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Homological Detection by Perfectoid Algebras

This paper establishes homological characterizations of modules and local rings in mixed characteristic using perfectoid algebras, providing analogues to known positive-characteristic results based on Frobenius morphisms.

Original authors: Mohsen Asgharzadeh, Ryo Ishizuka

Published 2026-07-21

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CC BY 4.0

Original authors: Mohsen Asgharzadeh, Ryo Ishizuka

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the shape of a mysterious, invisible city. In mathematics, this "city" is a structure called a ring, which is a set of numbers that follow specific rules for adding and multiplying. Some cities are smooth and orderly (called "regular"), some have sharp, jagged corners (called "singularities"), and some are built with a special kind of symmetry (called "Gorenstein"). For decades, mathematicians have had a magical flashlight to inspect these cities, but it only worked in one specific neighborhood: the world of positive characteristic. In this neighborhood, the flashlight was a tool called the Frobenius morphism, which acts like a special lens that reveals the true shape of the city by squaring, cubing, or raising numbers to the $p$ -th power.

However, there is a much larger, more complex neighborhood called mixed characteristic, where the rules of the city are a bit messier (involving prime numbers like 2, 3, or 5 in a different way). For a long time, the old flashlight didn't work here. Then, a new, super-powerful tool arrived: perfectoid algebras. Think of these as a "super-lens" that can see through the fog of mixed characteristic, acting like a universal translator that turns messy, complex structures into something as clean and perfect as the old neighborhoods. The big question mathematicians were asking was: "If this new lens can tell us if a city is smooth (regular), can it also tell us if the city has other hidden properties, like how many 'holes' it has or how symmetric it is?"

This paper, written by Mohsen Asgharzadeh and Ryo Ishizuka, answers that question with a resounding "yes." The authors show that perfectoid algebras are not just good at spotting smooth cities; they are also excellent detectives for finding injective dimensions (a measure of how "deep" or "complex" a structure is) and identifying Gorenstein rings (cities with a very specific, beautiful symmetry).

Here is how they did it. The team first established some ground rules for how these perfectoid lenses behave. They proved that if you take a perfectoid algebra and cut out a piece defined by a "radical ideal" (imagine removing a specific set of buildings that share a common root), the remaining piece doesn't get too messy. Specifically, they calculated that the "projective dimension" (a measure of how many steps it takes to build the piece from scratch) is limited to a specific number: less than or equal to $2(r + 1)$ , where $r$ is the number of generators of the ideal. They also showed that if the original lens had a finite "injective dimension" (a measure of its complexity), the cut piece would be just as simple or simpler.

With these rules in hand, the authors used the perfectoid lens to solve several long-standing puzzles.

Detecting Finite Injective Dimension: They proved that a module (a building block of the city) has a finite injective dimension if and only if a specific test involving perfectoid algebras yields zero results for all sufficiently large steps. It's like saying, "If you shine this perfectoid light on a building and it stops reflecting any 'noise' after a certain point, then the building is structurally sound."
Finding Gorenstein Rings: They showed that a local ring is Gorenstein (has that special symmetry) if and only if the perfectoid lens reveals a vanishing pattern in its mathematical "echoes" (Ext groups). This is a mixed-characteristic version of a famous result that previously only worked with the old Frobenius flashlight.
Identifying Regular Rings: Perhaps most importantly, they found that a ring is regular (perfectly smooth) if and only if there exists a perfectoid algebra that has a finite injective dimension. This is a direct mirror to a famous result by Bhatt, Iyengar, and Ma, but instead of looking at how "flat" the lens is, these authors looked at how "deep" or "injective" it is.

The paper also tackles the concept of Cohen–Macaulay modules, which are structures that are "balanced" in a specific way. The authors demonstrated that you can detect if a module is Cohen–Macaulay by checking if the perfectoid lens sees a "vanishing" of certain mathematical signals below a specific depth. If the signals disappear exactly where they should, the module is perfectly balanced.

In short, this paper proves that perfectoid algebras are a universal key. Just as the old Frobenius morphism could unlock the secrets of smoothness and symmetry in positive characteristic, these new perfectoid tools can unlock the same secrets in the more difficult mixed-characteristic world. The authors didn't just suggest this might be true; they provided rigorous mathematical proofs, establishing precise bounds and equivalences. They showed that if you have a perfectoid algebra that behaves nicely (has finite injective dimension), you can be certain the underlying ring is regular. Conversely, if the ring is regular, such an algebra must exist. This work bridges a gap in our understanding, showing that the deep, structural properties of these mathematical cities can be detected by the same powerful, modern tools, regardless of the characteristic of the world they live in.

Technical Summary: Homological Detection by Perfectoid Algebras

Problem Statement

The paper addresses the gap in understanding how perfectoid algebras, which have become fundamental in $p$ -adic geometry and mixed-characteristic singularity theory, interact with dual homological invariants. While previous work (notably Bhatt, Iyengar, and Ma) established that perfectoid algebras detect regularity via flat dimension in mixed characteristic (analogous to Kunz's and Peskine–Szpiro's results on the Frobenius morphism in positive characteristic), significantly less is known about injective dimension ( $\text{idim}_R(-)$ ) and related dual properties. The authors aim to develop a theory where perfectoid algebras detect finite injective dimension, Gorenstein properties, Cohen–Macaulayness, and regularity in mixed characteristic, providing analogues to classical results involving the Frobenius endomorphism.

Methodology

The authors employ a combination of homological algebra, spectral sequences, and the structural properties of perfectoid algebras.

Dimension Estimates for Quotients: The foundational step involves establishing bounds for the projective and injective dimensions of quotients of perfectoid algebras by radical ideals.
- They utilize the spectral sequence $E_2^{p,q} = \text{Ext}_S^p(S/J, \text{Ext}_R^q(S, M)) \Rightarrow \text{Ext}_R^{p+q}(S/J, M)$ to relate the vanishing of Ext groups over a perfectoid algebra $S$ to the vanishing over the residue field $k$ .
- They analyze the projective dimension of "perfectly principal ideals" (ideals generated by compatible systems of $p$ -power roots) using colimits and derived limits in the derived category $D(R)$ . Specifically, they show that the ideal generated by a compatible system of $p$ -power roots of a unit multiple of $p$ has projective dimension at most 1.
- By iterating these arguments over a generating set of a radical ideal, they derive an upper bound of $2(r+1)$ for the projective dimension of $A/\sqrt{I}A$ , where $I$ is generated by $r$ elements up to radical.
Dual Homological Characterizations:
- The authors leverage the Matlis duality functor to translate results between projective and injective dimensions.
- They utilize the Gorenstein projective dimension ( $\text{Gpd}_R(-)$ ) introduced by Enochs and Jenda to extend characterizations of Gorenstein rings to non-finitely generated modules.
- The proofs often rely on the fact that if a module $M$ has finite injective dimension, then $\text{Ext}_R^i(A, M) = 0$ for $i \gg 0$ for any algebra $A$ . The novelty lies in proving the converse: if this vanishing holds for a specific perfectoid algebra $A$ , then $M$ must have finite injective dimension.
Use of Perfectoid Conditions: The paper defines a specific set of conditions (Notation 3.1) for an $R$ -algebra $A$ to be "perfectoid-like" in the context of these theorems. This includes being a reduced algebra containing a compatible system of $p$ -power roots of a unit multiple of $p$ , such that the quotient by the ideal generated by these roots is perfect. This allows the authors to weaken the strict definition of perfectoid algebras while retaining the necessary homological properties.

Key Contributions and Results

1. Dimension Estimates (Section 2)

Proposition 2.3: For a reduced $R$ -algebra $A$ satisfying specific perfectoid-like conditions, and an ideal $I$ generated by $r$ elements up to radical (containing $p$ ), the projective dimension $\text{pd}_A(A/\sqrt{I}A)$ is bounded by $2(r+1)$ .
Proposition 2.5: If $A$ has finite injective dimension as an $R$ -module, then $A/\sqrt{I}A$ also has finite injective dimension, with $\text{idim}_R(A/\sqrt{I}A) \leq \text{idim}_R(A)$ .
Corollary 2.6: Establishes a vanishing criterion: if $\text{Ext}_R^i(A, M) = 0$ for a sufficiently long range of indices, then $\text{Ext}_R^i(k, M) = 0$ . This links the vanishing over the perfectoid algebra to the vanishing over the residue field.

2. Characterization of Finite Injective Dimension and Gorenstein Rings (Section 3.1)

Theorem 3.2: Provides a mixed-characteristic analogue of Herzog's criterion. For a Noetherian local ring $(R, \mathfrak{m})$ with $p \in \mathfrak{m}$ and a nonzero finitely generated or $\mathfrak{m}$ -adically complete module $M$ , the following are equivalent:
1. $M$ has finite injective dimension.
2. $\text{Ext}_R^i((R/pR)^{\text{perf}}, M) = 0$ for all $i \gg 0$ .
3. $\text{Ext}_R^i(A, M) = 0$ for all $i \gg 0$ for some perfectoid $R$ -algebra $A$ (with $\mathfrak{m}A \neq A$ ).
- Significance: This characterization is noted as new even in characteristic $p$ and applies to complete modules, not just finitely generated ones. It implies that $R$ is Cohen–Macaulay if these conditions hold.
Corollary 3.4: Characterizes Gorenstein local rings. $R$ is Gorenstein if and only if $\text{Ext}_R^i((R/pR)^{\text{perf}}, R) = 0$ for $i \gg 0$ , or equivalently, if there exists a perfectoid algebra $A$ with finite Gorenstein projective dimension. This serves as a mixed-characteristic analogue of results by Takahashi–Yoshino and Iyengar–Sather-Wagstaff.

3. Characterization of Cohen–Macaulay Modules (Section 3.2)

Theorem 3.7 & Corollary 3.10: A finitely generated (or complete) module $M$ is Cohen–Macaulay if and only if there exists a perfectoid algebra $A$ such that $\text{Ext}_R^i(A, M) = 0$ for all $i < \dim(M)$ .
Corollary 3.13: For a Cohen–Macaulay complete local ring, a torsion-free module $M$ is a finite direct sum of the canonical module if and only if $\text{Ext}_R^i(A, M) = 0$ for all $i \neq \dim(M)$ .

4. Regularity and Injective Dimensions (Section 3.3)

Theorem 3.15: Provides a dual to the Bhatt–Iyengar–Ma regularity criterion. A Noetherian local ring $R$ $R$ is regular if and only if:
1. The perfection $(R/pR)^{\text{perf}}$ has finite injective dimension.
2. There exists a perfectoid $R$ -algebra $A$ with finite injective dimension.
3. $R$ has an isolated singularity and there exists a perfectoid $A$ such that $\text{Ext}_R^i(A, A) = 0$ for $i \gg 0$ .
Corollary 3.16 & 3.18: If such a perfectoid algebra $A$ is also a big Cohen–Macaulay algebra and has finite injective dimension, then $\text{idim}_R(A) = \dim(R)$ . This establishes a dichotomy where the injective dimension is either the dimension of the ring or infinite.

Significance and Claims

The paper claims to establish a "mixed-characteristic analogue" of classical homological characterizations that rely on the Frobenius morphism in positive characteristic. By replacing the Frobenius endomorphism with perfectoid algebras (or their perfect closures), the authors demonstrate that these highly non-Noetherian objects can effectively control and detect standard homological invariants like injective dimension, Gorensteinness, and regularity.

Specifically, the authors highlight:

New Characterizations: The results provide new characterizations of finite injective dimension and Gorenstein rings, particularly for complete modules and in mixed characteristic.
Extension of Scope: The theory extends beyond finitely generated modules to include $\mathfrak{m}$ -adically complete modules and big Cohen–Macaulay modules.
Resolution of Conjectures: The work settles an implicit conjecture from previous literature regarding the injective dimension of the absolute integral closure ( $R^+$ ) in both prime and mixed characteristics, showing that if it is finite, the ring must be regular.
Duality: The paper successfully constructs a dual theory to the flat-dimension detection results of Bhatt, Iyengar, and Ma, thereby completing a broader picture of how perfectoid algebras interact with homological conjectures in commutative algebra.

The authors maintain a modest tone, noting that while the perfectoid condition can be weakened (to the conditions in Notation 3.1), the core results rely on the structural properties of these algebras to bound homological dimensions and detect vanishing Ext groups.