ac.commutative algebra - Class groups of normal domains over finite fields

As requested in the comments, here's an example of a local, normal $2$-dimensional domain R in positive characteristic such that $mathrm{Cl}(R)$ is not torsion: choose an elliptic curve $E subset mathbf{P}^2$ over a field $k$ such that $E(k)$ is not torsion, and take R to be the local ring at the origin of the affine cone on $E$ (i.e., $R = k[x,y,z]/(f)_{(x,y,z)}$ where $f$ is a homoegenous cubic defining $E$). This can be done over $k = overline{mathbf{F}_p(t)}$.

Proof: The normality follows from the fact that R is a hypersurface singularity (hence even Gorenstein) and isolated and $2$-dimensional (hence regular in codim 1). Blowing up at the origin defines a map $f:X to mathrm{Spec}(R)$. One can then show the following: $X$ is smooth, and $X$ can be identified with the Zariski localisation along the zero section of the total space of the line bundle $L = mathcal{O}_{mathbf{P^2}}(-1)|_E$ (these are general facts about cones). By Lipman's theorem, it suffices to show that $mathrm{Pic}^0(X)$ contains non-torsion elements. As $X$ is fibered over $E$ with a section, the pullback $mathrm{Pic}^0(E) to mathrm{Pic}^0(X)$ is a direct summand. As $mathrm{Pic}^0(E) simeq E(k)$ has non-torsion elements by assumption, so does $mathrm{Pic}^0(X)$.

Also, an additional comment: In general, Lipman's theorem tells you that $mathrm{Cl}(R)$ is torsion if and only if $mathrm{Pic}^0(X)$ is torsion. Now $mathrm{Pic}(X) simeq lim_n mathrm{Pic}(X_n)$ where $X_n$ is the $n$-th order thickening of the exceptional fibre $E$. Because we are blowing up a point, the sheaf of ideals $I$ defining $E$ is ample on $E$. The kernel and cokernel of $mathrm{Pic}(X_n) to mathrm{Pic}(X_{n-1})$ are identified with $H^1(E,I|_E^{otimes n+1})$ and $H^2(E,I|_E^{otimes n+1})$. As $I|_E$ is ample, it follows that the system "$lim_n mathrm{Pic}(X_n)$" is eventually stable. Thus, $mathrm{Pic}(X) simeq mathrm{Pic}(X_n)$ for $n$ sufficiently big. As $X_n$ is a proper variety, it follows that if we are working over a finite field (resp. an algebraic closure of a finite field), then $mathrm{Pic}^0(X)$ is finite (resp. ind-finite).

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