Almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings

This paper proves that any complete local domain of mixed characteristic possesses a weakly almost Cohen-Macaulay algebra by constructing it over the absolute integral closure using Fontaine rings and Witt vectors, thereby extending Heitmann's methods for the Monomial Conjecture.

The Gist

The Big Picture: The "Impossible" Puzzle

Imagine you are a mathematician trying to solve a giant, multi-dimensional puzzle. This puzzle represents the structure of numbers in a specific type of mathematical world called a Mixed Characteristic Local Ring.

• The World: Think of this world as a building with many floors (dimensions).
• The Goal: You want to prove that you can always find a special "key" (a specific sequence of numbers) that unlocks every floor perfectly without getting stuck. In math terms, you want to prove the existence of a Big Cohen-Macaulay Algebra.
• The Problem: For a long time, mathematicians could easily find these keys in "pure" worlds (where everything is either all even or all odd numbers). But in the "Mixed Characteristic" world (where you have a mix of zero and prime numbers, like the number 2 and 5 interacting), the puzzle seemed impossible to solve for buildings with 4 or more floors.

The Main Question: Does a perfect key exist for these mixed worlds? This is known as Hochster's Conjecture.

The Author's Strategy: Building a "Weak" Bridge

The author, Kazuma Shimomoto, doesn't try to build the perfect key immediately. Instead, he builds a "Weakly Almost Cohen-Macaulay Algebra."

Think of this like building a bridge across a river.

• The Perfect Bridge: A bridge where you can walk across without ever slipping, no matter how heavy the load. (This is the "Big Cohen-Macaulay Algebra").
• The "Weakly Almost" Bridge: A bridge that is almost perfect. If you try to walk across, you might slip a tiny, microscopic amount. But here is the magic: You can make that slip so small that it is practically zero.

Shimomoto proves that for any mixed-characteristic building, you can construct this "almost perfect" bridge.

The Tools: The "Time Machine" and the "Magic Ladder"

To build this bridge, Shimomoto uses two very advanced mathematical tools, which he treats like a time machine and a magic ladder.

1. The Time Machine (Fontaine Rings)

Imagine you are in a complex, messy room (the Mixed Characteristic world). It's hard to clean.

• The Trick: Shimomoto uses a "Time Machine" (called a Fontaine Ring) to send the messy room back in time to a simpler, cleaner era (Positive Characteristic).
• Why? In this simpler era, the rules of math are much easier. It's like taking a tangled ball of yarn and putting it in a freezer; the cold makes the fibers stiff and easy to untangle. In this "frozen" world, we know for a fact that perfect keys exist (thanks to previous work by Hochster and Huneke).
• The Result: He finds a perfect structure in this simple world.

2. The Magic Ladder (Witt Vectors)

Now he has a perfect structure in the simple world, but he needs it back in the messy, mixed world.

• The Trick: He uses a "Magic Ladder" (called Witt Vectors) to climb back up to the present.
• The Process: As he climbs back up, the perfect structure gets slightly distorted. It's like carrying a glass sculpture up a shaky ladder; it might get a few tiny cracks.
• The Result: When he reaches the top, the structure isn't perfectly rigid anymore. It has tiny flaws. However, he proves that these flaws are so small that they are "almost zero."

The "Almost" Concept: The Slippery Floor

The core of the paper is defining what "almost" means.

Imagine you are standing on a floor made of ice.

• Regular Math: If you push a block, it slides a specific distance.
• Almost Math: Shimomoto says, "I can push the block, and it will slide. But I can also push it with a force so tiny that the slide is smaller than a single atom."
• The "Weakly Almost" Sequence: He shows that you can find a sequence of numbers (the key) such that if you try to break the rules of the room, the "breakage" can be made arbitrarily small.

He calls this a Weakly Almost Regular Sequence. It's not a perfect lockpick, but it's a pick that works so well that the lock thinks it's perfect.

The "Bad" Sequence vs. The "Good" Sequence

The paper uses a concept called Algebra Modifications. Imagine you are trying to fix a broken machine.

• The Problem: Sometimes, the gears jam (a "bad" sequence).
• The Fix: Shimomoto's method involves adding new gears (modifications) to the machine to un-jam it.
• The Proof: He proves that no matter how many times you try to jam the machine, you can always add enough new gears to keep it running, provided you accept that the machine might vibrate slightly (the "almost" part).

Why Does This Matter? (The Monomial Conjecture)

The paper ends by connecting this "almost perfect" bridge to the Monomial Conjecture.

• The Conjecture: This is a famous rule in math that says, "In these mixed worlds, certain patterns of numbers can never be built from smaller pieces."
• The Connection: Shimomoto shows that if you have his "Weakly Almost" bridge, you are very close to proving the Monomial Conjecture.
• The Catch: To prove the conjecture completely, the bridge needs to be perfectly rigid (no slipping at all). Shimomoto's bridge slips a tiny bit. However, he argues that if you can find a way to measure that slip and prove it's truly negligible, the Monomial Conjecture is solved.

Summary Analogy

Imagine you are trying to build a skyscraper on a swamp (Mixed Characteristic).

1. The Old Way: People tried to build it directly on the swamp, but the ground kept shifting, and the building would tilt.
2. Shimomoto's Way:
   • He builds a perfect, solid skyscraper on a nearby mountain (Positive Characteristic/Fontaine Ring).
   • He then lowers this skyscraper onto the swamp using a crane (Witt Vectors).
   • As it touches the swamp, the bottom floor sinks a tiny, microscopic amount.
   • The Breakthrough: He proves that the building is so stable that even with that tiny sink, it stands firm enough to prove that the swamp can support a skyscraper.

In short: Shimomoto didn't solve the whole mystery of the "Big Cohen-Macaulay Algebra" in mixed characteristic yet, but he built a bridge that is so close to perfect that it proves the Monomial Conjecture is likely true, provided we can just tighten the final bolt.

Technical Summary

1. Problem Statement and Context

The Core Problem:
The paper addresses Hochster's Conjecture, which posits that every local Noetherian ring of mixed characteristic possesses a big Cohen-Macaulay algebra.

• A big Cohen-Macaulay algebra $B$ over a local ring $(R, \mathfrak{m})$ is an $R$-algebra such that $\mathfrak{m}B \neq B$ and a system of parameters $x_1, \dots, x_d$ of $R$ forms a regular sequence on $B$.
• Status: The conjecture is proven for equicharacteristic rings. For mixed characteristic, it was proven for dimension $\le 3$ (by Hochster using Heitmann's work) but remains open for dimensions $\ge 4$.

The Specific Gap:
While the existence of big Cohen-Macaulay algebras is the ultimate goal, Roberts introduced the concept of almost Cohen-Macaulay algebras, whose existence is sufficient to prove the Monomial Conjecture (and thus the Direct Summand Conjecture). Roberts' definition requires that the quotient $B/\mathfrak{m}B$ is not "almost zero" (elements are not annihilated by elements of arbitrarily small valuation).
Shimomoto aims to construct a weakly almost Cohen-Macaulay algebra in mixed characteristic. This is a slightly weaker notion where the "almost zero" condition on $B/\mathfrak{m}B$ is not explicitly assumed, but the regularity of the parameters is achieved in an "almost" sense via valuations.

2. Methodology

The paper employs a sophisticated blend of commutative algebra and arithmetic geometry, specifically utilizing Fontaine rings and Witt vectors to bridge the gap between mixed characteristic and positive characteristic.

A. Key Definitions

• Weakly Almost Regular Sequence: A sequence $x_1, \dots, x_d$ is weakly almost regular on an $R^+$-algebra $B$ if $\mathfrak{m}B \neq B$ and for any rational $\epsilon > 0$, there exists an element $b \in R^+$ with valuation $v(b) < \epsilon$ such that $b$ annihilates the Koszul homology of the sequence.
• Absolute Integral Closure ($R^+$): The integral closure of $R$ in an algebraic closure of its fraction field.

B. The Construction Strategy

The proof proceeds in three main stages:

1. Reduction to Positive Characteristic via Fontaine Rings:

   • Let $(R, \mathfrak{m})$ be a complete local domain of mixed characteristic $p$.
   • The author constructs the Fontaine ring $E(R^+)$, defined as the inverse limit of $R^+/pR^+$ under the Frobenius map:
     $$E(R^+) = \varprojlim (R^+/pR^+ \xleftarrow{F} R^+/pR^+ \xleftarrow{F} \dots)$$
   • $E(R^+)$ is a perfect ring of characteristic $p$. The author proves that $E(R^+)$ contains a complete regular local subring $E(R)^\times$ (constructed from a sequence of finite extensions of $R$) and that the parameters of $R$ lift to a regular sequence in $E(R^+)$ modulo the ideal generated by the lift of $p$.

2. Algebra Modifications (Hochster's Technique):

   • The author uses Hochster's method of algebra modifications to force parameters to become regular sequences.
   • Starting with $E(R^+)$, a sequence of algebra modifications is performed to trivialize relations among the parameters.
   • A crucial step involves localizing $E(R^+)$ at the product of the lifted parameters (excluding the lift of $p$) to ensure that the lift of $p$ remains a non-zero divisor.
   • This process yields a perfect algebra $S$ over $E(R^+)$ where the parameters form a "weakly almost" regular sequence.

3. Lifting to Mixed Characteristic via Witt Vectors:

   • The perfect algebra $S$ (characteristic $p$) is lifted to mixed characteristic using the ring of Witt vectors $W(S)$.
   • The author utilizes the surjective homomorphism $\psi: W(E(R^+)) \to \widehat{R^+}$ (the $p$-adic completion of $R^+$).
   • The target algebra $B$ is constructed as a quotient of $W(S)$ by the ideal generated by a specific element $\vartheta = \theta(\langle p \rangle) - p$, where $\theta$ is the Teichmüller lift.
   • This construction ensures that $B$ is an $R^+$-algebra in mixed characteristic inheriting the regularity properties of $S$.

3. Key Contributions and Results

Main Theorem (Theorem 5.3)

For a complete local domain $(R, \mathfrak{m})$ of mixed characteristic $p > 0$, there exists a system of parameters $p, x_2, \dots, x_d$ and a weakly almost Cohen-Macaulay $R^+$-algebra $B$ satisfying:

1. $(p, x_2, \dots, x_d)B \neq B$.
2. $x_2, \dots, x_d$ form a regular sequence on $B/pB$.
3. $p$ is not nilpotent in $B$.
4. The ideal $(0 :_B p)$ is annihilated by $p^\epsilon$ for any rational $\epsilon > 0$.

This establishes that the system of parameters is a weakly almost regular sequence on $B$.

Connection to the Monomial Conjecture (Corollaries 6.1 & 6.3)

The paper discusses the conditions under which this weakly almost Cohen-Macaulay algebra implies the Monomial Conjecture:

• If the constructed algebra $B$ satisfies the additional condition that $p^\epsilon \notin (p, x_2, \dots, x_d)B$ for some $\epsilon > 0$, then $B$ maps to a big Cohen-Macaulay algebra.
• The author suggests that if one can define a "pseudo-valuation" on $B$ (satisfying specific properties similar to a valuation on a domain), the Monomial Conjecture follows. This provides a potential pathway to proving the full conjecture, contingent on establishing the non-vanishing of certain elements (separatedness).

4. Significance

1. New Perspective on Mixed Characteristic: The paper moves away from direct manipulation of mixed characteristic rings by utilizing Fontaine rings to translate the problem into the realm of perfect rings of characteristic $p$, where powerful tools (like Hochster-Huneke's results on $R^+$) are available.
2. Bridging the Gap: It successfully constructs an algebra that is "almost" Cohen-Macaulay in mixed characteristic. While it does not fully resolve Hochster's conjecture for all dimensions (due to the "weakly" condition and the need for a separatedness argument), it provides a concrete construction that comes very close.
3. Methodological Innovation: The integration of Witt vector theory to lift the perfect algebra $S$ back to the mixed characteristic setting is a novel application in this specific context of Cohen-Macaulay algebras.
4. Foundation for Future Work: The paper identifies the precise obstruction to proving the Monomial Conjecture: the separatedness of the algebra (specifically, ensuring $B/\mathfrak{m}B$ is not "almost zero"). It frames the problem in terms of finding a valuation or pseudo-valuation on the constructed algebra, offering a clear direction for future research.

In summary, Shimomoto's work represents a significant step forward in the study of homological conjectures in mixed characteristic, demonstrating that the machinery of Fontaine rings and Witt vectors can be effectively harnessed to construct algebras with near-Cohen-Macaulay properties.