# Rosenlicht unit theorem-II

Recall the theorem (first part) : It says that if X is a k-variety, then $U_k(X)=\frac{k[X]^*}{k^*}$ is a free $\mathbb{Z}$ module of finite rank.

The first step is to be able to say “WLOG, we can assume X is a normal affine k-variety” .

For that, we should know what normalization is. Here’s a very well written post explaining what normalization actually is, with some worked out examples. And in case,the link becomes redundant at some point,  a pdf copy of the same post is here : normalization

Now if Y is any open affine subset of our given X, we construct the normalization of Y, and call it $Y '$. This is also affine, as the link above explains and $k[Y]\hookrightarrow k[Y ']$. And since $O_X$ is a sheaf, we have $O_X(X)=k[X]\to O_X(Y)\cong k[Y]$. In fact this is also injective because X is integral ?

So we have an injective map $k[X]\hookrightarrow k[Y']$, which induces an injection $U_k(X)\hookrightarrow U_k(Y')$. Hence if we show the theorem holds for affine normal varieties (like Y’), then because any subgroup of a free $\mathbb{Z}$ finite rank module is also free abelian and of finite rank, we are done !