Algebraization of absolute perfectoidization via section rings

Abstract: We construct and study a graded version of absolute perfectoidization for $G$-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization.

• The paper introduces a novel graded perfectoidization functor that is initial among derived graded complete perfectoid R-algebras.
• It demonstrates compatibility with ungraded constructions by showing that graded and p-adic completions coincide with traditional perfectoidization.
• The work algebraizes formal schemes via section rings, enabling explicit computations of mixed characteristic singularity invariants.

Algebraization of Absolute Perfectoidization via Section Rings: An Expert Analysis

Introduction and Motivation

The theory of perfectoid rings and their absolute perfectoidization represents a foundational machinery in modern arithmetic geometry, particularly in extending methods from characteristic $p$ geometry to mixed characteristic settings. While the original perfectoidization is formulated for $p$-adically complete commutative rings, recent developments—especially within prismatic cohomology—have motivated universal “absolute” versions, which admit invariants analogous to $F$-signature, Hilbert--Kunz multiplicity, and centers of $F$-purity for mixed-characteristic singularities.

The present work undertakes a substantial extension: constructing and investigating an absolute perfectoidization functor for $G$-graded adic rings, with $G$ a torsion-free abelian group, and applying this to give algebraizations of the absolute perfectoidization sheaf on formal schemes arising from projective geometry. This resolves the problem that previous incarnations of perfectoidization naturally produced only $p$-adic formal objects, and not coherent algebras on the original (algebraic) schemes themselves.

Construction of Absolute Graded Perfectoidization

For a $G$-graded adic ring $R$ with $p$-adically complete graded pieces, the authors construct an explicitly functorial assignment $p$0, the graded absolute perfectoidization, which is a limit in the $p$1-category of $p$2-graded $p$3-$p$4-algebras over the diagram of all graded perfectoid $p$5-algebras. The properties verified include:

• Universality: $p$6 is the initial object among derived $p$7-complete graded perfectoid $p$8-algebras.
• Compatibility with Ungraded Construction: The $p$9-adic derived completion of $F$0 coincides with the ungraded absolute perfectoidization $F$1 in the sense of Bhatt--Scholze.
• Graded Categorical Properties: The assignment preserves colimits, is compatible with graded base change along relatively perfect morphisms, and satisfies various descent and localization properties.

The graded structure requires new technical tools. Special attention is paid to the derived and gradedwise completions, and their interplay—a point at which the work leverages the authors' prior advancements on derived graded modules and the analysis of the forgetful and adjoint functors between graded and ungraded contexts.

Globalization and Algebraization: From Formal Schemes to Schemes

Building on the local (graded) machinery, the paper constructs a global perfectoidization sheaf $F$2 for adic formal schemes via arc-topological and prismatic techniques. This quasi-coherent $F$3-algebra is then shown, for projective-type formal schemes (notably those arising as formal completions of projective varieties along $F$4-adic thickenings), to descend to the algebraic category.

The main geometric result can be summarized as:

• Given a quasi-compact projective scheme $F$5 with an ample line bundle $F$6 and section ring $F$7, the global absolute perfectoidization $F$8 of the $F$9-adic formal completion $F$0 of $F$1 coincides (after $F$2-adic completion) with the sheaf associated to the absolute graded perfectoidization $F$3, denoted $F$4 via standard correspondences between graded modules and sheaves on $F$5.
• This algebraization is highly canonical: For $F$6 as above, there exists a quasi-coherent $F$7-algebra $F$8 on $F$9 such that its $G$0-adic completion yields $G$1. That is,

$G$2

as commutative ring objects in the derived category of quasi-coherent complexes.

A precise and functorial description of the global object in terms of the section ring is provided, relying heavily on the derived graded methods developed.

Notable Technical Assertions and Numerical Results

• The graded absolute perfectoidization functor is initial among all derived graded complete perfectoid $G$3-algebras, provided $G$4 admits a perfectoid base ring and the underlying absolute perfectoidization is concentrated in degree zero.
• Strong descent and exactness results are proved for functorial constructions (eg., the preservation of colimits by the graded perfectoidization functor, compatibility with localization and base change).
• In the global setting, the quasi-coherency of the perfectoidization sheaf is established under additional hypotheses (notably weak proregularity on the ideal of definition).

Theoretical and Practical Implications

Theoretical Implications

This work pushes the prismatic and perfectoid toolchain to a new level of granularity, allowing fine-grained invariants and operations to be defined and manipulated with respect to intrinsic gradings, such as those arising naturally from projective embeddings or representation-theoretic contexts.

The algebraization statements close an important gap: previous constructions of $G$5 for schemes $G$6 were necessarily filtered through the lens of the $G$7-adic formal completion. Now, genuinely algebraic (quasi-coherent) $G$8-algebras, linked functorially to projective data via section rings, are available.

This supplies new robust platforms for defining and studying perfectoid analogues of singularity invariants in mixed characteristic, as well as enhances the parallelism between characteristic $G$9 and mixed-characteristic singularity theory. In particular, it paves the way for local-global comparisons and finer analysis of singularities modeled on the $G$0-singularity paradigm, but in the perfectoid field.

Practical and Future Directions

• Explicit Computations and Invariants: With a functorial, algebraic description of $G$1 in terms of section rings, there is now the potential to compute invariants (eg., perfectoid signatures, Hilbert--Kunz-type numbers) in concrete cases, particularly for projective varieties with explicit coordinate descriptions.
• Singularity Theory in Mixed Characteristic: Extending the machinery of $G$2-singularity theory to mixed characteristic using these global perfectoidizations enables new structural results (such as local-global theorems) and quantitative measures of singular behavior in arithmetic geometry.
• Connections to Prismatic and Derived Deformation Theory: Since the constructions are compatible with prismatic and derived frameworks, one can expect further integration with prismatic cohomology calculations, especially as new prismatic techniques are developed for stacks and higher-dimensional singular loci.
• Generalizations and Stacky Extensions: The graded approach and corresponding algebraizations are likely adaptable to non-projective and possibly stack-theoretic settings; this would further unify the treatment of perfectoid and prismatic invariants in arithmetic geometry.

Conclusion

This paper develops a comprehensive and technically sophisticated theory of absolute perfectoidization in the graded context and demonstrates a powerful algebraization principle by linking the formal and algebraic worlds via section rings on projective-type (formal) schemes. The graded perfectoidization functor retains expected universal properties, interacts well with localization, colimits, and base change, and the algebraized global object recovers the formal perfectoidization upon $G$3-completion. This work substantially advances the prospects for effective mixed-characteristic singularity invariants and provides a robust platform for further developments in perfectoid and prismatic geometry.