Rational Analytic Syntomification in p-adic Cohomology

• Rational analytic syntomification is a p-adic geometric construction that replaces traditional analytic objects with stacks encoding prismatic, Nygaard-filtered, and syntomic data.
• It employs pushouts and coequalizers to glue de Rham and Hodge–Tate realizations, enabling the recovery of syntomic complexes and related cohomological structures.
• The framework bridges p-adic geometry with crystalline local systems and admissible filtered F‑isocrystals, providing insights into duality, Chern class formalism, and vector bundle correspondences.

Rational analytic syntomification is a $p$-adic geometric construction in which a rigid-analytic or formal object is replaced by a stack $X^{\mathrm{Syn}}$ whose quasicoherent or perfect complexes encode syntomic, de Rham, Hodge–Tate, Hyodo–Kato, and étale data in a single framework. For a partially proper rigid-analytic variety $X$ over $\mathbb{Q}_p$, the rational analytic syntomification $X^{\mathrm{Syn}}$ is built from the rational analytic prismatisation $X^\Prism$ and its Nygaard-filtered refinement $X^N$, and rational analytic syntomic cohomology is defined by

$$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$

For a smooth $p$-adic formal scheme $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$, a closely related syntomic stack $X^{\mathrm{Syn}}$0 is defined as a pushout of the Nygaard-filtered prismatization along de Rham and Hodge–Tate comparison maps (Hauck, 16 Apr 2026, Pentland, 19 Oct 2025).

1. Definition and terminological scope

Two explicit constructions underlie the current meaning of syntomification. In the formal $X^{\mathrm{Syn}}$1-adic setting, the syntomic stack $X^{\mathrm{Syn}}$2 is defined as the pushout of $X^{\mathrm{Syn}}$3 along two copies of $X^{\mathrm{Syn}}$4, via the maps $X^{\mathrm{Syn}}$5 and $X^{\mathrm{Syn}}$6; equivalently,

$X^{\mathrm{Syn}}$7

In the rigid-analytic setting, the rational analytic syntomification is defined as the coequalizer

$X^{\mathrm{Syn}}$8

where the two arrows are the embeddings $X^{\mathrm{Syn}}$9 (Pentland, 19 Oct 2025, Hauck, 16 Apr 2026).

Setting	Basic construction	Structural role
Smooth $X$0-adic formal scheme $X$1	Pushout defining $X$2 from $X$3 and $X$4	Geometrizes syntomic cohomology and $X$5-gauges
Partially proper rigid-analytic $X$6	Coequalizer $X$7	Produces rational analytic syntomic cohomology

In the rigid-analytic theory, “rational” refers to working after inverting $X$8, with coefficients in $X$9, and “analytic” refers to rigid-analytic, Berkovich, and Gelfand-stack geometry. In the formal theory, syntomification is the stack-theoretic operation that adds syntomic structure to prismatic and Nygaard-filtered data, so that derived global sections recover syntomic complexes (Hauck, 16 Apr 2026, Pentland, 19 Oct 2025).

2. Prismatisation, Nygaardification, and the gluing mechanism

The rigid-analytic construction begins from the rational analytic prismatisation $\mathbb{Q}_p$0. For a totally disconnected nilperfectoid Gelfand ring $\mathbb{Q}_p$1, one has a derived Berkovich space $\mathbb{Q}_p$2 with Frobenius

$\mathbb{Q}_p$3

a map $\mathbb{Q}_p$4, and a radius map

$\mathbb{Q}_p$5

A degree-$\mathbb{Q}_p$6 Cartier divisor $\mathbb{Q}_p$7 determines the prismatisation of the base, and $\mathbb{Q}_p$8 is obtained by taking morphisms $\mathbb{Q}_p$9. The de Rham stack satisfies

$X^{\mathrm{Syn}}$0

and $X^{\mathrm{Syn}}$1 inherits both a radius map $X^{\mathrm{Syn}}$2 and a Frobenius $X^{\mathrm{Syn}}$3 with $X^{\mathrm{Syn}}$4 (Hauck, 16 Apr 2026).

Nygaardification introduces filtered structure. The base $X^{\mathrm{Syn}}$5 is defined by a pullback involving $X^{\mathrm{Syn}}$6, the overconvergent closed unit disk, and its de Rham stack. For $X^{\mathrm{Syn}}$7, the Nygaardification $X^{\mathrm{Syn}}$8 is defined using a $X^{\mathrm{Syn}}$9-nilpotent thickening $X^\Prism$0 of the degree-$X^\Prism$1 divisor $X^\Prism$2, and

$X^\Prism$3

There is a natural projection

$X^\Prism$4

and two distinguished embeddings of $X^\Prism$5 into $X^\Prism$6.

The first embedding is the de Rham copy: $X^\Prism$7 arising from the locus $X^\Prism$8, where $X^\Prism$9. The second embedding is the Hodge–Tate copy: $X^N$0 arising from the locus $X^N$1, and it satisfies

$X^N$2

Away from the de Rham and Hodge–Tate loci, the Nygaardification is explicitly a cylinder: $X^N$3 This identifies $X^N$4 with the endpoint $X^N$5 and $X^N$6 with the Frobenius-twisted endpoint $X^N$7. The syntomification $X^N$8 is then the stack obtained by identifying these two copies of $X^N$9 inside $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$0 (Hauck, 16 Apr 2026).

The formal-stack construction exhibits the same pattern in a different language. There, $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$1 is the pushout of $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$2 along two maps from $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$3, and the resulting exact triangle

$$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$4

recovers syntomic complexes such as $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$5 (Pentland, 19 Oct 2025).

3. Cohomology, duality, and Chern classes

The basic coefficient object on $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$6 is the Breuil–Kisin twist $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$7. On $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$8 it refines the pullback of the corresponding line bundle from $$R\Gamma_{\mathrm{Syn}}(X,\mathbb{Q}_p(i)) := R\Gamma(X^{\mathrm{Syn}}, \mathcal{O}\{i\}).$$9, and its pullbacks along $p$0 and $p$1 agree, so $p$2 descends to $p$3. This yields rational analytic syntomic cohomology in weight $p$4: $p$5 More generally, for $p$6,

$p$7

defines syntomic cohomology with coefficients in perfect analytic $p$8-gauges (Hauck, 16 Apr 2026).

A central structural theorem is the fiber-square description

$p$9

valid for $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$0 a Berkovich smooth derived Berkovich space over $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$1. For the unit object $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$2, this recovers the usual syntomic comparison pattern between Hyodo–Kato and filtered de Rham realizations. A second fiber-square description relates syntomic, divisor, and Hodge–Tate stacks: $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$3 This underlies the truncated comparison

$\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$4

for the unit coefficient and, more generally, for vector bundle analytic $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$5-gauges with Hodge–Tate weights $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$6 (Hauck, 16 Apr 2026).

Rational analytic syntomic cohomology satisfies Poincaré duality. For the structural map

$\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$7

the dualizing complex is

$\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$8

If $\mathrm{X}/\mathrm{Spf}\,\mathcal{O}_K$9 is smooth proper of pure relative dimension $X^{\mathrm{Syn}}$00, then

$X^{\mathrm{Syn}}$01

and for $X^{\mathrm{Syn}}$02 smooth proper of dimension $X^{\mathrm{Syn}}$03,

$X^{\mathrm{Syn}}$04

The shift $X^{\mathrm{Syn}}$05 at the base is attributed to the extra topological dimension of the Nygaard interval direction (Hauck, 16 Apr 2026).

The theory also carries a strong first Chern class formalism: $X^{\mathrm{Syn}}$06 compatible with functoriality, additivity, pullback, and tensor product. The projective bundle formula is realized by the isomorphism built from powers of $X^{\mathrm{Syn}}$07 for $X^{\mathrm{Syn}}$08 (Hauck, 16 Apr 2026).

4. Vector bundles, de Rham coefficients, and the Fargues–Fontaine curve

The coefficient theory of syntomification is controlled by the relative Fargues–Fontaine curve. If $X^{\mathrm{Syn}}$09 is smooth partially proper over $X^{\mathrm{Syn}}$10, then

$X^{\mathrm{Syn}}$11

where the right-hand side is the full subcategory of vector bundles on the relative Fargues–Fontaine curve $X^{\mathrm{Syn}}$12 that are de Rham. Under this equivalence, de Rham $X^{\mathrm{Syn}}$13-local systems on the pro-étale site embed fully faithfully into $X^{\mathrm{Syn}}$14 (Hauck, 16 Apr 2026).

This description specializes cleanly at a point. For $X^{\mathrm{Syn}}$15,

$X^{\mathrm{Syn}}$16

the category of $X^{\mathrm{Syn}}$17-equivariant de Rham vector bundles on the Fargues–Fontaine curve. For de Rham representations $X^{\mathrm{Syn}}$18,

$X^{\mathrm{Syn}}$19

and in particular

$X^{\mathrm{Syn}}$20

Thus vector bundles on $X^{\mathrm{Syn}}$21 are not auxiliary coefficients but a geometric model for classical de Rham $X^{\mathrm{Syn}}$22-adic Hodge-theoretic objects (Hauck, 16 Apr 2026).

The same perspective explains the role of the Hyodo–Kato stack

$X^{\mathrm{Syn}}$23

An object of $X^{\mathrm{Syn}}$24 is simultaneously a Hyodo–Kato object on $X^{\mathrm{Syn}}$25, a filtered de Rham object on $X^{\mathrm{Syn}}$26, and a compatible object on $X^{\mathrm{Syn}}$27. This places syntomification at the intersection of Frobenius, monodromy, filtration, and de Rham comparison data (Hauck, 16 Apr 2026).

5. Formal-scheme syntomification, crystalline local systems, and rationalization

For a smooth $X^{\mathrm{Syn}}$28-adic formal scheme $X^{\mathrm{Syn}}$29, syntomification is formulated in the language of $X^{\mathrm{Syn}}$30-gauges. The heart $X^{\mathrm{Syn}}$31 contains a reflexive subcategory $X^{\mathrm{Syn}}$32, defined by the condition that the natural map

$X^{\mathrm{Syn}}$33

is an isomorphism. The étale realization functor is symmetric monoidal, respects duals, and induces an equivalence

$X^{\mathrm{Syn}}$34

where $X^{\mathrm{Syn}}$35 is the rigid generic fiber and the target is the category of pro-étale crystalline $X^{\mathrm{Syn}}$36-local systems. More generally, if $X^{\mathrm{Syn}}$37, then every

$X^{\mathrm{Syn}}$38

is a crystalline $X^{\mathrm{Syn}}$39-local system on $X^{\mathrm{Syn}}$40 (Pentland, 19 Oct 2025).

Rationalization removes integral syntomic torsion. The kernel of

$X^{\mathrm{Syn}}$41

consists precisely of coherent $X^{\mathrm{Syn}}$42-gauges killed by a power of $X^{\mathrm{Syn}}$43. After inverting $X^{\mathrm{Syn}}$44, one obtains

$X^{\mathrm{Syn}}$45

and the essential image of

$X^{\mathrm{Syn}}$46

is the full subcategory whose cohomology sheaves are crystalline $X^{\mathrm{Syn}}$47-local systems. When $X^{\mathrm{Syn}}$48 is smooth proper, there is a $X^{\mathrm{Syn}}$49-exact symmetric monoidal equivalence

$X^{\mathrm{Syn}}$50

where the target consists of perfect complexes whose cohomology sheaves are admissible filtered $X^{\mathrm{Syn}}$51-isocrystals (Pentland, 19 Oct 2025).

These results identify syntomification as a rational bridge between integral $X^{\mathrm{Syn}}$52-gauge geometry and analytic crystalline local systems. The role of rationalization is explicit: it passes from lattices and syntomic torsion to isogeny categories and admissible filtered $X^{\mathrm{Syn}}$53-isocrystals (Pentland, 19 Oct 2025).

6. Broader rational–analytic motifs and adjacent frameworks

Two adjacent literatures isolate structural themes that illuminate the phrase “rational analytic syntomification,” even though they are not themselves $X^{\mathrm{Syn}}$54-adic syntomic theories.

For relative semi-abelian schemes over an affine complex curve $X^{\mathrm{Syn}}$55, the study of analytic and rational sections develops a direct comparison between algebraic and analytic height formalisms. The same differential $X^{\mathrm{Syn}}$56-form $X^{\mathrm{Syn}}$57 gives the Néron–Tate height of a rational section,

$X^{\mathrm{Syn}}$58

and an analytic growth functional for a holomorphic section,

$X^{\mathrm{Syn}}$59

The theory proves, among other statements, that if $X^{\mathrm{Syn}}$60, then $X^{\mathrm{Syn}}$61 must be rational, and that transcendental holomorphic sections have Zariski closures with positive-dimensional stabilizer; for strictly transcendental sections, after finite base change, the closure is a translate of an abelian subscheme by a rational section (Corvaja et al., 2020).

For discrete analytic functions on the lattice $X^{\mathrm{Syn}}$62, rationality is redefined because pointwise multiplication does not preserve discrete analyticity. The relevant product is the convolution product $X^{\mathrm{Syn}}$63, rational discrete analytic functions are realized by

$X^{\mathrm{Syn}}$64

and discrete analytic Schur multipliers admit co-isometric realizations on de Branges–Rovnyak spaces. Rationality is equivalent to several finite-dimensional conditions, including the existence of a polynomial $X^{\mathrm{Syn}}$65 such that $X^{\mathrm{Syn}}$66 is polynomial and the finite-dimensionality of

$X^{\mathrm{Syn}}$67

(Alpay et al., 2021).

These frameworks suggest a recurrent structural pattern. Analytic data are encoded by auxiliary geometric or operator-theoretic objects—cohomological stacks, kernels, or colligations—and rationality is detected by finite-dimensionality, bounded growth, or compatibility with a distinguished product. A plausible implication is that syntomification, in the narrow $X^{\mathrm{Syn}}$68-adic sense of $X^{\mathrm{Syn}}$69, belongs to a wider class of constructions in which one replaces a naive analytic object by a more structured carrier that simultaneously records analytic, algebraic, and realization-theoretic information (Corvaja et al., 2020, Alpay et al., 2021).

Within the $X^{\mathrm{Syn}}$70-adic setting, that carrier is the syntomic stack itself. Its defining gluing identifies de Rham and Hodge–Tate incarnations of prismatised geometry; its line bundles produce rational analytic syntomic cohomology; its vector bundles recover de Rham bundles on the Fargues–Fontaine curve; and its rationalized perfect complexes recover crystalline local systems and admissible filtered $X^{\mathrm{Syn}}$71-isocrystals (Hauck, 16 Apr 2026, Pentland, 19 Oct 2025).