# Exploring the Grothendieck ring K0 of endomorphisms

Abstract: The aim of this note is to compute the Grothendieck group $K_0$ of the category of endomorphisms. This computation mostly plays with linear algebra. The main result is that in $K_0$, every endomorphism $f:P\to P$ is uniquely characterized by $P$ and its characteristic polynomial $\lambda_t(f)$. This computation was due to [1]. We will explain how to think about this computation, the reason for certain constructions and the “diagonalization” in this computation.

Edit history: fixed some mislabeling of diagrams (Mar 21 2022)

[1] Almkvist, G. (1974). The Grothendieck ring of the category of endomorphisms. Journal of Algebra28(3), 375-388.