Hirzebruch-Riemann-Roch and Lefschetz type formulas for finite dimensional algebras

TL;DR

This paper derives explicit Hirzebruch-Riemann-Roch and Lefschetz formulas for finite dimensional algebras, connecting cohomological and homological invariants with algebraic matrices and traces.

Contribution

It provides new explicit formulas for HRR and Lefschetz theorems in the context of finite dimensional algebras, including various cohomological and homological versions.

Findings

Explicit formulas for HRR and Lefschetz theorems for finite dimensional algebras.

Connection of Shklyarov pairing, Chern character, and Hattori-Stallings trace to algebraic matrices.

Comparison between formulas for finite dimensional algebras and dg algebras.

Abstract

The Hirzebuch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension are explicitly given. They have cohomological, homological, Hochschild cohomological and Hochschild homological four versions, and module, bimodule, module complex and bimodule complex four levels. For this, the dimension matrix of a bimodule (complex) and the trace matrix of a bimodule (complex) endomorphism are introduced. It is shown that Shklyarov pairing, Chern character and Hattori-Stallings trace can be concretely expressed by Cartan matrix, dimension vector and trace vector in this situation. Furthermore, the HRR and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension and dg algebras are compared.