[Paper Review] A Deligne-Riemann-Roch isomorphism I: Preliminaries on virtual categories

Dennis Eriksson|ArXiv.org|Apr 26, 2009

Algebraic Geometry and Number Theory19 references3 citations

TL;DR

This paper establishes foundational tools for a functorial, metrized Grothendieck-Riemann-Roch theorem by introducing and analyzing the virtual category of an exact category, using homotopy-theoretic methods. It constructs Adams and λ-operations on this category, proves rigidity results for lifting K₀-operations, and connects the framework to Chern intersection functors and rational Chow categories.

ABSTRACT

This is the first article in an upcoming series of papers. They have arisen through an attempt to answer open questions of Deligne proposed in "Le determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf. "Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm. Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary results on virtual categories which are the categories where the Deligne-Riemann-Roch theorem is originally formulated. Finally we compare our constructions to constructions of Franke on Chern intersection functors and Chow categories.

Motivation & Objective

• To develop a categorical framework for a functorial, metrized version of the Grothendieck-Riemann-Roch theorem, inspired by Deligne's open questions.
• To define and study the virtual category V(C) as a homotopy-theoretic truncation of K-theory, capturing K₀ and K₁ information.
• To construct Adams and λ-operations on the virtual category and prove their rigidity, enabling lifts of operations from K₀.
• To compare the virtual category construction with existing theories, particularly Franke's Chern functors and rational Chow categories.
• To lay the groundwork for a future integral version of the functorial Riemann-Roch theorem, extending Deligne's work on smooth curves.

Proposed method

• Define the virtual category V(C) as the fundamental groupoid of ΩBQ(C), where BQ(C) is the Quillen Q-construction of a small exact category C.
• Use simplicial sheaf models and A¹-homotopy theory to analyze the homotopical structure of V(C), particularly its automorphism groups isomorphic to K₁(C).
• Construct Adams operations ψᵏ and λ-operations on V(C) via universal properties and the splitting principle, proving their existence and compatibility with K₀.
• Prove rigidity theorems (Theorem 4.5) showing that operations on K₀ lift to V(C) after inverting integers, using homotopical algebra and model structures.
• Compare the virtual category with rational Chow categories and Chern intersection functors via cycle-theoretic constructions.
• Apply results from A¹-homotopy theory and algebraic stacks to fix foundational language and tools for schemes over regular bases.

Experimental results

Research questions

• RQ1 How can the virtual category V(C) be systematically constructed as a homotopy-theoretic refinement of K-theory for exact categories?
• RQ2 What is the role of Adams and λ-operations in the virtual category, and can they be lifted from K₀ in a canonical way?
• RQ3 To what extent do rigidity results ensure that K₀-operations extend to the virtual category, especially after inverting integers?
• RQ4 How does the virtual category relate to existing constructions in algebraic K-theory, such as Franke's Chern functors or rational Chow categories?
• RQ5 Can the framework be extended to an integral version of the functorial Riemann-Roch theorem, as sought by Deligne?

Key findings

• The virtual category V(C) is a groupoid whose isomorphism classes are in natural bijection with K₀(C), and whose automorphism groups are isomorphic to K₁(C).
• Adams operations ψᵏ and λ-operations on V(C) are constructed and shown to be compatible with the K₀-level operations via Propositions 3.8 and 3.10.
• The rigidity result (Theorem 4.5) establishes that any operation on K₀(C) that commutes with λ-operations lifts to V(C) after inverting integers.
• The virtual category construction is shown to be compatible with rational Chow categories and Chern intersection classes, as detailed in Section 5.
• The framework provides a natural setting for secondary invariants in Riemann-Roch theory, particularly those related to the determinant of cohomology.
• The approach via virtual categories is shown to be well-suited for capturing functorial and metrized Riemann-Roch data, especially in the context of determinant functors.