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Yes, this is true (at least in the excellent case). See the paper of J. Lipman Rational singularities with applications to algebraic surfaces and unique factorization. See in particular Proposition 17.1. In fact, Lipman proves that the divisor class group is locally finite for rational surface singularities.

When Lipman wrote that, he hadn't yet proved resolution of singularities for excellent 2-dimensional rings.

The converse statement is proven for Henselian local rings with algebraically closed residue fields in Theorem 17.4 (in other words, if the divisor class group locally is finite + those conditions, then the singularity is rational).

4 added 8 characters in body

Yes, this is true (at least in the excellent case). See the paper of J. Lipman Rational singularities with applications to algebraic surfaces and unique factorization. See in particular Proposition 17.1. In fact, Lipman proves that the divisor class group is locally finite for rational surface singularities.

When Lipman wrote that, he hadn't yet proved resolution of singularities for excellent 2-dimensional rings.

The converse statement is proven for Henselian local rings with algebraically closed residue fields in Theorem 17.4 (in other words, if the divisor class group is finite + those conditions, then the singularity is rational).

3 added 207 characters in body

Yes, this is true (at least in the excellent case). See the paper of J. Lipman Rational singularities with applications to algebraic surfaces and unique factorization. See in particular Proposition 17.1. In fact, Lipman proves that the divisor class group is locally finite for rational singularities.

When Lipman wrote that, he hadn't yet proved resolution of singularities for excellent 2-dimensional rings.

The converse statement is proven for Henselian local rings with algebraically closed residue fields in Theorem 17.4 (in other words, if the divisor class group is finite + those conditions, then the singularity is rational).

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