An application of the almost purity theorem to the homological conjectures

This paper establishes the existence of big Cohen-Macaulay algebras in mixed characteristic for certain special cases by applying the almost purity theorem of Davis and Kedlaya to advance the homological conjectures.

The Gist

The Big Goal: The "Perfect House" Problem

Imagine you are an architect working in a very strange neighborhood called Commutative Algebra. In this neighborhood, buildings are called Rings, and they have specific rules about how their rooms (elements) fit together.

For a long time, mathematicians have been trying to solve a famous puzzle called the Direct Summand Conjecture. In simple terms, this conjecture asks: If you have a sturdy, well-built house (a "Regular Ring"), and you build a new, slightly more complex house attached to it (a "Module-finite Algebra"), can you always find a way to "pull" the new house back to the old one without breaking anything?

To prove this, you need to build a specific type of "super-structure" called a Big Cohen-Macaulay Algebra. Think of this super-structure as a Universal Foundation. If you can build this foundation, it proves that the rooms in your new house line up perfectly with the rooms in the old house.

The problem? This is easy to do in some neighborhoods (like those with only "positive characteristic," which is like a sunny, dry climate), but it has been incredibly hard to do in Mixed Characteristic. This is a neighborhood where the ground is wet and unstable (involving the prime number $p$, like 2, 3, or 5, mixed with the number 0).

The New Tool: The "Almost Purity" Hammer

The author, Kazuma Shimomoto, uses a powerful new tool to solve this. This tool is called the Almost Purity Theorem.

Imagine you are trying to clean a very dirty room.

• Standard Purity: You want the room to be 100% spotless. This is often impossible in the "Mixed Characteristic" neighborhood.
• Almost Purity: You accept that the room might be 99.9% clean. The dust is so fine that for all practical purposes, the room is clean.

The Almost Purity Theorem (originally developed by Davis and Kedlaya) says: If you start with a clean room and make a small, smooth extension to it (an "étale" map), the new room will also be "almost" clean.

Shimomoto's paper is about taking this "almost clean" room and using it to build the "Universal Foundation" (the Big Cohen-Macaulay Algebra) that we need to solve the main puzzle.

The Journey: Step-by-Step

Here is how the paper works, using our construction analogy:

1. The Setup (The Tricky Neighborhood)

Shimomoto focuses on a specific type of building site: a Regular Local Ring in Mixed Characteristic.

• The Twist: The building is "finite" over a regular sub-ring, but only after you ignore the number $p$ (inverting $p$).
• The Analogy: Imagine you are building a house on a foundation that is solid everywhere except for a specific puddle of water (the prime $p$). If you drain the puddle (invert $p$), the ground is perfectly flat and the construction is smooth (étale).

2. Building the "Infinite Tower" (The $R_{p^\infty}$)

To handle the wet ground (the $p$-torsion), Shimomoto builds a special, infinite tower called $R_{p^\infty}$.

• How it works: He takes the original ring and keeps adding layers of roots, like taking a square root, then a fourth root, then an eighth root, forever.
• The Result: This creates a massive, non-finite structure that is "Witt-perfect."
• The Metaphor: Think of this as building a fractal staircase. It goes on forever, but it has a special property: if you try to "squash" it (apply the Frobenius map, which is like a mathematical compression), it doesn't crumble; it stays intact. This makes it a perfect candidate for a foundation.

3. The "Almost" Magic (Almost Cohen-Macaulay)

Using the Almost Purity Theorem, Shimomoto shows that if you attach your new building ($S$) to this infinite tower ($R_{p^\infty}$), the result is an "Almost Cohen-Macaulay" algebra.

• What this means: The rooms in this new structure are almost perfectly aligned. They are so close to being perfect that the tiny errors (the "almost zero" parts) don't matter.
• The Analogy: It's like a jigsaw puzzle where 99.9% of the pieces fit perfectly. The few that don't are so small you can't see them with the naked eye.

4. The Final Fix (Partial Algebra Modifications)

Now, we have a structure that is "almost" perfect. But we need it to be perfect (a true Big Cohen-Macaulay algebra).

• The Trick: Shimomoto uses a technique developed by Hochster called Partial Algebra Modifications.
• The Metaphor: Imagine you have a wobbly table. You can't fix it all at once, so you add a tiny shim under one leg, then another under the next. You keep adding these tiny shims (modifications) over and over.
• The Result: Eventually, the wobbles disappear, and the table becomes perfectly stable. In math terms, this process turns the "Almost" structure into a Big Cohen-Macaulay Algebra.

The Conclusion: Why It Matters

The paper proves a major result: In this specific "Mixed Characteristic" neighborhood, if your building is smooth after you drain the puddle (invert $p$), you can always build a Universal Foundation.

This leads to Corollary 1.1, which is the "Direct Summand Conjecture" for this specific case.

• The Takeaway: If you have a sturdy house ($R$) and you build a new one ($S$) that is connected smoothly (étale) once you ignore the water ($p$), then the new house is essentially a "split" version of the old one. You can always pull the new house back to the old one without breaking the connection.

Summary in One Sentence

Shimomoto uses a "near-perfect" cleaning theorem (Almost Purity) to build an infinite, fractal-like foundation, and then uses a series of tiny adjustments to turn that near-perfect foundation into a perfect one, finally solving a decades-old puzzle about how mathematical buildings fit together in wet, mixed-characteristic environments.

Technical Summary

1. Problem Statement and Context

The paper addresses the Hochster's Big Cohen-Macaulay Algebra Conjecture (Conjecture 1) in the context of mixed characteristic $p > 0$.

• The Conjecture: A Noetherian local ring $(R, \mathfrak{m}, k)$ admits an $R$-algebra $B$ such that $\mathfrak{m}B \neq B$ and every system of parameters of $R$ is a regular sequence on $B$. Such an algebra is called a balanced big Cohen-Macaulay algebra.
• The Specific Setting: The author focuses on the case where $R$ is a regular local ring of mixed characteristic $p > 0$, and $S$ is a torsion-free, module-finite $R$-algebra. The crucial condition is that the localization $R[1/p] \to S[1/p]$ is étale.
• Motivation: This setting arises from the "Almost Purity Theorem," originally developed by Faltings in $p$-adic Hodge theory and refined by Davis and Kedlaya. The goal is to leverage the "almost purity" properties to construct big Cohen-Macaulay algebras in mixed characteristic, a notoriously difficult area compared to equal characteristic.

2. Methodology

Shimomoto employs a multi-step strategy combining Almost Ring Theory, Witt Vector Theory, and Partial Algebra Modifications.

A. Almost Ring Theory and Witt-Perfect Rings

The core machinery is the Almost Purity Theorem (Davis-Kedlaya version). The author utilizes the following concepts:

• Almost Modules: Modules that are annihilated by "p-ideals" (ideals $I$ such that $I^m \subset pA$).
• Witt-Perfect Rings: A ring $A$ is Witt-perfect if the Witt-Frobenius map $F: W_{p^n}(A) \to W_{p^{n-1}}(A)$ is surjective for all $n \geq 2$.
• Almost Étale Maps: Ring maps $B \to C$ where $B[1/p] \to C[1/p]$ is finite étale. The theorem states that if $B$ is Witt-perfect, the integral closure $C'$ of $B$ in $C[1/p]$ is also Witt-perfect and is an almost finite projective $B$-module.

B. Construction of the "Big" Ring ($R_{p^\infty}$)

To apply the Almost Purity Theorem, the author constructs a specific non-Noetherian ring extension $R_{p^\infty}$ over the regular local ring $R$:

• Unramified Case ($p \notin \mathfrak{m}^2$): $R \cong W(k)[[x_2, \dots, x_d]]$. The extension is constructed by adjoining $p$-power roots of a uniformizer $\pi$ and $p$-power roots of the variables $x_i$.
• Ramified Case ($p \in \mathfrak{m}^2$): $R \cong W(k)[[t_1, \dots, t_d]]/(p-G)$. The extension involves taking $p$-power roots of the variables and manipulating the defining equation $p-G$.
• Properties: $R_{p^\infty}$ is shown to be a Witt-perfect, balanced big Cohen-Macaulay $R$-algebra. Crucially, the Frobenius map on $R_{p^\infty}/pR_{p^\infty}$ is surjective.

C. From "Almost" to "Big" Cohen-Macaulay

The construction yields an algebra that is almost Cohen-Macaulay (parameters form an almost regular sequence). To upgrade this to a true big Cohen-Macaulay algebra, the author uses Hochster's Partial Algebra Modifications:

• This technique involves iteratively modifying the algebra to "trivialize" relations among the parameters.
• The author proves that if an algebra is almost Cohen-Macaulay and satisfies certain separation and torsion-free conditions, it can be mapped to a true big Cohen-Macaulay algebra via these modifications.

3. Key Contributions and Results

Main Theorem (Theorem 4.13)

Let $R$ be a regular local ring of mixed characteristic $p > 0$, and let $S$ be a torsion-free module-finite $R$-algebra such that $R[1/p] \to S[1/p]$ is étale.
Result: $S$ admits a balanced big Cohen-Macaulay $R$-algebra.

• Significance: This establishes the existence of such algebras in a specific but important class of mixed characteristic extensions, bridging the gap between the almost purity theorem and the homological conjectures.

Corollary 1.1 (Direct Summand Conjecture)

Let $R$ be a Noetherian regular domain and $S$ a torsion-free module-finite $R$-algebra. If $R$ is $p$-torsion free and $R[1/p] \to S[1/p]$ is finite étale for some prime $p$, then the inclusion $R \hookrightarrow S$ splits as an $R$-module homomorphism.

• Context: This is a special case of the Direct Summand Conjecture (DSC). While Bhatt later proved a more general version using different methods, Shimomoto's proof relies specifically on the interplay between the Almost Purity Theorem and Hochster's algebra modifications.

Technical Lemmas

• Lemma 3.1 & Proposition 4.9: Establish that the constructed ring $R_{p^\infty}$ is Witt-perfect and a balanced big Cohen-Macaulay algebra.
• Proposition 4.12: Proves that if $B$ is an almost Cohen-Macaulay algebra and $B \to C$ is torsion-free and almost finite projective, then $C$ maps to a big Cohen-Macaulay algebra. This is the bridge between the "almost" world and the "true" world.

4. Significance and Impact

1. Bridging $p$-adic Hodge Theory and Commutative Algebra: The paper successfully translates deep results from $p$-adic Hodge theory (specifically the Davis-Kedlaya refinement of Faltings' Almost Purity Theorem) into the language of commutative algebra to solve homological conjectures.
2. Handling Mixed Characteristic: Proving the existence of big Cohen-Macaulay algebras in mixed characteristic is significantly harder than in equal characteristic. This work provides a concrete pathway for the case where the extension becomes étale after inverting $p$.
3. Refinement of the Direct Summand Conjecture: While the full DSC was eventually proven by Bhatt (using perfectoid spaces) and later by others, this paper contributes a specific proof for the "étale after inverting $p$" case using classical tools (Witt vectors) combined with almost ring theory, offering a different perspective on the problem.
4. Methodological Innovation: The paper demonstrates how to construct a "big" algebra from an "almost" one using partial algebra modifications, a technique that remains relevant in the study of homological conjectures.

In summary, Shimomoto's work is a rigorous application of the Almost Purity Theorem to construct balanced big Cohen-Macaulay algebras in mixed characteristic, thereby confirming the Direct Summand Conjecture for a specific class of module-finite extensions.