the twisted overconvergent tangent bundle

$F$03

This is an analytic analogue of the Hodge–Tate gerbe and of the geometric Sen or $F$04-adic Simpson constructions.

For smooth partially proper rigid analytic varieties, the main comparison identifies perfect complexes on the perfect analytic prismatization modulo Frobenius with perfect complexes on the relative Fargues–Fontaine curve. The construction is rational and does not provide an integral $F$05-theory. Its geometric input is the moduli of degree-one divisors on thickened Fargues–Fontaine discs, while its cohomological output recovers pro-étale $F$06-cohomology (Anschütz et al., 20 Oct 2026).

6. Spectral, condensed, and Robba-theoretic extensions

Several constructions extend prismatization without producing a rigid-analytic stack directly. Frobenius-untwisted topological cyclic homology yields even-periodic formal spectral stacks whose classical truncations are the Bhatt–Lurie and Drinfeld prismatizations: $F$07 The even-periodic homotopy sheaves recover Breuil–Kisin twists: $F$08 The Quillen formal group of the spectral enhancement agrees with the Drinfeld formal group. These are spectral and formal constructions over adic $F$09-rings, not analyticifications in the rigid or adic sense (Gregoric, 2 Nov 2025).

Another approach compares de Rham prismatization with de Rham Robba stackification. Over small $F$10-stacks, the two constructions are asserted to have equivalent solid quasicoherent categories: $F$11 Over small arc-stacks, the comparison is fully faithful rather than an equivalence. The distinction is interpreted through perfectoid descent: passage to $F$12-stacks perfectizes the local geometry, whereas arc-stacks retain imperfect analytic charts. The Robba spaces used in this comparison are pre–Fargues–Fontaine spaces; no Frobenius quotient is taken before the de Rham completion and stackification (Tong, 2024).

Analytic prismatic $F$13-crystals also interact with crystalline local systems, shtukas, $F$14-pairs, and Tannakian structures. Their analytic locus captures the region

$F$15

while the omitted codimension-two locus records integral extension data. This accounts for the difference between analytic and ordinary prismatic crystals: analytic objects encode the étale region, Hodge–Tate boundary, and Frobenius comparison, whereas ordinary objects additionally extend over the full integral prism.

The various meanings of analytic prismatization can therefore be organized by the retained structure:

Construction	Primary objects	Main output
$F$16-prismatization	$F$17-adic rigid thickenings	$F$18-divided de Rham and pro-étale cohomology
Analytic prismatic $F$19-crystals	Vector bundles on $F$20	Crystalline $F$21-local systems
Perfect-prismatic analytification	Perfectoid evaluations and $F$22-bundles	Shtukas with fixed legs
Rational analytic prismatization	Degree-one divisors on analytic $F$23-spaces	$F$24-adic twistor stacks and Fargues–Fontaine comparisons
Spectral prismatization	Even-periodic formal spectral stacks	Spectral refinements of prismatic, Nygaard, and syntomic stacks
Robba comparison	Solid quasicoherent categories	De Rham Robba–prismatic equivalences

No single construction subsumes all the others. Integral prismatic stacks retain $F$25-structures, Cartier divisors, and Frobenius; $F$26-prisms emphasize characteristic-zero infinitesimal geometry; perfect-prismatic constructions emphasize perfectoid descent and shtukas; and rational analytic prismatization emphasizes degree-one divisors and Frobenius quotients on Fargues–Fontaine-type spaces. Taken together, they form a hierarchy from formal algebraic prismatization to analytic period geometry.