[Paper Review] Determinants of perfect complexes and Euler characteristics in relative K_0-groups

Manuel Breuning|ArXiv.org|Dec 8, 2008

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper establishes that Euler characteristics in relative algebraic K₀-groups can be constructed using determinant functors on triangulated categories of perfect complexes, generalizing earlier methods based on exact categories. The key result shows that such Euler characteristics defined via triangulated category functors agree with classical constructions, providing a more natural and conceptually robust framework for arithmetic and algebraic geometry applications.

ABSTRACT

We study the K_0 and K_1-groups of exact and triangulated categories of perfect complexes, and we apply the results to show how determinant functors on triangulated categories can be used for the construction of Euler characteristics in relative algebraic K_0-groups.

Motivation & Objective

• To generalize the construction of Euler characteristics in relative K₀-groups beyond exact categories to triangulated categories of perfect complexes.
• To demonstrate that determinant functors on triangulated categories can replace those on exact categories for defining refined Euler characteristics.
• To establish a canonical isomorphism between K₀-groups of triangulated categories of perfect complexes and classical K-groups, ensuring consistency with existing theory.
• To provide a conceptual framework for Euler characteristics in relative K₀-groups that is compatible with applications in arithmetic algebraic geometry.
• To resolve foundational issues in K-theory by showing that the K₁-group of a triangulated category of perfect complexes maps surjectively (and isomorphically if R is regular) to the K₁-group of the ring R.

Proposed method

• Leverages universal determinant functors on the triangulated category of perfect complexes over a ring R, denoted D^perf(R), to define Euler characteristics.
• Uses the canonical homomorphism from K₀(R) to K₀(D^perf(R)) and the isomorphism for i=0 and surjection for i=1 to relate classical and derived K-theory.
• Applies the theory of homotopy fibers of monoidal functors between Picard categories to relate relative K₀-groups K₀(R,S) to the fundamental group of a category of determinant functors.
• Constructs a commutative diagram involving exact and triangulated categories, with compatible determinant functors and monoidal functors, to relate the Euler characteristic in the derived setting to the classical one.
• Establishes an isomorphism between the fundamental group of the category of determinant functors and the relative K₀-group K₀(R,S), enabling the definition of refined Euler characteristics.
• Proves that the Euler characteristic defined via triangulated categories (χ^tri) coincides with the classical Euler characteristic (χ) when the complex becomes acyclic after base change to S.

Experimental results

Research questions

• RQ1 Can determinant functors on triangulated categories of perfect complexes be used to define Euler characteristics in relative K₀-groups?
• RQ2 Is the construction of Euler characteristics in K₀(R,S) via triangulated categories equivalent to the classical construction via exact categories?
• RQ3 What is the relationship between the K₀ and K₁-groups of the triangulated category of perfect complexes and the classical K-groups of the ring R?
• RQ4 Does the canonical map K₁(R) → K₁(D^perf(R)) remain an isomorphism when R is not regular?
• RQ5 How do the relative K₀-groups K₀(R,S) relate to the fundamental group of the category of determinant functors associated with a monoidal functor between Picard categories?

Key findings

• The canonical homomorphism K₀(R) → K₀(D^perf(R)) is an isomorphism, and K₁(R) → K₁(D^perf(R)) is surjective, with isomorphism if R is regular.
• The Euler characteristic defined via determinant functors on triangulated categories (χ^tri) agrees with the classical Euler characteristic (χ) in K₀(R,S) when the complex becomes acyclic over S.
• The relative K₀-group K₀(R,S) is canonically isomorphic to the fundamental group π₀(F(M)) of the category of determinant functors associated with a monoidal functor M.
• The construction of refined Euler characteristics in K₀(R,S) is independent of the choice of determinant functor, relying only on the universal property of the universal determinant functor.
• For a non-regular ring R, the map K₁(R) → K₁(D^perf(R)) is not injective, showing that the isomorphism result from [1] does not extend to general exact categories.
• The isomorphism K₁(A) ≅ K₁(D^b(A)) for abelian categories A does not generalize to exact categories, as shown by a counterexample in the paper.