# Checking the ring k[x,y,u,v]/(xy-uv) is a normal domain

Let $A= k[x,y,u,v]/(xy=uv)$, is this ring normal (integral closed in its field of fraction)?

Edit (see comments below): Because $p$ is of height 1, then there exists one among $x,y,u,v,$ say $x$ such that $f(x)\notin p$ for any $f(x)\in k[x]$. Otherwise, if there are polynomials in one variable for each of the variables $x,y,u,v,$ contained in $p$, then $p$ is of height $>1$. Thus $f(x)$ becomes invertible in $A_p$ for any $f(x)\in k[x]$.

# 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.