Cohomology classes of complex approximable algebras

This paper demonstrates that over the complex numbers, the infinite Weil divisor associated with any approximable graded algebra necessarily possesses a finite cohomology class.

The Gist

Here is an explanation of the paper, translated from the complex language of algebraic geometry into a story about building with blocks, using some creative analogies.

The Big Picture: The "Approximation" Puzzle

Imagine you are an architect trying to build a massive, perfect cathedral (a mathematical object called a Big Line Bundle). You have a set of blueprints, but they are infinite and messy. You can't build the whole thing at once because you don't have enough bricks, and the plans keep changing.

In the 1990s, a mathematician named Fujita discovered a clever trick: even if you can't build the perfect cathedral immediately, you can build a slightly smaller, simpler version of it that looks almost exactly like the real thing. You can get as close to the perfect shape as you want, just by tweaking your smaller model.

Later, another mathematician named Huayi Chen asked a big question: "If I give you a set of rules for building these 'almost-perfect' models (which he called Approximable Algebras), can I always find a real cathedral (a specific geometric shape) that these rules describe?"

The Plot Twist

The author of this paper, Catriona Maclean, previously proved that the answer is NO. You cannot always find a standard cathedral. Sometimes, the rules describe something stranger: an "infinite cathedral" made of an endless sum of divisors (think of it as a structure built from an infinite number of different types of bricks, some of which are imaginary or infinitely small).

However, Maclean also showed that while you can't always find a standard cathedral, you can always find this "infinite cathedral" structure.

The New Discovery: The "Converging" Proof

This specific paper is the final piece of the puzzle. Maclean proves the converse: If you have a set of rules that works (an approximable algebra), then the "infinite cathedral" it describes isn't just a chaotic mess. It has a finite, stable shape.

Here is the breakdown using our analogies:

1. The Infinite Brick Pile (The Infinite Weil Divisor)

Imagine you are piling up bricks to build a tower.

• The Problem: In the past, we knew that for any "good" set of rules, you could pile up an infinite number of bricks ($D_1 + D_2 + D_3 + \dots$).
• The Fear: What if the pile grows so fast that it never settles? What if the total "weight" or "shape" of the pile becomes infinite or undefined? That would mean the structure is unstable and doesn't really exist in a meaningful way.
• The Paper's Result: Maclean proves that for any valid set of rules, this infinite pile of bricks always settles down. The total weight and shape converge to a specific, finite number. The pile doesn't explode; it stabilizes into a recognizable form.

2. The "Shadow" Test (Cohomology Classes)

How do we know the pile is stable without counting every single brick?

• The Analogy: Imagine shining a giant spotlight on your infinite brick tower. The shadow it casts on the ground is called the Cohomology Class.
• The Logic: If the shadow keeps growing forever, the tower is unstable. But Maclean proves that if your rules for building are "good" (approximable), the shadow will always stop growing. It will reach a specific size and stop.
• The Metaphor: Think of it like a river flowing into a lake. You might think the water will keep rising forever, but Maclean proves that the lake has a bottom. The water level (the numerical class) will rise and then stop at a specific height.

3. The "Recipe" vs. The "Dish"

• The Recipe (The Algebra): This is the list of instructions. "Take 1 cup of flour, add 2 eggs, mix..."
• The Dish (The Geometry): This is the actual cake you bake.
• The Discovery: Maclean proves that if you have a recipe that works (you can bake a cake that tastes good), then there is definitely a real cake in the oven. Even if the cake is made of infinite layers, the total volume of the cake is finite and measurable.

Why Does This Matter?

In the world of mathematics, specifically Birational Geometry (which studies how shapes can be twisted and stretched into other shapes), knowing that something is "finite" is crucial.

• Before this paper: We knew these infinite structures existed, but we weren't sure if they were "well-behaved." They could have been mathematical monsters with infinite energy.
• After this paper: We know they are tame. They have a finite "numerical class." This means mathematicians can now treat these infinite structures just like they treat normal, finite shapes. They can measure them, compare them, and use them to solve bigger problems in arithmetic and geometry.

Summary in One Sentence

Catriona Maclean proves that any mathematical structure that can be "approximated" by simple rules is actually built upon a foundation that, while infinite in parts, has a finite and stable total shape, ensuring it is a valid and usable object in the universe of geometry.

Technical Summary

Here is a detailed technical summary of the paper "Cohomology class of complex approximable algebras" by Catriona Maclean.

1. Problem Statement and Context

The paper addresses a fundamental question in algebraic geometry regarding the relationship between approximable graded algebras and geometric objects (specifically, divisors on projective varieties).

• Background: The Fujita Approximation Theorem states that the section ring of a big line bundle on an algebraic variety can be approximated arbitrarily well by finitely generated algebras. Huayi Chen extended this concept to the arithmetic setting, introducing the notion of approximable graded algebras (Definition 1.2). These are algebras satisfying specific dimension growth and approximation conditions that allow for a Fujita-type theorem.
• The Open Question: Chen asked whether every approximable graded algebra is a subalgebra of the section ring of a big line bundle (a finite divisor).
• Previous Results:
  • Maclean (2017) provided a counter-example, showing that an approximable algebra is not necessarily associated with a finite line bundle. Instead, it corresponds to the section ring of an infinite Weil divisor (an infinite formal sum of divisors with real coefficients).
  • Maclean (2017, subsequent paper) proved that any approximable algebra is a full-dimensional subalgebra of the section ring of such an infinite divisor $D(B) = \sum a_i D_i$. Furthermore, it was shown that if the numerical class of this infinite divisor converges to a finite class $[D]$, then the algebra is approximable.
• The Specific Problem: The converse remained unproven: If an algebra is approximable, does the associated infinite Weil divisor necessarily have a convergent numerical class in the Néron–Severi space?

2. Methodology

The paper employs a combination of birational geometry, valuation theory, and asymptotic analysis of graded algebras.

• Construction of Geometric Objects: The author utilizes the construction from her previous work ([Mac17b]) to associate a smooth complex projective variety $X(B)$ and an infinite Weil divisor $D(B)$ to a given approximable algebra $B$.
  • $X(B)$ is defined via the homogeneous field of fractions of $B$.
  • $D(B)$ is defined as the limit of the sequence of effective divisors $D_m/m$, where $D_m$ represents the "poles" of sections in $B_m$.
• Numerical Criteria for Convergence: The core of the proof relies on establishing a numerical criterion for the convergence of the sequence of divisor classes $[D_m/m]$ in the Néron–Severi space $NS(X)$.
  • Lemma 3.1: Establishes a lower bound relating the intersection number of a pseudo-effective divisor with an ample divisor ($E \cdot H^{d-1}$) to its norm in $NS(X)$. This implies that if the intersection numbers converge, the cohomology classes converge.
  • Lemma 3.2: Proves that the sequence $[D_m/m]$ converges in $NS(X)$ if and only if the numerical sequence $(D_m \cdot H^{d-1})/m$ converges.
• Asymptotic Dimension Analysis: The proof leverages the asymptotic properties of approximable algebras established by Chen. Specifically, it uses the fact that for an approximable algebra, the dimension of graded pieces grows polynomially ($\dim B_n \sim M n^d$).
• Polynomial Representation: A crucial technical step (Lemma 3.3) involves showing that elements in high-degree graded pieces of the algebra can be represented as ratios of polynomials in symmetric powers of a fixed lower-degree piece ($S_n(B_p)$). This allows the author to bound the poles of these elements.

3. Key Contributions and Results

The main result of the paper is Theorem 1.3, which resolves the converse question posed by the previous work.

Theorem 1.3: Let $B$ be an approximable algebra over $\mathbb{C}$. Let $X(B)$ and $D(B) = \sum_{i=1}^\infty a_i D_i$ be the associated smooth variety and infinite Weil divisor constructed in [Mac17b]. Then, the series of cohomology classes $\sum_{i=1}^\infty a_i [D_i]$ in the Néron–Severi space $NS(X)$ converges to a finite real big cohomology class.

Proof Outline:

1. Reduction: The author assumes without loss of generality that $B_1 \neq 0$.
2. Boundedness of Poles: Using the approximability condition, the author proves that for sufficiently large $m$, any element $b_m \in B_m$ can be written as a ratio $T_1/T_2$ where $T_1, T_2$ are polynomials in a fixed symmetric power $S_n(B_p)$.
3. Numerical Bound: By analyzing the poles of these rational functions, the author derives an inequality for the intersection number:
   $$(D_m \cdot H^{d-1}) / m \leq 2k (D_p \cdot H^{d-1})$$
   where $k$ and $p$ are fixed constants independent of $m$.
4. Convergence: This inequality proves that the sequence of numerical values $(D_m \cdot H^{d-1})/m$ is bounded. Since the sequence of divisors is monotone increasing, the boundedness implies convergence.
5. Conclusion: By Lemma 3.2, the convergence of the numerical sequence implies the convergence of the cohomology classes $[D_m/m]$ in $NS(X)$.

4. Significance

This paper completes a significant theoretical framework connecting abstract graded algebras to geometric divisors:

• Characterization of Approximable Algebras: It establishes a precise equivalence: An algebra is approximable if and only if it is a full-dimensional subalgebra of the section ring of an infinite Weil divisor whose numerical class converges.
• Geometric Interpretation: It confirms that the "approximability" property is intrinsically linked to the existence of a well-defined, finite numerical class for the associated infinite divisor. This bridges the gap between the algebraic definition of approximability and the geometric reality of the underlying variety.
• Foundation for Arithmetic Geometry: Since Chen's original motivation was arithmetic Fujita approximation, this result strengthens the theoretical underpinnings for applying these geometric techniques to arithmetic settings, ensuring that the "infinite divisors" arising in these contexts behave well numerically.

In summary, Maclean proves that the geometric object associated with any approximable algebra is not just an arbitrary infinite sum, but one that possesses a stable, finite cohomological limit, thereby fully characterizing the geometric nature of these algebras.