Additive invariants of finite dimensional algebras of finite global dimension

TL;DR

This paper demonstrates that for finite dimensional algebras with finite global dimension, additive invariants depend solely on the number of simple modules, using noncommutative motives theory.

Contribution

It introduces a method to compute additive invariants of such algebras based only on simple module count, leveraging noncommutative motives.

Findings

Additive invariants depend only on the number of simple modules.

Established compatibility between pairings and Galois actions on Grothendieck groups.

Proved a transfer result in noncommutative motives.

Abstract

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value of every additive invariant at A only depends on the number of simple modules. Examples of additive invariants include algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Along the way we establish two results of general interest. The first one concerns the compatibility between bilinear pairings and Galois actions on the Grothendieck group of every proper dg category. The second one is a transfer result in the setting of noncommutative motives.