I have been perusing Harthorne for some time, and I noticed something: it is well known that the class group on $\mathbb{P}^n\_k$ is $\mathbb{Z}$. But as I look at Harthorne's proof it seems to me that it works in much greater generality. Namely if I consider any projective scheme $X=\operatorname{Proj}(A)$, where $A$ is a graded UFD in such a way that there exists an irreducible element of degree $1$, then the exact same reasoning shows that the class group of $X$ is also $\mathbb{Z}$, and generated by the prime divisor $(a)$. Is this true?



Divisors on Proj(UFD)
Well, if you read on to Chapter 2, exercise 6.3, then it is stated that:
$$Cl(A) \cong Cl(X)/\mathbb Z[H]$$
here $[H]$ represents the hyperplane section. So the answer is yes.


There is a less well-known but very nice generalization. Suppose that $X$ is smooth. Let $R=A\_m$ be the local ring of A at the irrelevant ideal. Then one has a (graded) isomorphism of $\mathbb Q$- vector spaces:


$$CH(X)\_{\mathbb Q}/[H]CH(X)\_{\mathbb Q} \cong A\_\*(R)\_{\mathbb Q}$$


Here $CH(X)$ is the Chow *ring* of $X$ and $A\_\*(R)$ is the total Chow group of $R$. Details can be found in this [paper](https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-48/issue-1/A-remark-on-the-Riemann-Roch-formula-on-affine-schemes/10.2748/tmj/1178225414.full) by Kurano.


