Perfectoid Pure Singularities in Mixed Characteristic

• Perfectoid pure singularities are mixed-characteristic analogues of F-pure singularities, defined via pure maps into perfectoid algebras and absolute perfectoidization.
• They extend classical F-singularity theory and interact with Du Bois and log canonical singularities upon inverting p, linking arithmetic and geometric methods.
• Computable criteria, including Fedder-style techniques and splitting-order sequences, provide explicit thresholds to assess perfectoid purity and singularity invariants.

Perfectoid pure singularities are a mixed-characteristic analogue of $F$-pure and $F$-split singularities in characteristic $p$, obtained by replacing Frobenius and perfection with perfectoid algebras and absolute perfectoidization. For a Noetherian ring with $p$ in its Jacobson radical, the basic question is whether the ring admits a pure map into a perfectoid algebra, or equivalently in a weaker canonical form whether it is pure in its absolute perfectoidization. This framework extends the characteristic-$p$ $F$-singularity package, interacts with Du Bois and log canonical singularities after inverting $p$, and is part of the broader perfectoid program in which almost purity, tilting, and perfectoid big Cohen–Macaulay methods supply mixed-characteristic replacements for Frobenius techniques (Bhatt et al., 2024, Scholze, 2011, Andreé, 2018).

1. Definitions and formalism

Let $R$ be a Noetherian ring with $p$ in its Jacobson radical. The foundational definition is that $R$ is perfectoid pure if there exists a perfectoid $F$0-algebra $F$1 such that $F$2 is pure, while $F$3 is lim-perfectoid pure if the canonical map

$F$4

is pure in $F$5. In this setting, a map in the derived category is called pure if it can be written as a filtered colimit of split maps. The distinction is therefore between an existential purity condition, using some perfectoid target, and a canonical purity condition, using the absolute perfectoidization (Bhatt et al., 2024).

The target objects are perfectoid rings in the sense recalled in mixed-characteristic singularity theory: a $F$6-adically complete ring $F$7 is perfectoid if Frobenius on $F$8 is surjective, the map $F$9 has principal kernel, and there exists $p$0 with $p$1 for some unit $p$2. For module maps, purity means injectivity after tensoring with every module. In characteristic $p$3, a ring is perfectoid iff it is perfect, so perfectoid purity becomes exactly $p$4-purity (Baily et al., 3 Apr 2025).

The absolute perfectoidization admits several equivalent descriptions. It can be written as the inverse limit over all maps from $p$5 to perfectoid rings,

$p$6

computed in $p$7, and it also satisfies

$p$8

These descriptions make precise the idea that $p$9 is the mixed-characteristic replacement for the perfect closure $p$0 in characteristic $p$1 (Bhatt et al., 2024).

2. Comparison with $p$2-purity, $p$3-injectivity, and log-canonical behavior

The current theory distinguishes four closely related singularity notions.

Notion	Definition	Characteristic-$p$4 analogue
perfectoid pure	$p$5 perfectoid $p$6 with $p$7 pure	$p$8-pure
lim-perfectoid pure	$p$9 pure in $p$0	$p$1-pure
perfectoid injective	$p$2 perfectoid $p$3 with $p$4 injective	$p$5-injective
lim-perfectoid injective	$p$6 injective	$p$7-injective

In general one has

$p$8

In quasi-Gorenstein or Gorenstein settings the converse implications hold in the corresponding pure/injective pairs, and if $p$9 is LCI then all four notions coincide. In characteristic $F$0, the theory recovers the usual equal-characteristic picture: $F$1

$F$2

This makes perfectoid purity a genuine mixed-characteristic extension of the classical $F$3-singularity package (Bhatt et al., 2024).

The LCI case is particularly rigid. If $F$4 is a complete intersection and $F$5 is a nonzerodivisor such that $F$6 is perfectoid injective, then $F$7 is perfectoid injective, hence $F$8 is perfectoid injective; because the four notions agree for LCI rings, this is an inversion-of-adjunction statement for perfectoid pure singularities. The same framework gives a basic lifting principle: if

$F$9

is a complete intersection and $p$0 is $p$1-pure, then $p$2 is perfectoid pure (Bhatt et al., 2024).

The theory also links to characteristic-zero birational singularities after inverting $p$3. If $p$4 is a lim-perfectoid injective Noetherian local ring of mixed characteristic $p$5, essentially of finite type over a mixed-characteristic DVR, then $p$6 has Du Bois singularities. If $p$7 is normal, $p$8-Gorenstein, and perfectoid pure, then $p$9 is log canonical; if the $R$0-Gorenstein index is not divisible by $R$1, then $R$2 itself is log canonical. The same paper also proves that lim-perfectoid injective rings are reduced and weakly normal (Bhatt et al., 2024).

3. Compatible ideals and centers of perfectoid purity

A more refined local theory is furnished by centers of perfectoid purity, introduced as mixed-characteristic analogues of centers of $R$3-purity and log canonical centers. For a complete Noetherian local ring $R$4, after choosing a mixed-characteristic Noether normalization $R$5 and the associated perfectoidization $R$6, an ideal $R$7 is called $R$8-compatible for a map $R$9 if $p$0, and it is uniformly perfectoid compatible if this holds for every such $p$1. A prime ideal is then called a center of perfectoid purity for the pair $p$2 or for $p$3 itself in the uniform setting (Fayolle, 22 Apr 2025).

These compatible ideals behave formally like their characteristic-$p$4 counterparts. They are closed under intersections and sums, and when the relevant map is surjective they are radical. If $p$5 is perfectoid pure, the minimal primes of a uniformly perfectoid compatible ideal are again uniformly perfectoid compatible, and there are only finitely many uniformly perfectoid compatible ideals. The conductor ideal is a nonzero uniformly perfectoid compatible ideal, so the absence of nontrivial uniformly perfectoid compatible ideals detects normality (Fayolle, 22 Apr 2025).

The same paper constructs a mixed-characteristic analogue of the splitting prime of Aberbach–Enescu. Its role is to isolate the largest center where perfectoid purity fails, and it is explicitly designed to detect perfectoid purity of the ring. This is the singularity-theoretic mechanism behind statements that perfectoid purity can be read off from compatible ideals rather than from a chosen perfectoid algebra (Fayolle, 22 Apr 2025).

The relation to birational geometry is strong. In a complete Noetherian normal quasi-Gorenstein local ring of residue characteristic $p$6, every log canonical center is uniformly perfectoid compatible, and the multiplier ideal is uniformly perfectoid compatible as well. The theory is also stable under finite étale morphisms: splitting primes, compatible cores, and test ideals along primes commute with finite étale base change in the precise sense proved in the paper (Fayolle, 22 Apr 2025).

4. Computable criteria and threshold invariants

A central development is that perfectoid purity admits Fedder-style numerical criteria. For hypersurfaces in the ambient ring

$p$7

with Frobenius lift $p$8, an element $p$9 such that $R$0 is a regular sequence, and the $R$1-operator

$R$2

one defines an inductive splitting-order sequence

$R$3

The main theorem states that if $R$4 for every $R$5, then $R$6 is perfectoid pure and

$R$7

The same work proves that if $R$8 is a regular local ring with $R$9 and $F$00 is $F$01-finite, then $F$02 is a rational number (Yoshikawa, 22 Oct 2025).

A parallel computation method uses quasi-$F$03-splitting in mixed characteristic. For a $F$04-torsion free Noetherian local $F$05-algebra $F$06, one considers Witt-vector modules $F$07, defines $F$08-quasi-$F$09-split and the quasi-$F$10-splitting height $F$11, and strengthens this to quasi-$F$12-splitting. If $F$13 is a complete intersection, quasi-$F$14-split, and $F$15, then $F$16 is perfectoid pure and

$F$17

This criterion is made explicit by a Fedder-type recursion for ideals $F$18 built from the Frobenius-lift calculus (Yoshikawa, 10 Feb 2025).

The threshold theory is already rich in explicit families. For $F$19 and $F$20, the splitting-order sequence satisfies

$F$21

hence $F$22 is perfectoid pure. In particular, for

$F$23

and $F$24, one obtains

$F$25

These are presented as new and unexpected examples of perfectoid pure singularities (Yoshikawa, 22 Oct 2025).

For lifts of rational double points, the perfectoid pure threshold becomes a highly structured discrete invariant. If

$F$26

then $F$27, $F$28 satisfies ACC, and in characteristic $F$29 it contains all reciprocals $F$30 with $F$31; moreover, every accumulation point of $F$32 is $F$33. This demonstrates that perfectoid pure thresholds are not merely ad hoc numerical devices, but form a discrete singularity spectrum analogous to threshold invariants in other singularity theories (Takamatsu et al., 26 Mar 2026).

5. Deformation, local–global correspondence, and examples

Perfectoid purity also deforms upward from characteristic $F$34 under strong hypotheses. If $F$35 is a complete local ring of mixed characteristic $F$36 and $F$37 is an $F$38-pure Gorenstein domain, then $F$39 is perfectoid pure; more generally, analytically irreducible $F$40-pure Gorenstein special fiber suffices after reduction. The proof passes through splinters: local splinters of mixed characteristic are perfectoid pure, and the deformation problem for perfectoid purity is reduced to a deformation problem for splinters via a Heitmann-style completion construction and the perfectoid $F$41-completion of $F$42 (Baily et al., 3 Apr 2025).

A global geometric counterpart is given by lim-perfectoid splitting. For a quasi-compact separated scheme $F$43, one asks whether the canonical morphism

$F$44

is ind-split on the $F$45-adic completion $F$46. On affines this matches the local notions: $F$47

$F$48

For a flat projective $F$49-scheme $F$50 with section ring

$F$51

under Cohen–Macaulay and Calabi–Yau/Fano/canonical hypotheses, injectivity of

$F$52

implies that $F$53 is lim-perfectoid split, and converses hold in the stated cases. If $F$54 is a complete intersection in projective space, then

$F$55

This is the mixed-characteristic analogue of the local–global correspondence between Frobenius splitting and $F$56-purity of section rings (Ishizuka et al., 28 Apr 2026).

The examples produced by these methods go beyond earlier complete-intersection or splinter-type constructions. Every $F$57-lift of an elliptic curve over a perfect field $F$58 is perfectoid split, so the $F$59-adic completion of its section ring is lim-perfectoid pure. The canonical $F$60-lift of an ordinary abelian variety is perfectoid split. For $F$61, every smooth quartic in $F$62 is lim-perfectoid split, and the corresponding section rings are lim-perfectoid pure; the Fermat quartic K3 surface yields lim-perfectoid pure section rings even for ample line bundles not of the form $F$63 (Ishizuka et al., 28 Apr 2026).

Other explicit families arise from the LCI and Frobenius-liftable theories. If

$F$64

has $F$65, then it is perfectoid pure. More generally, mixed-characteristic lifts of $F$66-pure complete intersections, as well as certain Frobenius-liftable quasi-Gorenstein singularities with $F$67-split special fiber, furnish concrete examples of perfectoid pure or perfectoid injective rings (Bhatt et al., 2024).

6. Relation to broader perfectoid singularity theory

Perfectoid pure singularities sit inside a wider mixed-characteristic hierarchy. A stronger notion is perfectoid BCM-regularity, defined by requiring that every map from $F$68 to a perfectoid big Cohen–Macaulay $F$69-algebra be pure. This is the mixed-characteristic analogue of strong $F$70-regularity. In graded settings with log Fano-type hypotheses, quasi-$F$71-splitting implies perfectoid BCM-regularity, and in characteristic $F$72 the converse holds in the theorem proved there (Yoshikawa, 10 Feb 2025).

The theory also admits quantitative invariants analogous to $F$73-signature and Hilbert–Kunz multiplicity. Using Bhatt–Scholze perfectoidization and Faltings’s normalized length, one defines the perfectoid signature $F$74 and perfectoid Hilbert–Kunz multiplicity $F$75. In equal characteristic $F$76 these coincide with classical $F$77-signature and Hilbert–Kunz multiplicity; in mixed characteristic,

$F$78

Moreover, positivity of perfectoid signature implies weakly BCM-regular behavior, the signature transforms under split quasi-étale maps by the same degree formula as $F$79-signature, and the theory yields finiteness of local étale fundamental groups and torsion in divisor class groups for BCM-regular rings (Cai et al., 2022).

Conceptually, this entire singularity theory depends on the foundational properties of perfectoid rings and spaces. Perfectoid geometry supplies the tilting equivalence, the almost purity theorem, and the principle that finite étale covers of perfectoid objects remain perfectoid and are almost finite étale on integral structures. In commutative algebra, perfectoid techniques also underlie the direct summand conjecture, the existence of perfectoid big Cohen–Macaulay algebras, and the first mixed-characteristic bridge from Frobenius-based singularity theory toward characteristic-zero birational singularities. Perfectoid pure singularities are one of the main local manifestations of that bridge (Scholze, 2011, Andreé, 2018).