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Formalizing multi-graded Brenner-Schröer Proj schemes and dilatations of rings in Lean4

This paper presents a detailed formalization in Lean4 of multigraded algebraic geometry constructions, specifically focusing on the Brenner-Schröer Proj construction and the algebraic dilatations of rings.

Original authors: Arnaud Mayeux, Jujian Zhang

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Arnaud Mayeux, Jujian Zhang

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 build a complex city out of mathematical blocks. Usually, architects (mathematicians) have a very specific rulebook for how to stack these blocks: they must be arranged in neat, single-file lines (like the natural numbers 1, 2, 3...) or in a simple back-and-forth pattern (like integers ...-2, -1, 0, 1, 2...).

This paper is about a team of architects who decided to break those rules. They wanted to build cities using blocks that can be stacked in much stranger, more chaotic, and more flexible patterns (using "monoids" and "groups" that are more general than just simple numbers).

Here is the story of what they built, explained without the heavy math jargon:

1. The Blueprint: "Multi-Graded" Geometry

In standard math, a "graded ring" is like a library where books are sorted strictly by shelf number (1, 2, 3).
The authors are working with Multi-Graded Rings. Imagine a library where books are sorted not just by shelf, but by a combination of shelf, color, and author's birth year all at once. It's a much more complex way of organizing information.

They focused on a specific, tricky way of building a geometric space called the Brenner-Schröer Proj construction.

The Analogy: Think of "Proj" as a way to look at a massive, infinite library and only see the "interesting" parts of it, ignoring the empty shelves. The Brenner-Schröer method is a new, sophisticated lens that lets you see interesting structures even when the books are sorted in that chaotic, multi-dimensional way mentioned above.

2. The Tool: "Potions"

To build these spaces, the authors invented a tool they whimsically named "Potions."

What is a Potion? In math, you often take a ring (a collection of numbers) and "localize" it. This is like taking a specific set of ingredients and saying, "From now on, we can divide by these ingredients."
The Magic: A "Potion" is the result of this process, but specifically looking at the "degree zero" part (the part that stays balanced). The authors realized that if you mix these Potions together correctly, you can glue them side-by-side to build a complete geometric shape (a "scheme").
The "Good Potion Ingredients": Not every mix works. They defined "Good Potion Ingredients" as specific types of ingredient sets that, when mixed, create a stable, usable potion. They proved that if you have a bunch of these good ingredients, you can mix them in any order, and the result is always a valid potion.

3. The Glue: Stitching the City Together

Once they had their Potions, they needed to stick them together to make a whole city (a Scheme).

The Glue: They showed that if you take two different Potions (say, Potion A and Potion B), you can create a "transition map" that tells you how to walk from the neighborhood of A to the neighborhood of B without falling off the edge.
The Result: By proving these maps work perfectly (they commute and form a consistent loop), they successfully glued all the individual Potion neighborhoods together into one giant, coherent geometric object. This object is their version of the Proj Scheme.

4. The Expansion: "Dilatations"

The paper also formalized a concept called Dilatations of rings.

The Analogy: Imagine you have a map of a city, but some streets are blocked or too narrow. A "dilatation" is like a magical construction crew that takes a specific intersection (an ideal) and a specific building (an element) and "blows up" that intersection. They expand the area, creating new, wider roads that allow you to navigate around the blockage.
The Universal Property: The authors proved that this expansion is the only way to do it that satisfies a specific set of rules. If you want to expand the city in a way that keeps certain rules intact, the Dilatation is the unique blueprint you must use.

5. The Big Achievement: The Lean4 Prover

Why does this paper matter? Because they didn't just write these ideas on paper; they translated them into code using a computer program called Lean4.

The Challenge: Math is full of tiny, easy-to-miss details. A human might skip a step in a proof because it "seems obvious." A computer doesn't skip steps.
The Victory: The authors took these complex, abstract geometric ideas and forced the computer to check every single logical step. If the computer said "Yes, this is true," then it is undeniably true. They built a digital foundation for this new type of geometry.

Summary

In short, this paper is a construction manual for a new kind of mathematical city.

They introduced a flexible way to organize mathematical blocks (Multi-graded rings).
They created "Potions" to turn these blocks into usable building materials.
They figured out how to glue these materials together to form a complete shape (The Proj Scheme).
They also built a tool to expand and fix parts of these shapes (Dilatations).
Most importantly, they wrote a computer-verified manual for all of this, ensuring that every brick is placed exactly where it should be, with no room for human error.

This work doesn't just describe the math; it builds a digital fortress around it, making it ready for other mathematicians to use as a solid foundation for future discoveries.

Technical Summary: Formalizing Multi-Graded Brenner-Schröer Proj Schemes and Dilatations of Rings in Lean4

Problem Statement
The paper addresses the formalization of advanced constructions in multi-graded algebraic geometry within the Lean4 theorem prover. While classical algebraic geometry often relies on rings graded by the natural numbers ( $\mathbb{N}$ ) or integers ( $\mathbb{Z}$ ), modern developments—such as those in Lie theory, geometric representation theory, and the minimal model program—require rings graded by more general monoids or groups. Specifically, the authors focus on the Brenner–Schröer Proj construction, which generalizes the classical Proj construction to multi-graded settings, and the theory of ring dilatations, an operation related to localization and blowups. The challenge lies in implementing these structures in a type-theoretic framework (Lean4) where the distinction between a graded ring $A$ and its direct sum decomposition $\bigoplus_{i \in \iota} A_i$ must be managed rigorously without sacrificing ergonomic usability.

Methodology
The authors adopt an algebraic approach consistent with existing literature (specifically following the treatment in [9]) to formalize these concepts in Lean4. The methodology involves:

General Setup for Graded Structures: The formalization defines graded rings and modules where the grading monoid $\iota$ is an arbitrary abelian group. To handle type-theoretic constraints, the authors distinguish between the ring $A$ and its graded components $A_i$ , utilizing SetLike structures to treat components as subgroups while maintaining the ability to switch between the ring and its direct sum decomposition.
Homogeneous Localization and "Potions": A central technical innovation is the definition of "potions" ( $\vartheta(S)$ ), defined as the degree-zero part of a homogeneous localization $A(S)$ . Unlike standard localization, potions are not defined as subrings of the localized ring but as quotients of triples $(i, a, b)$ where $a, b$ are homogeneous elements of the same degree and $b \in S$ . This structure facilitates easier induction and access to numerators and denominators.
Relevance and Good Potion Ingredients: The authors formalize the concept of "relevant" homogeneous submonoids, where the subgroup generated by the degrees of elements in the submonoid has finite index (or torsion quotient) in the grading group. They define "good potion ingredients" as submonoids that are both relevant and finitely generated.
Gluing and Proj Construction: The Proj scheme is constructed by gluing affine schemes $\text{Spec}(\vartheta(S_i))$ associated with a family of good potion ingredients. The gluing data relies on isomorphisms between tensor products of potions (e.g., $\vartheta(ST) \otimes_{\vartheta(S)} \vartheta(SR) \cong \vartheta(RST)$ ), which are established via "potion generators."
Dilatations: The paper formalizes dilatations of rings with respect to a multicenter (a collection of ideals and elements). This involves defining equivalence classes of symbolic fractions $m/a^\nu$ and proving the universal property characterizing dilatations.

Key Contributions and Results
The paper presents a comprehensive library of definitions, lemmas, and theorems in Lean4:

Formalization of Multi-Graded Proj: The authors provide a complete implementation of the Brenner–Schröer Proj construction. This includes the definition of homogeneous localization, the theory of potions, and the gluing of affine schemes to form the Proj scheme.
Potion Generators and Isomorphisms: A significant contribution is the formalization of "potion generators," which allow the authors to prove that for relevant $S$ and finitely generated $T$ , the morphism $\text{Spec}(\vartheta(ST)) \to \text{Spec}(\vartheta(S))$ is an open immersion. They also establish the cocycle conditions required for the gluing construction.
Functoriality: The Proj construction is shown to be a contravariant functor from the category of graded rings to the category of schemes. The authors carefully address type-theoretic subtleties where the image of a family under a ring homomorphism is not definitionally equal to the target family, requiring the use of specific morphisms (projHomOfLE) to establish functoriality.
Dilatations: The paper formalizes the definition of ring dilatations and proves their universal property: for any ring homomorphism $\chi: A \to B$ where the images of the center elements are non-zero-divisors and generate the images of the corresponding ideals, there exists a unique morphism from the dilatation to $B$ .
Isomorphism of Enlarged Families: The authors prove that enlarging a family of good potion ingredients (by including products of existing ingredients) results in an isomorphic Proj scheme, ensuring the robustness of the construction against the choice of the generating family.

Significance
The paper claims significance primarily as a foundational step in the formalization of advanced algebraic geometry. It serves as:

Documentation: A detailed account of the definitions, design choices, and main results underlying the formalization, intended to guide future developers.
Foundation: A reusable library for future extensions in formalized algebraic geometry, particularly for projects requiring multi-graded structures or dilatations.
Verification: A fully formal implementation that verifies the correctness of the Brenner–Schröer construction and dilatation theory within a proof assistant, ensuring rigorous adherence to the algebraic definitions found in the literature.

The authors explicitly state that the corresponding code is publicly available to support further developments in formalizing advanced geometric structures. They do not claim to have solved open problems in algebraic geometry but rather to have successfully translated existing complex theories into a machine-checked format.