Hattori-Stallings trace and character

TL;DR

This paper introduces the Hattori-Stallings character, a new homomorphism linking algebraic K-theory and Hochschild homology, providing novel proofs of the strong no loop conjecture for finite-dimensional algebras.

Contribution

It defines the Hattori-Stallings character and demonstrates its application in proving the Igusa-Liu-Paquette Theorem through complex and module-level approaches.

Findings

Hattori-Stallings trace induces a homomorphism to Hochschild homology.

Provides new proofs of the strong no loop conjecture.

Studies traces of projective bimodules for algebraic insights.

Abstract

It is shown that Hattori-Stallings trace induces a homomorphism of abelian groups, called Hattori-Stallings character, from the $K_1$-group of endomorphisms of the perfect derived category of an algebra to its zero-th Hochschild homology, which provides a new proof of Igusa-Liu-Paquette Theorem, i.e., the strong no loop conjecture for finite-dimensional elementary algebras, on the level of complexes. Moreover, the Hattori-Stallings traces of projective bimodules and one-sided projective bimodules are studied, which provides another proof of Igusa-Liu-Paquette Theorem on the level of modules.