A local-global correspondence for perfectoid purity

Abstract: We introduce (lim-)perfectoid splitting, which is a global variant of (lim-)perfectoid purity. Our main result establishes a correspondence between the lim-perfectoid splitting of projective schemes and the lim-perfectoid purity of their Gorenstein section rings. As an application, we construct a new supply of examples of lim-perfectoid pure rings that go beyond the previously known complete intersection or splinter-type cases.

• The paper introduces a local-global correspondence linking lim-perfectoid splitting with purity of section rings, setting necessary and sufficient criteria for mixed-characteristic schemes.
• It employs derived graded algebra methods and explicit fiber sequences to replace classical Frobenius techniques in contexts lacking a natural Frobenius endomorphism.
• The framework extends vanishing theorems and singularity theory, broadening applications to Calabi–Yau, Fano, and other projective schemes in mixed characteristic.

Local-Global Correspondence for Perfectoid Purity

Introduction and Motivation

The paper "A local-global correspondence for perfectoid purity" (2604.25265) presents a rigorous advancement in the study of singularities in mixed-characteristic algebraic geometry, formulating a theory that generalizes the well-established relationship between Frobenius splitting and $F$-singularities to the context of perfectoid techniques. The work links local properties, specifically perfectoid purity and its variants (such as lim-perfectoid purity), to global geometric structures via new splitting concepts for projective schemes.

Mixed-characteristic counterparts to $F$-singularities are challenging due to the absence of a Frobenius endomorphism. The role of the Frobenius is supplanted through maps to perfectoid rings, following the paradigm initiated by Bhatt, Scholze, and collaborators. The authors bridge global geometric and local algebraic phenomena by introducing and analyzing (lim-)perfectoid splitting conditions for projective schemes and relating them formally to purity properties of section rings.

Main Results and Structural Contributions

The authors introduce the notion of (lim-)perfectoid splitting for (formal) schemes as a categorical analogue to classical $F$-splitting. The main results provide a two-way correspondence between the purity of section rings (local, algebraic) and splitting conditions on schemes (global, geometric):

• Theorem A (Section LocalGlobal): For a flat projective scheme $X$ over a $p$-torsion-free discrete valuation ring $V$ and section ring $R$, under appropriate Cohen-Macaulay and (anti-)canonical or Calabi–Yau hypotheses, injectivity of the local cohomology morphism $H^{d+1}_{(p, R_+)}(R) \to H^{d+1}_{(p, R_+)}(R_{\perfd})$ implies $X$ is lim-perfectoid split. The converse holds in dimension at least 3 in the Calabi–Yau and Fano cases, and equivalence of perfectoid split and lim-perfectoid split is established for the complete intersection case.
• Theorem (Fiber Sequence): Exploiting derived graded algebra methods, the authors develop a homotopical fiber sequence relating completed local cohomology of the section ring's absolute perfectoidization to the global cohomology of the structure sheaf’s absolute perfectoidization with rational twists. This sequence replaces the role of classical comparison theorems in positive characteristic.
• Examples and Applications: Application of the local-global framework yields new constructions of lim-perfectoid pure rings outside the complete intersection and splinter-type loci, realized as section rings of projective Calabi–Yau or Fano lifts. The work further demonstrates that lim-perfectoid and perfectoid splitting in mixed characteristic exhibit phenomena with no direct analogues in positive characteristic.

Technical Methodology

The authors' approach combines advanced tools from derived algebraic geometry, perfectoid theory, and graded ring theory:

• They utilize the absolute graded perfectoidization functor for graded rings, controlling the local-to-global passage via the cohomology of the associated formal schemes.
• The machinery involves careful consideration of derived completions, the algebraization of perfectoidizations, and interaction with local cohomology via explicit fiber sequences.
• The proofs leverage duality techniques, comparison of ind-split morphisms in stable $\infty$-categories, and the careful analysis of graded module categories, including behavior under localization and completion.
• Essential to their arguments are new vanishing theorems, extension of Matlis/Macaulay duality to the derived context, and technical innovations for handling rational twists not present in classical positive characteristic settings.

Numerical and Structural Consequences

• The paper establishes that, for a sizable class of projective schemes, injectivity of a single local cohomology morphism precisely characterizes the global lim-perfectoid splitting property.
• The framework applies to a broad variety of examples, including K3 surfaces, abelian and Calabi–Yau varieties, and smooth Fano hypersurfaces, including instances where previously only splinter-type or complete intersection cases were understood.
• In the Gorenstein case, the condition that the degree zero part of the local cohomology transfers injectively to the absolute perfectoidization is both necessary and sufficient for lim-perfectoid purity, paralleling but sharply extending the classical $F$0-split/Calabi–Yau correspondence to mixed characteristic.

Theoretical Implications and Directions

This work substantiates lim-perfectoid splitting as the correct replacement for $F$1-splitting in mixed characteristic for both theoretical and computational purposes. The local-to-global principle implies that local computations on section rings (where effective criteria, such as the analogues of Fedder’s criterion, are known) can be transferred to global geometric properties of the scheme.

The results have the following implications:

• They enable the proof of mixed-characteristic Kodaira- and Kawamata–Viehweg type vanishing theorems using perfectoid technology, as lim-perfectoid splitting provides an entry point for such results (cf. Bhatt’s work on vanishing for lim-perfectoid split schemes).
• The classification and construction of mixed-characteristic singularities is substantially broadened beyond complete intersection and splinter cases, impacting the minimal model program and birational algebraic geometry in mixed characteristic.
• The explicit comparison criteria and fiber sequence developed here point toward future extension of singularity theory, tight closure, and birational invariants to the mixed-characteristic setting, and potential interaction with prismatic and prytopological cohomological invariants.
• The delicate behavior of the fiber sequence and the obstructions present for the converse direction underscore foundational differences between mixed and equal characteristic, suggesting lines of investigation into the structure of derived categories and their local cohomology in arithmetic contexts.

Conclusion

This paper develops a comprehensive local-global framework for perfectoid purity in mixed characteristic, establishing precise correspondences between lim-perfectoid splitting of projective schemes and purity of section rings. The results provide both foundational insights and practical criteria, as well as a potent extension of vanishing theorems and singularity theory. This work applies sophisticated derived and perfectoid techniques to resolve open questions regarding purity and singularities, and indicates rich avenues for further exploration of arithmetic geometry in mixed characteristic.