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Rational analytic prismatization

Published 20 Oct 2026 in math.AG and math.NT | (2610.27918v1)

Abstract: We develop a general theory of analytic prismatization with Qp\mathbb{Q}_p-coefficients in rigid analytic geometry over Qp\mathbb{Q}_p. We prove an equivalence of perfect complexes on the perfect analytic prismatization modulo Frobenius with perfect complexes on the relative Fargues--Fontaine curve for any smooth partially proper rigid analytic variety. In particular, we explain how this equivalence conceptualizes the decompletion techniques by passing to locally analytic vectors which appear in (geometric) Sen theory

Summary

  • The paper develops a $Q_p$-linear analogue of prismatization using rational analytic spaces with nontrivial overconvergent structure.
  • The rational analytic prismatization functor places Gelfand stacks between rigid analytic and perfectoid geometry, encoding $(\varphi,\Gamma)$-modules and geometric Sen theory.
  • The central result is a categorical equivalence of perfect complexes on the $p$-adic twistor stack with those on the relative Fargues–Fontaine curve for smooth rigid spaces.

Analytic setting and central construction

“Rational analytic prismatization” develops a QpQ_p-linear analogue of prismatization for rigid and Berkovich geometry. Its primary objective is to construct a geometric object whose coherent and perfect-coherent sheaf theories encode relative (φ,Γ)(\varphi,\Gamma)-modules, pro-étale QpQ_p-cohomology, and geometric Sen theory. The paper works throughout with rational coefficients and postpones the integral ZpZ_p-theory, refined filtered constructions, and syntomification.

The geometric framework is that of Gelfand rings and Gelfand stacks. This choice is essential: Gelfand geometry contains Banach QpQ_p-algebras, admits a descendable topology with nilperfectoid rings as a basis, and retains enough nonreduced analytic structure to model overconvergent thickenings of perfectoid spaces. The perfectoidization functor

(−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}

forgets precisely the overconvergent thickening data, while the analytic de Rham functor is its right adjoint. The resulting formalism places Gelfand stacks between rigid analytic geometry and arc-stack/perfectoid geometry.

The basic analytic analogue of a δ\delta-ring is a rational δ\delta-space (S,X)(S,\mathcal X). Here SS is a perfectoid space in characteristic (φ,Γ)(\varphi,\Gamma)0, (φ,Γ)(\varphi,\Gamma)1 is a Berkovich space with Frobenius, and

(φ,Γ)(\varphi,\Gamma)2

induces an isomorphism after perfectoidization. Thus (φ,Γ)(\varphi,\Gamma)3 is an overconvergent or derived thickening of the punctured relative Fargues–Fontaine disc (φ,Γ)(\varphi,\Gamma)4. A rational analytic prism is a (φ,Γ)(\varphi,\Gamma)5-space equipped with a degree-(φ,Γ)(\varphi,\Gamma)6 Cartier divisor (φ,Γ)(\varphi,\Gamma)7. The degree condition means that the divisor restricts on (φ,Γ)(\varphi,\Gamma)8 to the divisor associated with an untilt of (φ,Γ)(\varphi,\Gamma)9.

For a Gelfand stack QpQ_p0, the analytic prismatic site consists of quadruples

QpQ_p1

with coverings induced by the QpQ_p2-topology on QpQ_p3. The rational analytic prismatization QpQ_p4 is the realization of the corresponding analytic QpQ_p5-stack. Its perfect form and its quotient by Frobenius are

QpQ_p6

The latter is called the QpQ_p7-adic twistor stack of QpQ_p8.

The construction is genuinely richer than the perfectoidization of QpQ_p9. The underlying arc-stack of ZpZ_p0 depends only on ZpZ_p1, but the Gelfand stack ZpZ_p2 retains the nontrivial overconvergent structure of the original analytic space. This distinction is crucial: the paper explicitly notes that ZpZ_p3 is generally not the relative Fargues–Fontaine curve of ZpZ_p4, even when ZpZ_p5 is perfectoid.

Thickened Fargues–Fontaine geometry

The analytic ZpZ_p6-space attached to a nilperfectoid ZpZ_p7-algebra ZpZ_p8 is constructed by a pushout. Writing ZpZ_p9 and QpQ_p0, the associated thickening QpQ_p1 is obtained from

QpQ_p2

by identifying QpQ_p3 with its Frobenius translates in the punctured Fargues–Fontaine disc. This is the analytic counterpart of the Witt-vector construction. The use of QpQ_p4-coefficients substantially simplifies the resulting algebra: the contribution of the QpQ_p5-nilradical splits as a product indexed by QpQ_p6, and the difference between analytic and perfect analytic prismatization is largely the replacement of QpQ_p7 by QpQ_p8.

This pushout description is not merely a technical device. It isolates the precise source of the additional analytic information. The classical punctured Fargues–Fontaine disc accounts for the perfectoid or “reduced” part, while the copies of QpQ_p9 indexed by Frobenius encode the overconvergent nilpotent directions. The degree-(−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}0 divisor then intersects these pieces in a finite collection of Frobenius translates, allowing the authors to reduce mapping problems into the prismatization to gluing problems involving ordinary cones.

The paper formalizes this reduction through the notion of (−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}1-cohesiveness. A Gelfand stack is (−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}2-cohesive when it converts the relevant pushouts of Gelfand rings into pullbacks of anima. Berkovich spaces, categories of perfect complexes, and analytifications of perfect strongly Tannakian stacks satisfy this condition. It is precisely the hypothesis needed to obtain the pullback description of prismatization and to compare the nonperfect and perfect constructions.

The Hodge–Tate locus and the (−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}3-adic twistor stack

The prismatization has a canonical morphism to the stack of Cartier divisors,

(−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}4

The fiber over the zero divisor defines the analytic Hodge–Tate stack

(−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}5

For smooth partially proper rigid spaces, this locus is not generally a split cotangent or Higgs stack. Instead, the paper proves that there is a natural morphism

(−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}6

which exhibits (−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}7 as a gerbe banded by the Breuil–Kisin-twisted overconvergent tangent bundle

(−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}8

The gerbe is generally non-split. This is a significant distinction from the classical complex Hodge stack and from some formulations of the (−)⋄:GelfStk⟶ArcStkQp(-)^\diamond:\mathrm{GelfStk}\longrightarrow \mathrm{ArcStk}_{Q_p}9-adic Simpson correspondence: the twisting is intrinsic and cannot in general be removed without additional choices.

The Hodge–Tate stack is controlled by the Faltings extension. In particular, the cotangent-theoretic structure of δ\delta0 recovers the extension governing arithmetic and geometric Sen operators. The paper therefore interprets the canonical Higgs fields on geometric δ\delta1-vector bundles as a consequence of their descent from the Hodge–Tate gerbe, rather than as an external structure imposed on them.

The twistor stack lies between the relative Fargues–Fontaine curve and its analytic de Rham stack: δ\delta2 The second morphism is an isomorphism away from the Hodge–Tate neighborhood. Consequently, the genuinely new geometry of δ\delta3 is concentrated near the Hodge–Tate locus. Generically, the twistor stack agrees with the Hyodo–Kato stack, whereas near the Hodge–Tate divisor it records the nontrivial deformation data encoded by the Faltings extension.

The comparison theorem for perfect complexes

The principal theorem concerns a smooth partially proper rigid space δ\delta4 over δ\delta5. The morphism

δ\delta6

is proved to be prim with trivial codualizing complex. Its pullback on quasi-coherent sheaves is fully faithful, and it induces an equivalence

δ\delta7

This is the paper’s central categorical result. It asserts that the twistor stack and the relative Fargues–Fontaine curve have the same perfect complexes, despite the fact that the former contains additional nontrivial Hodge–Tate and overconvergent geometry. The implication is not that the stacks are isomorphic; rather, their difference is invisible to dualizable objects under the stated hypotheses.

The result also holds after restriction to suitable open arc-substacks and on the Hodge–Tate stack and the completed perfectoidization. For vector bundles, the equivalence identifies vector bundles on the Fargues–Fontaine curve—that is, relative δ\delta8-modules—with vector bundles on the δ\delta9-adic twistor stack. On the Hodge–Tate locus, it becomes an equivalence between δ\delta0-vector bundles and twisted Higgs bundles on the non-split Hodge–Tate gerbe.

The theorem has an immediate cohomological consequence: δ\delta1 Thus the coherent cohomology of the δ\delta2-adic twistor stack geometrizes pro-étale δ\delta3-cohomology. This should be distinguished from the corresponding assertion for arbitrary nonreduced Berkovich spaces, where the two sides can differ. The smoothness and partial properness assumptions are therefore structural, not cosmetic.

For δ\delta4 with δ\delta5 finite, the theorem specializes to a decompletion statement for δ\delta6-modules over the Robba ring. More generally, the local computation on a torus expresses the twistor stack through locally analytic vectors. If δ\delta7, the relevant group is

δ\delta8

and the quotient description of the twistor stack is obtained from the perfected torus by passing from continuous to locally analytic functions. This identifies analytic prismatization with a geometric form of Sen-theoretic decompletion.

Locally analytic decompletion and geometric Sen theory

A major independent contribution is the general decompletion theorem for a Berkovich space δ\delta9 with a continuous action of a (S,X)(S,\mathcal X)0-adic Lie group (S,X)(S,\mathcal X)1. The authors compare

(S,X)(S,\mathcal X)2

The natural map between them is prim with trivial codualizing complex, yielding a fully faithful pullback on derived categories. If (S,X)(S,\mathcal X)3 is affinoid and classically decompleted, then the category of modules on the locally analytic quotient is identified with modules over the derived algebra of locally analytic vectors (S,X)(S,\mathcal X)4.

Under the stronger local decompletion hypothesis, pullback induces an equivalence on dualizable objects: (S,X)(S,\mathcal X)5 The two technical inputs are the vanishing of higher locally analytic vectors and the comparison between continuous and locally analytic descent. These hypotheses are satisfied in the toric and cyclotomic situations used to prove the main theorem.

The geometric significance is that locally analytic vectors are not merely an analytic approximation to continuous vectors. They provide the correct coefficient algebra for the twistor stack. The passage from (S,X)(S,\mathcal X)6 to (S,X)(S,\mathcal X)7 is precisely the mechanism through which a perfectoid Fargues–Fontaine object descends to the overconvergent analytic prismatization.

This framework also yields a geometric interpretation of Sen operators. A pro-étale (S,X)(S,\mathcal X)8-torsor on (S,X)(S,\mathcal X)9 decompacts to a SS0-torsor on SS1, and the resulting morphism to SS2 induces maps on cotangent complexes whose infinitesimal behavior recovers arithmetic and geometric Sen operators. The paper leaves open whether, under suitable decompletion hypotheses, the entire twistor stack admits a direct presentation as

SS3

It proves the corresponding statement in several basic charts and obtains the associated formula for global sections in general.

Cone constructions and de Rham theory

The paper introduces a cone construction for Gelfand stacks. For a stack SS4, the cone SS5 over SS6 evaluates a Cartier divisor by restricting SS7 to its zero locus. Relative cones recover the deformation to the normal bundle. Their special fibers are shifted and SS8-twisted tangent bundles: SS9

This construction supplies the local geometric model for analytic prismatization. Its special fiber represents square-zero lifting problems. More precisely, for a Cartier divisor (φ,Γ)(\varphi,\Gamma)00 and a map (φ,Γ)(\varphi,\Gamma)01, the special fiber of the relative cone over (φ,Γ)(\varphi,\Gamma)02 is a principally homogeneous space whose sections classify lifts from (φ,Γ)(\varphi,\Gamma)03 to the first infinitesimal thickening (φ,Γ)(\varphi,\Gamma)04 inside (φ,Γ)(\varphi,\Gamma)05. The obstruction is therefore expressed directly by the cotangent complex.

For a formally smooth morphism (φ,Γ)(\varphi,\Gamma)06, the formal cone has a sheaf-theoretic description. The pushforward of its unit is, after the relevant shearing, the de Rham complex

(φ,Γ)(\varphi,\Gamma)07

For smooth rigid spaces this recovers the classical de Rham differential and the Hodge filtration. Consequently, filtered flat connections satisfying Griffiths transversality embed fully faithfully into modules over the formal cone. In the relative divisor setting, the corresponding object is a (φ,Γ)(\varphi,\Gamma)08-de Rham complex, and vector bundles on the formal cone correspond to (φ,Γ)(\varphi,\Gamma)09-connections whose reduction modulo (φ,Γ)(\varphi,\Gamma)10 is nilpotent.

This cone formalism clarifies the relationship between prismatization, filtered de Rham stacks, and Hodge–Tate geometry. The prismatization can be viewed as a Fargues–Fontaine analogue of deformation to the normal bundle, with the Hodge–Tate fiber replacing the classical Hodge fiber and the Breuil–Kisin twist replacing the ordinary Tate weight.

Breuil–Kisin twists and the pullback description

The Frobenius translates of the Hodge–Tate divisor are pairwise disjoint on the thickened Fargues–Fontaine curve. Their tensor product defines the Breuil–Kisin ideal

(φ,Γ)(\varphi,\Gamma)11

Over (φ,Γ)(\varphi,\Gamma)12 this infinite product is well-defined because locally only finitely many translates are nontrivial. The resulting Breuil–Kisin divisor provides a morphism

(φ,Γ)(\varphi,\Gamma)13

For a (φ,Γ)(\varphi,\Gamma)14-cohesive Gelfand stack (φ,Γ)(\varphi,\Gamma)15, the paper proves a cartesian diagram

(φ,Γ)(\varphi,\Gamma)16

Hence (φ,Γ)(\varphi,\Gamma)17 is (φ,Γ)(\varphi,\Gamma)18-formally étale. This is a conceptual reformulation of the prismatization: it is the analytic cone of (φ,Γ)(\varphi,\Gamma)19 pulled back along the Breuil–Kisin divisor, subject to the de Rham condition determined by the underlying perfectoid geometry.

The result explains why the Breuil–Kisin twist appears systematically in duality, in the Hodge–Tate gerbe, and in the relative tangent complex. It also isolates the point at which the rational theory differs from the integral theory. Over (φ,Γ)(\varphi,\Gamma)20, the infinite product of divisor ideals requires no delicate convergence analysis; integrally, extending the product across the locus (φ,Γ)(\varphi,\Gamma)21 is expected to be substantially more difficult.

Cohomological smoothness and duality

The paper establishes cohomological smoothness statements for the universal divisor stack and for relative prismatizations. In particular,

(φ,Γ)(\varphi,\Gamma)22

is prim with trivial codualizing sheaf and has dualizing complex

(φ,Γ)(\varphi,\Gamma)23

where (φ,Γ)(\varphi,\Gamma)24 is the first Breuil–Kisin twist.

If (φ,Γ)(\varphi,\Gamma)25 is a rigid smooth morphism of relative dimension (φ,Γ)(\varphi,\Gamma)26, then the induced morphism

(φ,Γ)(\varphi,\Gamma)27

is cohomologically smooth and satisfies

(φ,Γ)(\varphi,\Gamma)28

Combining this with the identification of twistor cohomology with pro-étale cohomology yields Poincaré duality for geometric and arithmetic pro-étale (φ,Γ)(\varphi,\Gamma)29-cohomology. The dualizing shift and twist are not added externally; they are intrinsic to the geometry of the degree-(φ,Γ)(\varphi,\Gamma)30 divisor stack.

The same formalism provides a setting for coefficients. Although the paper does not develop a complete six-functor theory for all prismatizations, it relates the (φ,Γ)(\varphi,\Gamma)31-adic Riemann–Hilbert functor to vector bundles on overconvergent and formal neighborhoods of the Hodge–Tate locus. The twistor stack therefore supports both the coefficient categories arising from Fargues–Fontaine geometry and the filtered or connection-theoretic categories arising from the cone construction.

Limitations and open questions

The results are deliberately restricted to the rational setting. Both the geometry and the coefficients are over (φ,Γ)(\varphi,\Gamma)32, and the paper does not construct the integral (φ,Γ)(\varphi,\Gamma)33-analytic prismatization. This restriction removes the convergence difficulties associated with Witt vectors, the Breuil–Kisin ideal, and extension across the characteristic-(φ,Γ)(\varphi,\Gamma)34 locus. It also means that the precise comparison with integral and sheared prismatization remains unresolved.

The paper does not develop filtered prismatization, syntomification, or other refined variants. These constructions are mentioned as related developments but are not incorporated into the main theory. In addition, the authors do not establish in general the expected presentation of the twistor stack by locally analytic vectors on an arbitrary perfectoid torsor. They prove the required comparisons in explicit cyclotomic and toric charts, and obtain general cohomological formulas, but the global geometric identification remains open.

Several hypotheses in the main comparison theorem are essential in the current arguments. Smoothness and partial properness are used to obtain the required local charts, decompletion properties, cohomological smoothness, and descent for perfect complexes. The paper does not claim that the equivalence of perfect complexes extends to arbitrary non-smooth, non-partially-proper, or nonreduced analytic spaces.

Finally, the non-split nature of the Hodge–Tate gerbe is structural. The comparison with twisted Higgs bundles is therefore weaker than a full untwisted (φ,Γ)(\varphi,\Gamma)35-adic Simpson correspondence. The specific question left open is when the Breuil–Kisin twist can be canonically trivialized, or equivalently when the twisted Higgs and (φ,Γ)(\varphi,\Gamma)36-vector-bundle descriptions can be converted into an untwisted equivalence.

Conclusion

The paper constructs a (φ,Γ)(\varphi,\Gamma)37-linear analytic prismatization by replacing (φ,Γ)(\varphi,\Gamma)38-rings with overconvergent Frobenius-equivariant thickenings of relative Fargues–Fontaine discs. Its central result identifies perfect complexes on the resulting (φ,Γ)(\varphi,\Gamma)39-adic twistor stack with perfect complexes on the relative Fargues–Fontaine curve for smooth partially proper rigid spaces. The proof is based on a geometric form of locally analytic decompletion and establishes a direct connection between prismatization, Sen theory, (φ,Γ)(\varphi,\Gamma)40-modules, and pro-étale cohomology.

The construction also supplies a unified geometric framework for the Hodge–Tate gerbe, Breuil–Kisin twists, filtered de Rham theory, and Poincaré duality. Its main unresolved issue is the integral theory, where the rational simplifications disappear and the interaction between analytic thickenings, Witt vectors, and divisor convergence must be treated directly.

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Explain it Like I'm 14

1. What is this paper about?

This paper develops a new mathematical framework called rational analytic prismatization.

The main goal is to create a bridge between several important areas of modern number theory:

  • pp-adic geometry, which studies numbers and shapes using the prime number pp;
  • prismatic cohomology, a way of collecting different kinds of information about geometric objects;
  • Fargues–Fontaine curves, special geometric spaces used in pp-adic Hodge theory;
  • pp-adic Hodge theory, which compares different ways of studying shapes defined over pp-adic numbers;
  • Sen theory, which studies how pp-adic Galois groups act on mathematical objects.

The paper focuses on geometry over the pp-adic numbers Qp\mathbb{Q}_p. Its main achievement is to construct a new space, called the pp-adic twistor stack, and show that it contains essentially the same information as the relative Fargues–Fontaine curve for a large class of rigid analytic spaces.

In simple terms, the authors build a new “translation device” between different mathematical languages and prove that the information survives the translation.

2. What questions are the authors asking?

The paper investigates several connected questions:

  1. Can prismatic geometry be extended from algebraic or formal objects to analytic spaces? Earlier versions of prismatization mainly worked with algebraic objects defined using power series and formal schemes. The authors ask whether a similar construction can be made for analytic spaces over Qp\mathbb{Q}_p.
  2. Can this construction connect analytic geometry with Fargues–Fontaine curves? These curves are central objects in modern pp-adic mathematics. The authors want to understand whether the new analytic prismatization naturally produces them.
  3. Does the new construction explain decompletion? In pp-adic mathematics, “decompletion” means recovering simpler, locally analytic information from a more complicated completed object. The authors ask whether this process has a geometric explanation.
  4. Can the construction organize pp-adic cohomology and coefficient objects? They want to see whether vector bundles, local systems, and cohomology theories can all be studied on one common geometric space.
  5. Can it provide a geometric version of Sen theory? Sen theory studies how continuous group actions become locally analytic. The paper asks whether this change can be understood as moving between two related geometric spaces.

3. How did the authors approach the problem?

The paper is mainly theoretical. It does not use experiments or numerical data. Instead, the authors define new mathematical objects and prove theorems about them.

Building a suitable kind of analytic geometry

The authors work with Gelfand stacks, which are a flexible form of analytic space. A useful analogy is:

  • an ordinary geometric space is like a map or landscape;
  • a stack is like a landscape that also remembers symmetries and different ways of describing the same point.

Gelfand stacks allow the authors to handle both ordinary analytic spaces and more complicated objects that appear in pp-adic geometry.

Defining rational δ\delta-spaces

The authors introduce rational δ\delta-spaces. These are analytic objects equipped with a version of the Frobenius map.

The Frobenius is a transformation that, roughly speaking, raises quantities to their ppth power. It is one of the most important symmetries in pp-adic mathematics.

A rational δ\delta-space contains:

  • a perfectoid space in characteristic pp;
  • an analytic thickening of a Fargues–Fontaine-type space;
  • a Frobenius operation connecting these pieces.

The word rational means that the construction uses Qp\mathbb{Q}_p-coefficients rather than integral Zp\mathbb{Z}_p-coefficients. This makes the theory technically simpler.

Defining analytic prisms and prismatization

An analytic prism is a rational δ\delta-space together with a special divisor of degree one. A divisor can be thought of as a chosen subspace inside a larger space, similar to choosing a surface inside a three-dimensional object.

Given an analytic space XX, the authors construct its analytic prismatization, written informally as X△X^\triangle.

The idea is to study all possible prisms that map into XX. This is somewhat like studying a shape by looking at all the different “probes” that can be placed inside it.

They then repeatedly apply Frobenius and take a quotient by Frobenius. The resulting object is called the pp-adic twistor stack, written as XDiv1X^{\mathrm{Div}^1}.

The word twistor comes from an analogy with twistor spaces in complex geometry. Here, it is a space that connects different pp-adic viewpoints.

Studying locally analytic vectors

A major technical tool is the study of locally analytic vectors.

Suppose a pp-adic Lie group acts on a space. A function on that space might only be continuous, or it might vary in a particularly smooth and analytic way near each point. Locally analytic vectors are the objects with this stronger type of behavior.

The authors compare:

  • objects with a continuous group action;
  • objects obtained by keeping only their locally analytic parts.

This process is called decompletion. The authors prove conditions under which decompletion does not lose important information, especially for perfect complexes.

An everyday analogy is compressing a large file into a more organized format. Normally compression might lose information, but under the paper’s hypotheses, the important mathematical information can still be recovered.

Comparing the twistor stack with the Fargues–Fontaine curve

For a smooth, partially proper rigid space XX, the authors construct a natural map

αX:FFX⋄⟶XDiv1,\alpha_X: FF_{X^\diamond} \longrightarrow X^{\mathrm{Div}^1},

where FFX⋄FF_{X^\diamond} is a relative Fargues–Fontaine curve.

They then study how mathematical objects such as vector bundles and perfect complexes move along this map.

4. What are the main results?

The twistor stack and the Fargues–Fontaine curve contain the same perfect-complex information

The central theorem says that, for a smooth partially proper rigid space XX over Qp\mathbb{Q}_p,

αX∗:Perf⁡(XDiv1)  ≃  Perf⁡(FFX⋄).\alpha_X^*: \operatorname{Perf}(X^{\mathrm{Div}^1}) \;\simeq\; \operatorname{Perf}(FF_{X^\diamond}).

Here, Perf⁡\operatorname{Perf} means the category of perfect complexes. These are well-behaved algebraic objects that can be thought of as finite-dimensional, nicely controlled systems of equations or vector spaces varying over a space.

The symbol ≃\simeq means that the two categories are equivalent: although the spaces look different, they support the same kind of perfect mathematical data.

This is the paper’s most important conclusion.

The construction explains decompletion

The authors prove a general decompletion theorem for a pp-adic Lie group GG acting on a Berkovich space YY.

Under suitable assumptions, perfect complexes on the quotient by the continuous action are equivalent to perfect complexes built from locally analytic vectors. In simplified form,

Perf⁡(Y/Gcont)≃Perf⁡(Yla/Gla).\operatorname{Perf}(Y/G^{\mathrm{cont}}) \simeq \operatorname{Perf}(Y^{\mathrm{la}}/G^{\mathrm{la}}).

This gives a geometric explanation for why locally analytic vectors can replace completed objects without changing the essential perfect-complex information.

The Hodge–Tate stack behaves like a twisted version of the original space

The authors define the analytic Hodge–Tate stack XHTX^{HT}.

For a smooth rigid space XX, this stack is a gerbe over

X×QpHT.X \times \mathbb{Q}_p^{HT}.

A gerbe is a space that looks locally like another space but may be globally twisted by hidden symmetry. An analogy would be a collection of identical-looking puzzle pieces that cannot be assembled into one globally consistent picture without making extra choices.

This result connects the Hodge–Tate stack with:

  • twisted Higgs bundles;
  • the Faltings extension;
  • the pp-adic Simpson correspondence;
  • Sen operators.

The twistor stack computes pro-étale cohomology

For a smooth partially proper rigid space XX, the authors obtain

Γ(XDiv1,O)≃RΓproeˊt(X,Qp).\Gamma(X^{\mathrm{Div}^1},\mathcal{O}) \simeq R\Gamma_{\mathrm{pro\acute{e}t}}(X,\mathbb{Q}_p).

In plain language, the cohomology of the structure sheaf on the pp-adic twistor stack agrees with the pro-étale Qp\mathbb{Q}_p-cohomology of XX.

Cohomology measures hidden features of a space, such as holes, twists, or ways of gluing local pieces together. Thus, the twistor stack gives a geometric home for important pp-adic cohomology information.

The paper proves a form of Poincaré duality

The authors also prove cohomological smoothness and a version of Poincaré duality.

Poincaré duality is a principle saying that cohomology in one degree is related to cohomology in a complementary degree. It is similar to how the holes of different dimensions in a sphere are related to one another.

In this setting, the duality includes a pp-adic twist and a degree shift. For a smooth map of relative dimension dd, the dualizing object has the form

O{d}[2d].\mathcal{O}\{d\}[2d].

The notation represents a pp-adic version of changing weights and shifting degrees.

5. Why are these results important?

The paper brings several theories into one framework.

The main relationships can be summarized as follows:

Mathematical viewpoint What it helps describe
Analytic prismatization A geometric version of prismatic ideas
Fargues–Fontaine curves (φ,Γ)(\varphi,\Gamma)-modules and pp-adic Hodge theory
Hodge–Tate stack Twisted Higgs-type information
Locally analytic vectors Decompletion and Sen theory
Twistor stack A bridge between these viewpoints
Pro-étale cohomology pp-adic topological information

The result is important because mathematicians often have several different ways to describe the same object. One description may be good for calculating, while another may make hidden structures easier to see.

This paper shows that the pp-adic twistor stack can act as a common meeting point for these descriptions.

6. Possible impact of the research

The work could have several long-term effects.

First, it may provide a more geometric explanation of pp-adic Hodge theory. Instead of treating different cohomology theories as separate tools, mathematicians may be able to see them as different views of one larger geometric object.

Second, it may help organize pp-adic local systems, (φ,Γ)(\varphi,\Gamma)-modules, Higgs bundles, and related coefficient systems. This could make it easier to move between these theories.

Third, the decompletion results may clarify parts of geometric Sen theory and the pp-adic Simpson correspondence, which connect group actions, differential equations, and vector bundles.

Fourth, the cohomological results may lead to new versions of duality, Riemann–Hilbert correspondences, and six-functor formalisms in pp-adic geometry.

The authors also clearly state the limits of the paper. Their theory is currently developed with Qp\mathbb{Q}_p-coefficients. They do not yet give the full integral theory with Zp\mathbb{Z}_p-coefficients, nor do they develop all refined versions of prismatization.

Simple conclusion

The paper creates a new pp-adic geometric space called the twistor stack. It shows that this space and the Fargues–Fontaine curve describe the same well-behaved mathematical objects, even though they are built in different ways.

The deeper message is that several difficult theories—prismatic cohomology, pp-adic Hodge theory, Sen theory, and pro-étale cohomology—are not isolated subjects. They can be connected through one common geometric framework. This may give mathematicians a clearer way to understand and calculate with complicated pp-adic objects in the future.

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Knowledge Gaps

Knowledge gaps, limitations, and open questions

  • Integral prismatization is not developed. The constructions are restricted to Qp\mathbb{Q}_p-coefficients and rational geometry; an integral theory over Zp\mathbb{Z}_p—including the required Witt-vector and nilpotent-thickening machinery—remains open.
  • The relationship with sheared prismatization is not established. The paper identifies analytic prismatization as being conceptually closer to sheared prismatization but does not provide a precise comparison or equivalence.
  • Characteristic-pp analytic geometry is largely unexplored. The rational theory for analytic spaces in characteristic pp, and its relation to rigid cohomology of schemes over characteristic pp, is deferred to future work.
  • Filtered and syntomic refinements are absent. The paper does not construct or analyze filtered prismatization, syntomification, Nygaard-type structures, or other refined variants within the analytic setting.
  • The scope of the geometric category is limited. The main prismatization construction is formulated for Berkovich spaces, with only a conditional extension to certain †\dagger-cohesive Gelfand stacks; a general theory for analytic stacks, singular spaces, and broader derived Gelfand stacks is not provided.
  • The dependence on the auxiliary Gelfand/arc formalism is not fully clarified. The paper does not determine how its constructions compare with alternative analytic, diamond, condensed, or v-stack formulations, nor whether the resulting stacks are independent of the chosen presentation.
  • The prismatization of non-reduced spaces remains poorly understood. The paper explicitly notes that twistor-stack cohomology and pro-étale Qp\mathbb{Q}_p-cohomology differ for arbitrary non-reduced Berkovich spaces, but does not characterize this discrepancy or identify the correct cohomological theory for such spaces.
  • The equivalence on perfect complexes has restrictive hypotheses. The main equivalence is proved for smooth, partially proper rigid spaces, leaving open whether analogous results hold for non-smooth, non-partially-proper, non-quasi-compact, or singular spaces.
  • The extension from perfect complexes to larger coefficient categories is incomplete. It remains unclear whether the equivalence between XDiv1X^{Div^1} and FFX⋄FF_{X^\diamond} extends to all quasi-coherent complexes, ind-coherent objects, coherent sheaves, or suitable derived categories beyond dualizable/perfect objects.
  • A full six-functor formalism is not established. Although smooth pullbacks, dualizing complexes, and Poincaré duality are proved, the paper does not construct the complete six operations, including general f!f_!, f!f^!, tensor, internal Hom, and base-change theorems for arbitrary morphisms.
  • Functoriality under general morphisms is unresolved. The behavior of analytic prismatization and twistor stacks under proper, étale, open, closed, and non-smooth morphisms is not systematically established.
  • The global locally analytic quotient description is conjectural in general. The authors do not know whether, for a general perfectoid pro-étale torsor X~⋄→X⋄\widetilde{X}^{\diamond}\to X^\diamond, one has an isomorphism

XDiv1≃FFX~la/Gla.X^{Div^1}\simeq FF_{\widetilde{X}^{la}/G^{la}}.

Only related statements about global sections and certain Hodge–Tate cases are proved.

  • The hypotheses ensuring local analytic decompletion are not fully characterized. The strong decompletion conditions used to obtain equivalences of perfect complexes are technical and verified only in selected examples; intrinsic geometric or representation-theoretic criteria for these conditions are not known.
  • Higher locally analytic vectors are not understood in general. The vanishing of higher locally analytic vectors is assumed or verified in particular settings, but a general theory describing when higher derived locally analytic vectors vanish is lacking.
  • The relation between continuous and locally analytic quotient stacks needs broader validation. The comparison between Y/GcontY/G^{\mathrm{cont}} and Yla/GlaY^{la}/G^{la} is developed under strong assumptions, leaving open its behavior for non-affinoid spaces, non-compact groups, and more general pp-adic Lie actions.
  • The full geometric Sen theory is not developed. The paper gives a stack-theoretic interpretation of Sen operators and decompletion, but does not establish a comprehensive geometric Sen correspondence for arbitrary pp-adic Lie groups, coefficients, or families.
  • The non-abelian pp-adic Hodge-theoretic consequences remain incomplete. The relationship between vector bundles on the twistor stack, (φ,Γ)(\varphi,\Gamma)-modules, pro-étale local systems, and pp-adic Simpson objects is described as a weak form of the correspondence; criteria for untwisting the gerbe and recovering a full pp-adic Simpson correspondence are not given.
  • The Hodge–Tate gerbe is not generally shown to split. The paper identifies XHTX^{HT} as a gerbe banded by the twisted tangent bundle, but does not determine when this gerbe is neutral, how splittings are classified, or what geometric conditions guarantee a splitting.
  • The relation to the Faltings extension is not fully generalized. The Faltings extension controls the Hodge–Tate neighborhood in the settings considered, but its precise relative, integral, singular, and non-proper generalizations remain open.
  • Relative base-change compatibility is incomplete. A relative version over a perfectoid base is mentioned, but systematic compatibility with arbitrary base change, descent, and families is not established.
  • The theory of coefficients beyond Qp\mathbb{Q}_p is missing. The paper does not treat integral coefficients, torsion coefficients, general pp-adic coefficient rings, or ℓ\ell-adic analogues, so the extent to which the constructions depend essentially on rational coefficients remains unresolved.
  • Comparison with other approaches to analytic thickenings is incomplete. No precise comparison is made with inscribed v-sheaves, the Simpson gerbe formalism, or other theories of finite-flat or nilpotent thickenings of Fargues–Fontaine curves.
  • Motivic and realization-theoretic properties are not addressed. Although the paper discusses analogies with Hyodo–Kato and Betti theories, it does not establish motivic functoriality, realization maps, or a universal cohomological characterization of the twistor stack.
  • The behavior of the construction under singularities and derived structures is unclear. The paper focuses on smooth spaces in its main applications and does not determine how singularities, derived intersections, or nontrivial derived nilpotents affect the geometry and cohomology of analytic prismatizations.
  • The geometric meaning of the radius map and its fibers is incomplete. The generic Hyodo–Kato identification and Hodge–Tate behavior are described, but a detailed classification of intermediate-radius fibers and their associated cohomological or differential structures is not provided.
  • Explicit calculations are limited to basic examples. Beyond GSpec(Qp)GSpec(\mathbb{Q}_p), the rigid multiplicative group, and affine-space-type charts, there is no broad catalogue of explicit prismatizations, twistor stacks, or coefficient categories for important classes such as curves, abelian varieties, or general proper rigid varieties.
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Practical Applications

Immediate Applications

  • Computational framework for relative pp-adic Hodge theory — academia and mathematical software.
    • pp-adic étale and pro-étale cohomology;
    • de Rham and Hyodo–Kato-type constructions;
    • Fargues–Fontaine curves;
    • Hodge–Tate and Sen-theoretic structures;
    • relative (φ,Γ)(\varphi,\Gamma)-modules.
    • A practical output would be a formalized library in Lean, Coq, or a specialized computer-algebra system implementing rational δ\delta-spaces, Frobenius quotients, and perfect-complex pullbacks. This is deployable as a research infrastructure project, although the underlying mathematics remains highly specialized.
  • A geometric workflow for computing pro-étale QpQ_p-cohomology — arithmetic geometry and computational number theory. For a smooth, partially proper rigid space XX, the paper identifies

Γ(XDiv1,O)≃Γproeˊt(X,Qp).\Gamma(X^{Div^1},\mathcal O)\simeq \Gamma_{\mathrm{pro\acute et}}(X,\mathbb Q_p).

Thus, instead of computing pro-étale cohomology directly, one can: 1. construct the pp-adic twistor stack; 2. replace it by the relative Fargues–Fontaine geometry FFX⋄FF_{X^\diamond}; 3. compute coherent or perfect-complex cohomology there. This suggests algorithms and symbolic workflows for explicit rigid spaces, especially tori, affinoid charts, and spaces admitting perfectoid covers.

  • Geometric organization of (φ,Γ)(\varphi,\Gamma)-modules — pp-adic representation theory. The equivalence

P(XDiv1)≃P(FFX⋄)P(X^{Div^1})\simeq P(FF_{X^\diamond})

allows relative (φ,Γ)(\varphi,\Gamma)-modules to be represented as perfect complexes on the pp-adic twistor stack. In concrete terms, this can improve the organization of: - pp-adic Galois representations; - families of pp-adic representations over rigid spaces; - Robba-ring modules; - cohomological invariants of relative representations. For X=Spa⁡(K)X=\operatorname{Spa}(K), with K/QpK/\mathbb Q_p finite, the result gives a geometric interpretation of decompletion for (φ,Γ)(\varphi,\Gamma)-modules over the Robba ring.

  • Locally analytic decompletion of equivariant analytic data — representation theory and pp-adic functional analysis. The locally analytic vectors theorem gives a systematic method for replacing continuous GG-actions by locally analytic GlaG^{la}-actions when GG is a pp-adic Lie group. Under the paper’s decompletion hypotheses, perfect complexes on

Y/GcontY/G^{\mathrm{cont}}

can be recovered from perfect complexes on

Yla/Gla.Y^{la}/G^{la}.

This can be used immediately in research workflows involving: - locally analytic representations; - completed cohomology; - pp-adic automorphic forms; - analytic vectors in Banach representations; - equivariant sheaves on perfectoid covers.

  • Geometric interpretation of Sen operators and canonical Higgs fields — pp-adic Hodge theory.
    • extracting Sen operators from torsors;
    • interpreting geometric Sen theory via cotangent complexes;
    • studying twisted Higgs bundles;
    • comparing vv-vector bundles with Hodge–Tate data.
    • The construction is immediately usable conceptually and in explicit examples, but not yet a general-purpose computational tool.
  • Poincaré duality for pro-étale cohomology — arithmetic geometry. The cohomological smoothness results provide a geometric explanation for dualizing complexes of the form

O{d}[2d]\mathcal O\{d\}[2d]

for relative dimension dd. This can be incorporated into proofs and calculations of: - geometric pro-étale Poincaré duality; - arithmetic pro-étale duality; - duality for families of smooth rigid spaces; - compactly supported cohomology and trace maps. The practical benefit is a reusable geometric formalism rather than a new numerical algorithm.

  • A reusable cyclotomic and toric local model — education, research, and software testing.
    • validate implementations of analytic prismatization;
    • test locally analytic quotient constructions;
    • illustrate the relation between continuous and locally analytic vectors;
    • build examples for graduate courses or research seminars in pp-adic geometry.
    • The cyclotomic prism and prismatic logarithm are particularly suitable as benchmark examples.
  • Interpretation of pp-adic Riemann–Hilbert constructions — non-abelian pp-adic Hodge theory.
    • pp-adic local systems;
    • Higgs-type objects;
    • vector bundles on the twistor stack;
    • analytic DD-modules on Fargues–Fontaine-type spaces.
    • This can guide the construction and classification of coefficient objects in current pp-adic Hodge-theoretic research.

Long-Term Applications

  • A general-purpose pp-adic Hodge-theoretic coefficient platform — healthcare, finance, and other applied sectors indirectly.
    • cryptographic parameter spaces;
    • arithmetic statistics;
    • pp-adic dynamical systems;
    • formal verification of number-theoretic protocols;
    • mathematical models involving pp-adic differential equations.
    • Such applications would require substantial translation from abstract rigid geometry into domain-specific algorithms, as well as efficient implementations of derived and stack-theoretic operations.
  • Algorithms for families of pp-adic Galois representations — computational number theory and arithmetic geometry.
    • construction of relative (φ,Γ)(\varphi,\Gamma)-modules;
    • comparison between étale, de Rham, crystalline, and Hodge–Tate realizations;
    • computation of cohomology in parameterized families;
    • detection of decompletion and local analyticity conditions.
    • Dependencies include effective descriptions of X⋄X^\diamond, computable affinoid coverings, algorithms for locally analytic vectors, and manageable models for derived categories.
  • A geometric form of the pp-adic Simpson correspondence — non-abelian geometry. The Hodge–Tate gerbe and its twisted Higgs-bundle interpretation suggest a route toward a broader pp-adic Simpson correspondence. A mature theory could provide a pipeline

$\text{%%%%50%%%%-adic local systems} \longleftrightarrow \text{relative %%%%51%%%%-modules} \longleftrightarrow \text{twisted Higgs bundles}.$

Potential outputs include moduli spaces of pp-adic local systems, deformation theories, and geometric classifications of non-abelian pp-adic representations. Major dependencies are the resolution of twisting and integrality issues, together with stronger conditions under which the relevant gerbes can be split.

  • Integral analytic prismatization over Zp\mathbb Z_p — arithmetic geometry and mixed-characteristic applications.
    • integral pp-adic Hodge theory;
    • crystalline and prismatic cohomology with torsion;
    • integral comparison theorems;
    • mod-pp and mixed-characteristic deformation problems;
    • integral families of Galois representations.
    • This requires controlling Witt vectors, torsion, nilpotent thickenings, and the difference between analytic and perfect analytic prismatization.
  • Six-functor formalisms and derived coefficient systems — advanced mathematical infrastructure.
    • pullback, pushforward, and exceptional functors;
    • compactly supported pp-adic cohomology;
    • base-change theorems;
    • intersection-style constructions for analytic stacks;
    • functorial operations on pp-adic local systems and (φ,Γ)(\varphi,\Gamma)-modules.
    • Feasibility depends on extending the current results beyond perfect complexes and establishing robust finiteness, descent, and base-change theorems.
  • Non-abelian and locally analytic pp-adic Langlands programs — representation theory.
    • moduli of GlaG^{la}-torsors;
    • geometric parameter spaces for locally analytic representations;
    • cohomological constructions of pp-adic automorphic representations;
    • geometric Sen operators for non-abelian families.
    • This is speculative and depends on extending decompletion results to broader classes of groups, spaces, and coefficient categories.
  • Relative and family-valued pp-adic Hodge theory — moduli and arithmetic geometry.
    • eigenvarieties and eigenfamilies;
    • deformation spaces of Galois representations;
    • moduli of pp-adic local systems;
    • variation of Hodge–Tate weights and Sen operators;
    • parameterized Riemann–Hilbert correspondences.
    • Required developments include stronger relative descent, effective control of perfectoid bases, and compatibility with integral and filtered structures.
  • Extension to characteristic pp and rigid cohomology — arithmetic geometry.
    • rigid cohomology of varieties over characteristic pp;
    • Frobenius modules and slope decompositions;
    • comparisons between characteristic-pp and mixed-characteristic realizations.
    • This depends on constructing the appropriate characteristic-pp analytic theory and proving comparison theorems.
  • Formalized and scalable computational geometry for pp-adic stacks — software engineering.
    • Prismatization(X);
    • TwistorStack(X);
    • HodgeTateStack(X);
    • LocallyAnalyticVectors(Y, G);
    • PerfectComplexes(FF_X).
    • Such tooling would require finite or symbolic representations of Berkovich spaces, perfectoid covers, Frobenius actions, derived sheaves, and quotient stacks. Without these advances, the theory remains primarily a conceptual and research-level framework rather than an industrial software technology.
  • Applications to ordinary daily life — currently none directly. The findings do not presently yield deployable consumer applications, medical procedures, policy instruments, or household workflows. Any eventual practical impact would most plausibly be indirect, through advances in arithmetic geometry, cryptography, formal verification, or computational number theory. The main feasibility constraints are the highly abstract mathematical setting, the restriction to smooth partially proper rigid spaces in key theorems, and the current absence of efficient computational realizations.
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Glossary

  • Analytic de Rham stack: A stack encoding de Rham-theoretic information in an analytic or derived-geometric setting. “Its right adjoint, the analytic de Rham stack functor (−)dR(-)^dR, is especially interesting”
  • Analytic prismatization: A geometric construction that represents prismatic cohomological data for analytic spaces. “We develop a general theory of analytic prismatization with QpQ_p-coefficients in rigid analytic geometry over QpQ_p.”
  • Arc-stack: A stack associated with the arc or perfectoidization perspective on non-Archimedean geometry. “Any Gelfand stack gives rise to an arc-stack over M(Qp)M(Q_p)”
  • Berkovich space: A type of non-Archimedean analytic space defined using multiplicative seminorms. “X\mathcal{X} is a Berkovich space with an endomorphism φX\varphi_\mathcal{X}”
  • Breuil–Kisin twist: A pp-adic Hodge-theoretic twist associated with the cyclotomic character and Breuil–Kisin modules. “where {1}\{1\} refers to the Breuil--Kisin twist on QpHTQ_p^HT”
  • Cartier divisor: A codimension-one geometric subobject locally defined by a single non-zero-divisor equation. “A (derived) Cartier divisor (s ⁣:L→OX)(s\colon \mathcal{L}\to \mathcal{O}_{\mathcal{X}}) on X\mathcal{X} is of degree $1$”
  • Cohomologically smooth: A property ensuring a theory of smooth pullbacks and dualizing complexes for derived or stacky cohomology. “The morphism Div1→GSpec(Qp)Div^1\to GSpec(Q_p) is prim with trivial codualizing sheaf and cohomologically smooth”
  • Codualizing complex: A complex governing the duality behavior of a morphism in derived geometry. “The morphism αX ⁣:FFX⋄→XDiv1\alpha_X \colon FF_{X^\diamond}\to X^{Div^1} is prim with trivial codualizing complex”
  • Condensed anima: A homotopy-theoretic form of condensed mathematical objects, used here to model underlying topological information. “the perfectoidization functor is really taking the ‘underlying topological space’ (or, rather, condensed anima)”
  • Crystalline cohomology: A pp-adic cohomology theory for schemes in characteristic pp that captures infinitesimal structure. “crystalline cohomology (of the special fiber)”
  • δ\delta-ring: A ring equipped with an arithmetic Frobenius lift encoded by a δ\delta-operation. “This relies classically on the key definitions of δ\delta-rings and prisms.”
  • δ\delta-space: A geometric object modeled on a Frobenius-equipped analytic space and a punctured Fargues–Fontaine disc. “A rational δ\delta-space is a triple”
  • Decompletion: The process of recovering locally analytic or finite-level objects from completed pp-adic representations or modules. “we explain how this equivalence conceptualizes the decompletion techniques”
  • Derived Berkovich space: A derived enhancement of a Berkovich analytic space. “the category of derived Berkovich spaces over QpQ_p”
  • Derived Cartier divisor: A Cartier divisor defined in a derived geometric setting, allowing derived structure in its defining map. “A (derived) Cartier divisor (s ⁣:L→OX)(s\colon \mathcal{L}\to \mathcal{O}_{\mathcal{X}})”
  • Derived category: A category that retains chain-complex and homological information rather than merely morphisms of objects. “Let Rla∈CAlg⁡(D⁡(BGla))R^{la}\in \operatorname{CAlg}(\operatorname{D}(BG^{la})) denote the Gelfand algebra”
  • Diamond: A geometric object in pp-adic geometry obtained by sheafifying perfectoid spaces in the pro-étale topology. “inducing an isomorphism of diamonds DS×,⋄→∼X⋄D_S^{\times,\diamond}\xrightarrow{\sim}\mathcal{X}^{\diamond}”
  • Dualizable object: An object in a monoidal category possessing a categorical dual, analogous to a finite-dimensional vector space. “induces an equivalence of dualizable objects”
  • Fargues–Fontaine curve: A pp-adic geometric curve constructed from a perfectoid space and used to study pp-adic representations. “The quotient of the perfect δ\delta-space associated to TT by Frobenius will be denoted FFTFF_T”
  • Fargues–Fontaine disc: The punctured or unpunctured analytic space underlying the construction of a Fargues–Fontaine curve. “occasionally call them the (punctured) Fargues--Fontaine disc relative to SS.”
  • Faltings’ extension: An extension encoding the relationship between differentials and pp-adic Hodge–Tate structures. “the Hodge--Tate stack, and even the overconvergent neighborhood of XHTX^HT inside XDiv1X^{Div^1}, is controlled by a familiar object in pp-adic Hodge theory: the Faltings' extension of XX.”
  • Frobenius: An endomorphism acting as a pp-power map or its geometric analogue in characteristic pp. “called ‘the Frobenius of the δ\delta-space’”
  • Gelfand ring: A solid QpQ_p-algebra used as the foundational ring object in Gelfand analytic geometry. “The category of Gelfand rings is a subcategory of the category of solid QpQ_p-algebras”
  • Gelfand stack: A sheaf-like geometric object on Gelfand rings for the descendable topology. “a Gelfand stack is (in first approximation) a sheaf of anima on #1GelfRing\#1{GelfRing}”
  • Gerbe: A stack locally equivalent to a classifying stack, often encoding twisted forms or torsors. “πX\pi_X realizes XHTX^HT as a gerbe over X×QpHTX\times Q_p^HT”
  • Hodge–Tate stack: The Cartier-divisor locus of analytic prismatization associated with Hodge–Tate phenomena. “The analytic Hodge-Tate stack of XX is the Gelfand stack”
  • Hyodo–Kato stack: A stack encoding Hyodo–Kato cohomological data in the analytic setting. “the quotient by Frobenius on this locus is the Hyodo--Kato stack of XX.”
  • Locally analytic vectors: Vectors in a representation whose orbit maps are locally analytic for a pp-adic Lie group. “decompletions by (derived) locally analytic vectors”
  • Nilperfectoid ring: A ring whose quotient by its dagger-nilradical is perfectoid. “a Gelfand ring AA is nilperfectoid if its quotient by its †\dagger-nilradical is perfectoid.”
  • Nygaard filtration: A filtration on prismatic or related cohomology reflecting Frobenius divisibility. “encoding extra structures on prismatic cohomology, such as the Nygaard filtration or the Frobenius.”
  • Perfect complex: A derived complex locally quasi-isomorphic to a bounded complex of finite projective modules. “We prove an equivalence of perfect complexes”
  • Perfectoid space: A highly complete non-Archimedean space with Frobenius-surjective tilt, central to modern pp-adic geometry. “The perfectoid space T♭T^{\flat} over FpF_p is the tilt of TuT^u”
  • Perfectoidization: A functor replacing an analytic object by its associated perfectoid or diamond object. “The reader can think of the perfectoidization of a Gelfand stack as analogous to taking the topological manifold underlying a complex manifold.”
  • Prism: A pair consisting of a δ\delta-ring and a distinguished ideal satisfying conditions that support prismatic cohomology. “the key definitions of δ\delta-rings and prisms.”
  • Prismatic cohomology: A pp-adic cohomology theory unifying de Rham, crystalline, and étale cohomologies. “Prismatic cohomology has been defined in \cite{bhatt_scholze_prisms_and_prismatic_cohomology} as a new cohomology theory for pp-adic formal schemes”
  • Prismatization: A stack-valued construction whose coherent cohomology recovers prismatic cohomology. “the coherent cohomology of the stack X∧X^\wedge recovers the prismatic cohomology of XX.”
  • Pro-étale cohomology: Cohomology computed on the pro-étale site, particularly suited to perfectoid and rigid-analytic geometry. “the pp-adic twistor stack geometrizes pro-étale QpQ_p-cohomology”
  • Quasi-coherent sheaf: A sheaf of modules locally presented by modules over the structure sheaf, generalized to stacks and derived spaces. “the pullback αX∗\alpha_X^* on quasi-coherent sheaves is fully faithful”
  • Rigid analytic space: A non-Archimedean analytic space defined using admissible affinoid coverings. “for any smooth partially proper rigid analytic variety.”
  • Sen theory: The theory relating pp-adic Galois representations to locally analytic group actions and Sen operators. “which appear in (geometric) Sen theory.”
  • Sheafification: The process of converting a presheaf into a sheaf satisfying descent. “the sheafification in Gelfand stacks”
  • Solid algebra: A generalized topological algebraic object designed to encode completed tensor products and analytic phenomena. “The category of Gelfand rings is a subcategory of the category of solid QpQ_p-algebras”
  • Spectral stack: A stack built from ring spectra or derived rings, retaining homotopical information. “The realization functor on δ\delta-stacks is the colimit preserving functor”
  • Syntomification: A refined prismatic construction incorporating syntomic or related filtration data. “such as the filtered prismatization or the syntomification.”
  • Tilt: The characteristic-pp perfectoid space or ring associated with a mixed-characteristic perfectoid object. “The perfectoid space T♭T^{\flat} over FpF_p is the tilt of TuT^u.”
  • Twistor stack: A stack interpolating between Hodge–Tate, de Rham, and related pp-adic structures. “the pp-adic twistor stack of XX”
  • Uniform completion: A completion of a ring with respect to its uniform analytic structure, producing a Banach algebra. “any Gelfand ring AA admits a uniform completion AuA^u”
  • v-vector bundle: A vector-bundle-like object on a perfectoid or diamond space in the v-topology. “an equivalence between twisted Higgs bundles ... and v-vector bundles”
  • Witt vectors: A construction associating a characteristic-zero ring to a characteristic-pp ring while retaining Frobenius information. “the Witt vectors functor R↦W(R)R\mapsto W(R) on pp-nilpotent rings.”
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  1. Rational analytic prismatization (preprint) (111 points, 13 comments)