[Paper Review] The Combinatorial Norm of a Morphism of Schemes

Feng-Wen An|ArXiv.org|Jan 17, 2008

Topological and Geometric Data Analysis17 references3 citations

TL;DR

This paper introduces a covariant functor from the category of schemes to the category of graphs, enabling the assignment of combinatorial invariants—such as the combinatorial norm—to morphisms of schemes. The key contribution is the definition of the combinatorial norm, which quantifies morphism behavior via graph-theoretic length and specialization structures, with morphisms of norm ≤1 characterized via injectivity and length-preservation in catenary schemes.

ABSTRACT

In this paper we will prove that there exists a covariant functor from the category of schemes to the category of graphs. This functor provides a combination between algebraic varieties and combinatorial graphs so that the invariants defined on graphs can be introduced to algebraic varieties in a natural manner. By the functor, we will define the combinatorial norm of a morphism of schemes. Then we will obtain some properties of morphisms of norm not great than one. The topics discussed here can be applied to study the discrete Morse theory on arithmetic schemes and Kontsevich's theory of graph homology.

Motivation & Objective

• To establish a functorial bridge between algebraic geometry (schemes) and graph theory, enabling the transfer of graph invariants to algebraic varieties.
• To define and analyze the combinatorial norm of a morphism of schemes as a measure of its structural complexity using graph-theoretic length and specialization relations.
• To characterize morphisms of norm ≤1, particularly identifying conditions under which such morphisms are injective or length-preserving.
• To lay foundational tools for applying discrete Morse theory and Kontsevich’s graph homology to arithmetic schemes via combinatorial invariants.

Proposed method

• Construct a covariant functor Γ from the category of schemes to the category of graphs, assigning to each scheme X a graph Γ(X) whose vertices are points of X and edges represent specializations.
• Define the combinatorial norm ||f|| of a morphism f:X→Y as the supremum of the ratio of graph distances in Γ(Y) to those in Γ(X), over all pairs of points.
• Utilize the concept of catenary schemes and length functions l(x) to define and compute distances in the graph Γ(X), ensuring consistency with specialization chains.
• Apply the notion of Sp-type morphisms and level-separated maps to control fiber behavior and preserve graph structure under morphisms.
• Leverage the property that in catenary schemes, l(ξ,x) = dim X − l(x) for generic point ξ, to relate topological dimension to graph-theoretic distance.
• Prove injectivity of morphisms via contradiction: assuming f(x)=f(y) for x≠y leads to a contradiction in length preservation and specialization chains.

Experimental results

Research questions

• RQ1 How can a functor be constructed from schemes to graphs that preserves morphism structure and enables transfer of graph invariants?
• RQ2 What is the combinatorial norm of a morphism of schemes, and how does it quantify morphism behavior in terms of graph distance and specialization?
• RQ3 Under what conditions is a morphism of norm ≤1, and how do injectivity and length-preservation relate in this context?
• RQ4 Can the combinatorial norm framework be applied to discrete Morse theory on arithmetic schemes and graph homology theories such as Kontsevich’s?
• RQ5 What role does the catenary property play in ensuring consistency of length functions and norm computation in the graph model?

Key findings

• There exists a covariant functor Γ from the category of schemes to the category of graphs, assigning to each scheme X a graph Γ(X) whose vertices are points of X and edges represent specializations.
• The combinatorial norm ||f|| of a morphism f:X→Y is defined as the supremum of the ratio of graph distances in Γ(Y) to those in Γ(X), and it can take any non-negative integer value.
• A morphism f:X→Y has norm ≤1 if and only if it is injective and of Sp-type, as shown in Theorem 2.7.
• Length-preserving morphisms between catenary schemes are injective if they are level-separated and Sp-type, as established in Theorem 2.8.
• For an injective morphism f:X→Y with Y of finite dimension, ||f||=1 holds if f is also length-preserving, as per Corollary 2.10.
• The graph Γ(Spec(ℤ)) is a tree (star-shaped with the generic point as center), contradicting earlier illustrations that misrepresented its structure.