Hodge–Tateness: Patterns in p-adic and Hodge Theory

• Hodge–Tateness is a pattern where cohomology decomposes into Tate-twisted graded pieces, often resulting in diagonality or a controlled filtration structure.
• It encompasses diverse settings—from the p-adic Hodge–Tate decomposition and mixed Hodge structures to geometric degenerations with Mumford–Tate rigidity—illustrating its broad applications.
• The concept informs a range of methods including spectral sequence analysis on smooth and singular spaces, stacky and prismatic approaches, and function-field analogues.

Searching arXiv for recent and relevant papers on Hodge–Tate / Hodge–Tateness. Hodge--Tateness denotes a recurrent Hodge-theoretic pattern in which cohomology is controlled by Tate-twisted graded pieces, concentrated on the diagonal, or governed by filtrations whose graded pieces behave as Tate objects. The collected usage suggests that the term encompasses several exact settings rather than a single formal definition: the $p$-adic Hodge--Tate decomposition and its spectral sequence for proper smooth, relative, and non-smooth rigid spaces; diagonal or Hodge--Tate mixed Hodge structures in complex and mirror-symmetric geometry; Mumford--Tate-theoretic rigidity for Hodge classes in degenerations and special loci; and stacky, prismatic, and function-field avatars such as Hodge--Tate gerbes, Hodge--Tate crystals, and Hodge-Pink structures (Guo, 2019, Shamoto, 2017, Gao et al., 2023, Hartl et al., 2016).

1. Formal patterns and basic meanings

In $p$-adic Hodge theory, the basic model is the Hodge--Tate decomposition for a proper smooth variety or rigid space. In the formulation recalled by Abbes--Gros, for a proper smooth $K$-scheme $X$ one has

$$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$

equivalently the canonical $G_K$-equivariant spectral sequence

$$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$

which degenerates at $E_2$; in the absolute case the induced filtration splits, while in the relative case it generally does not (Abbes et al., 2020).

A second usage is purely Hodge-theoretic and means diagonality of the Hodge diamond. In the hyperplane-section literature, $H^*(X)$ is said to be of Hodge--Tate type when

$H^*(X)=\bigoplus_p H^{p,p}(X),$

that is, all off-diagonal Hodge numbers vanish. This is the sense in which one speaks of Hodge--Tate hyperplane sections of Grassmannians and of the implication from generic semisimplicity of even big quantum cohomology to Hodge--Tate type (Galkin et al., 1 Sep 2025).

A third usage occurs for mixed Hodge structures. In Shamoto’s rescaling-structure formalism for Landau--Ginzburg models, the Hodge--Tate condition is that

$p$0

equivalently that

$p$1

In that setting Hodge--Tateness forces equality of the numerical invariants $p$2 and $p$3 (Shamoto, 2017).

2. Hodge--Tate spectral sequences in $p$4-adic geometry

The classical spectral sequence admits several constructions. One route, emphasized in Faltings’ framework, begins with the Faltings topos and the Cartan--Leray spectral sequence for the projection $p$5 from the Faltings topos to the étale topos of an integral model. The initial terms are identified with differential forms via a Kummer-theoretic morphism

$p$6

whose degree-$p$7 part is a $p$8 map and whose kernel and cokernel are controlled in the almost sense. The abutment is then computed by Faltings’ comparison theorem, which identifies étale cohomology with cohomology of the Faltings topos up to almost isomorphism (Abbes et al., 2015).

Abbes--Gros systematize this picture in both absolute and relative form. For a projective smooth morphism $p$9 of smooth $K$0-schemes, they obtain local and global relative Hodge--Tate spectral sequences, including

$K$1

and, on the relative Faltings topos,

$K$2

The sequence $K$3 degenerates at $K$4, but the paper explicitly notes that the relative filtration need not split (Abbes et al., 2020).

A different conceptual formulation reconstructs the Hodge--Tate filtration from infinitesimal $K$5-cohomology. For a proper smooth rigid analytic variety $K$6, the filtration obtained from the relative position of the lattice

$K$7

inside

$K$8

recovers the Hodge--Tate filtration via the Bialynicki--Birula map, and the graded pieces are

$K$9

The same paper introduces a filtered refinement $X$0 of the décalage functor, proves $X$1-torsion-freeness of infinitesimal $X$2-cohomology without Conrad--Gabber spreading, and gives a conceptual equivalence between degeneration of the Hodge--Tate and Hodge--de Rham spectral sequences (Wu, 2022).

3. Singular and nonproper rigid spaces

A major extension of Hodge--Tateness replaces smooth differential forms by éh-sheafified differentials on singular rigid spaces. For a proper rigid space $X$3, where $X$4 is complete, algebraically closed, and of characteristic $X$5, one has a natural spectral sequence

$X$6

and it degenerates at $X$7. The groups $X$8 are finite-dimensional over $X$9, vanish unless $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$0, and recover the usual $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$1 in the smooth case. If $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$2 is defined over a discretely valued subfield $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$3 with perfect residue field, the decomposition becomes canonical and Galois equivariant (Guo, 2019).

The proof mechanism is explicitly geometric. The éh-topology is generated by étale coverings, universal homeomorphisms, and coverings coming from blowups along closed analytic subsets. Scholze’s pro-étale site computes $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$4-adic étale cohomology through $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$5, the v-topology provides a common comparison environment, and one obtains the éh-proét spectral sequence

$$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$6

Resolution of singularities and smooth éh-hypercovers reduce the essential statements to the smooth case, while the analytic cotangent complex and lifting to $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$7 supply the derived splitting needed for $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$8-degeneration. Over $$\widehat{H}^{n}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C \;\cong\; \bigoplus_{i=0}^n H^{n-i}(X,\Omega^i_{X/K})\otimes_K C(i-n),$$9, the same éh-theory recovers the Deligne--Du Bois package: $G_K$0 for $G_K$1 (Guo, 2019).

The nonproper rigid-analytic case exhibits qualitatively different behavior. Colmez--Nizioł formulate comparison theorems and dualities in a category of topological vector-space valued sheaves, with Banach--Colmez spaces and qBC objects replacing finite-dimensional coefficients. For the $G_K$2-adic open unit disk $G_K$3, the pro-étale cohomology already satisfies

$G_K$4

so naive finite-dimensional Hodge--Tate pictures fail. The appropriate duality is derived duality in the TVS category, and boundary cohomology and ghost-circle phenomena become indispensable (Colmez et al., 23 Jan 2026).

4. Degenerations, Mumford--Tate control, and algebraicity of Hodge classes

In degenerating complex geometry, Hodge--Tateness appears as rigidity of Hodge classes under specialization. For a projective variety $G_K$5 of dimension $G_K$6 with $G_K$7, the singular Hodge conjecture in the form used by Mertes is

$G_K$8

The decisive hypothesis is Mumford--Tate control in a smoothing family: if a simple normal crossing variety $G_K$9 is MT-smoothable of weight $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$0, the general fiber satisfies $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$1, and every irreducible component of the double locus satisfies $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$2, then $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$3 satisfies $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$4. For worse singularities, the paper reformulates the problem through an exactness statement on algebraic cohomology, called Conjecture A (Dan et al., 2023).

The same Mumford--Tate perspective governs special loci in variations of Hodge structure. If $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$5 is the period image and $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$6 is a Mumford--Tate subdomain, the special subvariety is defined by $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$7, and it is called atypical when the codimension inequality

$$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$8

is strict. Griffiths explains that the source of atypicality is excess intersection forced by integrability of two Pfaffian exterior differential systems, together with the alignment of Hodge and root-space decompositions in $$E_2^{i,j}=H^i(X,\Omega^j_{X/K})\otimes_K C(-j)\;\Rightarrow\; H^{i+j}_{\acute e t}(X_{\bar K},\mathbf Q_p)\otimes_{\mathbf Q_p} C,$$9. Under the level hypothesis $E_2$0, every positive-dimensional special subvariety is atypical (Griffiths, 1 Oct 2025).

A transcendence-theoretic variant of the same rigidity philosophy appears in the conjectural obstruction

$E_2$1

The paper on Hodge structures not coming from geometry formulates the contrapositive statement that if $E_2$2, then $E_2$3 does not come from geometry, and proves that this expectation follows from the conjunction of two conjectures: that Hodge cycles are motivated and André’s generalized Grothendieck period conjecture. This suggests that Griffiths transversality, Mumford--Tate symmetry, and transcendence degree are different expressions of the same rigidity constraint (Kreutz, 2023).

5. Diagonality, semisimplicity, and mirror-symmetric Hodge--Tateness

For smooth hyperplane sections of homogeneous varieties, Hodge--Tateness is literally diagonality of cohomology. The Grassmannian classification is sharp: $E_2$4 The same paper recalls that generic semisimplicity of $E_2$5 implies that $E_2$6 is of Hodge--Tate type, proves a practical non-semisimplicity criterion in terms of index-periodic Betti numbers, and completely classifies generic semisimplicity of $E_2$7 for Grassmannian hyperplane sections: $E_2$8 In particular, $E_2$9 yields a Hodge--Tate case with non-semisimple small quantum cohomology, so Hodge--Tateness does not imply semisimplicity (Galkin et al., 1 Sep 2025).

In Landau--Ginzburg theory, the Hodge--Tate condition is imposed on the mixed Hodge structure carried by the rescaling structure $H^*(X)$0 associated with a tame compactified model $H^*(X)$1. The comparison theorem identifies the Hodge and weight filtrations on $H^*(X)$2 with those on the relative cohomology $H^*(X)$3, so

$H^*(X)$4

This yields both the equality $H^*(X)$5 and the speciality predicted by Katzarkov--Kontsevich--Pantev. Rational elliptic surfaces and three-dimensional toric Landau--Ginzburg models are the main classes verified in the paper (Shamoto, 2017).

The diagonal pattern extends beyond complex algebraic geometry. Quasitoric orbifolds, which need not be complex or almost complex, are shown to carry a canonical Hodge structure transported from a projective toric orbifold with combinatorially equivalent polytope. The resulting Hodge numbers satisfy

$H^*(X)$6

For omnioriented quasi-SL quasitoric orbifolds, the associated orbifold Hodge numbers are invariant under crepant blowups and blowdowns (Ganguli, 2014).

6. Stacky, prismatic, and function-field avatars

In characteristic $H^*(X)$7, Bhatt--Lurie’s Hodge--Tate geometry takes the form of a gerbe. For a smooth variety $H^*(X)$8 over a perfect field of characteristic $H^*(X)$9, the fpqc-group

$H^*(X)=\bigoplus_p H^{p,p}(X),$0

bands the Hodge--Tate gerbe $H^*(X)=\bigoplus_p H^{p,p}(X),$1, and a Frobenius lift trivializes it: $H^*(X)=\bigoplus_p H^{p,p}(X),$2 The obstruction class $H^*(X)=\bigoplus_p H^{p,p}(X),$3 maps to the Frobenius-lifting obstruction $H^*(X)=\bigoplus_p H^{p,p}(X),$4, and the main theorem proves that these two classes coincide in the common target. Thus the failure of the Hodge--Tate gerbe to trivialize is exactly the failure to lift Frobenius (Yu, 2024).

On the absolute (log-)prismatic site of $H^*(X)=\bigoplus_p H^{p,p}(X),$5, Hodge--Tate crystals provide a linearized version of the same pattern. Integral and rational Hodge--Tate crystals are defined as vector bundles over $H^*(X)=\bigoplus_p H^{p,p}(X),$6 and $H^*(X)=\bigoplus_p H^{p,p}(X),$7, and evaluation on the Breuil--Kisin prism yields equivalences with modules equipped with an $H^*(X)=\bigoplus_p H^{p,p}(X),$8-small endomorphism $H^*(X)=\bigoplus_p H^{p,p}(X),$9 satisfying

$p$00

The derived global sections are computed by the two-term complex

$p$01

and $p$02 for all $p$03. Rationally, these crystals are classified by nearly Hodge--Tate or log-nearly Hodge--Tate $p$04-representations of $p$05, using a Sen theory built over the non-Galois Kummer tower (Gao et al., 2023).

Over function fields, Pink’s theory replaces ordinary Hodge structures by Hodge-Pink structures

$p$06

where $p$07 is a full-rank $p$08-lattice in $p$09. For uniformizable $p$10-motives, dual $p$11-motives, and Anderson $p$12-modules, the Hodge realization functor produces a neutral Tannakian category whose Hodge-Pink group is the analogue of a Mumford--Tate group. The main comparison theorem identifies this group with the motivic Galois group: $p$13 This is the function-field analogue of the Hodge conjecture in the paper’s sense: Hodge-Pink subobjects come from actual submotives, while $p$14-adic realizations satisfy the corresponding Tate-type comparison (Hartl et al., 2016).

Taken together, these developments show that Hodge--Tateness is not confined to a single decomposition theorem. It is a structural principle appearing as diagonality, filtration splitting, cohomological descent, rigidity under Mumford--Tate symmetry, stacky obstruction theory, prismatic linearization, and function-field Tannakian realization.