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

Abstract: We present a detailed formalization in Lean4 of some multigraded algebraic geometry constructions, focusing on the Brenner--Schröer Proj construction and algebraic dilatations of rings.

• The paper formalizes multi-graded Proj schemes and ring dilatations in Lean4 through rigorous definition of graded ring structures, enabling machine verification of advanced algebraic geometry.
• It establishes explicit constructs for homogeneous localization, potion generators, and gluing of spectra, ensuring compatibility with universal properties and descent arguments.
• The work extends classical frameworks to arbitrary abelian monoids and paves the way for further formalization of birational geometry and relative Proj constructions.

Formalization of Multi-Graded Brenner-Schröer Proj Schemes and Ring Dilatations in Lean4

Overview

The paper "Formalizing multi-graded Brenner-Schröer Proj schemes and dilatations of rings in Lean4" (2606.01438) addresses the formalization of advanced commutative algebra and algebraic geometry concepts in Lean4. The two principal themes are: (1) the multi-graded Proj construction à la Brenner–Schröer for rings graded by abelian monoids beyond classical $\mathbb{N}$- or $\mathbb{Z}$-gradings, and (2) the formalization of dilatations of rings, an operation closely related to localization and algebraic blow-ups. The formalization is not merely a direct translation of mathematical definitions but also encompasses categorical, functorial, and gluing aspects essential for rigorous machine verification.

Multi-Graded Algebras and Morphisms

The construction of multi-graded Proj schemes necessitates a foundation of graded rings indexed by general abelian monoids, together with their modules and morphisms. The formalization defines graded ring structures by treating the grading as data $(\iota \to \text{Subgroup}~A)$, ensuring compatibility with Lean’s type-theoretic constraints and supporting efficient manipulation of both "internal" and "external" gradings.

A comprehensive formalization is provided for graded ring homomorphisms, including their kernels and isomorphisms, ensuring the infrastructure necessary for the subsequent geometric constructions. The key property—closure of the kernel under the grading—ensures that the algebraic geometry built atop these rings will faithfully respect the underlying algebraic structures.

Homogeneous and Relevant Submonoids

Central to the multi-graded Proj construction is the management of homogeneous and relevant submonoids. The formalization rigorously defines homogeneous submonoids and their relevance with respect to the grading monoid. Relevance is characterized (when $\iota$ is finitely generated) via the finiteness of the torsion quotient $\iota/\iota[S]$, where $\iota[S]$ is generated by the degrees of $S$.

This relevance condition is critical for ensuring the "affineness" and effectiveness of the corresponding open subsets in the geometric setting. The theory is developed to allow transfer along graded ring homomorphisms and to handle ideals of relevant homogeneous elements. Notably, the formalization proves that the radical ideals of the images under surjective graded ring homomorphisms coincide, which is crucial for descent arguments and categorical constructions.

Homogeneous Localization and Potions

The notion of homogeneous localization is generalized beyond classical cases to handle arbitrary homogeneous submonoids, leading to the definition of "potions," i.e., the degree-zero parts of such localizations. The formalization constructs the structure of homogeneous localizations, verifies locality in the case of localizations at complements of homogeneous prime ideals, and proves compatibility with morphisms.

Potions serve as the building blocks of the multi-graded Proj, yielding coordinate rings for basic open sets in the Proj topology. The paper develops explicit algebraic structures and morphisms between these potions, using "potion generators" to encode algebraic information required for handling different localizations and gluings. The construction of finite potion generators under relevance and finite generation assumptions provides the foundation for constructing the topological space and structure sheaf of the scheme.

Multi-Graded Proj Construction

By gluing spectra of potions along well-controlled overlaps powered by explicit potion generators, the paper formalizes the construction of the multi-graded Proj scheme. The gluing is carried out via explicit ring isomorphisms arising from the universal properties of the constructions, with the necessary cocycle and compatibility conditions meticulously checked. This enables the definition of a scheme structure on the Proj and establishes its functoriality in the graded ring, including compatibility with arbitrary families of "good potion ingredients" (relevant, finitely generated submonoids).

The formalism includes a robust infrastructure for enlarging covering families and transitioning to larger collections of basic opens, with corresponding open immersions of schemes yielding isomorphisms topologically. This flexibility is essential for practical work in the theory, especially in handling refinements and base changes.

Tensor Products and Structural Results

The formalization extends to the graded tensor product of graded algebras, verifying that the product grading is respected and that relevance is preserved under tensor products of relevant homogeneous elements. It is shown that the submodule generated by the tensor products of relevant elements is the radical of the ideal of relevant elements in $A \otimes_R B$, recovering a key property for descent and base change compatibilities in geometric settings.

Dilatations of Rings

Moving to commutative algebra, the paper formalizes dilatations of rings with multicenters, encoding classes of symbolic fractions subject to a specific equivalence relation capturing the algebraic operation akin to blow-ups. The construction produces a new commutative ring $A\left[\frac{M_i}{a_i}\right]_{i \in I}$ together with a universal morphism from $A$, satisfying a universal property with respect to $\mathbb{Z}$0-algebra maps under prescribed nonzerodivisor and generating set hypotheses. The formalization handles both ring and semiring contexts, provides concrete operations, and verifies the key universal property in a purely type-theoretic manner.

Implications and Future Development

This formalization contributes a highly structured and general algebraic geometry environment within Lean4, achieving several technical milestones:

• Support for genuinely multi-graded settings: The constructions are not restricted to the previously standard $\mathbb{Z}$1-grading but allow arbitrary abelian monoids, facilitating applications to generalized projective geometries, representation theory, and equivariant settings.
• Automation and verifiability: By encoding the definitions, morphisms, and gluing data in Lean4, the work enables machine verification of advanced geometric arguments, potentially reducing errors and supporting formal proofs of complex theorems in algebraic geometry.
• Framework extensibility: The infrastructural choices, exemplified by the modular handling of gradings and potion generators, establish a foundation for further development—such as the formalization of relative Proj, the minimal model program, and sophisticated operations in birational geometry.

Potential next steps include the integration with other categorical and cohomological constructions, extension to algebraic stacks and derived geometry, and more systematic support for functoriality and base change at the level of structured categories. As formalized mathematics expands, the precise and flexible mechanisms introduced here lay groundwork for further mechanization of advanced algebraic and geometric theories.

Conclusion

The paper achieves a comprehensive formalization of multi-graded Proj schemes and dilatations of rings in Lean4, providing explicit and functional infrastructure for advanced algebraic geometry. The rigorous development of gradings, localizations, gluings, and universal properties is poised to have significant impact on both the mechanization of mathematics and the future of computational algebraic geometry (2606.01438).