Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives

TL;DR

This paper explores how Lefschetz fixed point theorems and Hirzebruch-Riemann-Roch formulas for noncommutative motives can be derived from their motivic foundations, linking recent advances in noncommutative geometry.

Contribution

It demonstrates the formal derivation of noncommutative Lefschetz and Hirzebruch-Riemann-Roch formulas from motivic principles, unifying recent results in the field.

Findings

Formal connection between motivic and noncommutative formulas

Derivation of Lefschetz fixed point theorems for dg algebras

Noncommutative Hirzebruch-Riemann-Roch theorem explained

Abstract

V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.