On local divisor class groups of complete intersections

Abstract

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel--Grothendieck Theorem and prove the following generalization: Let $A$ be a local Cohen--Macaulay ring. (i) $A$ is a normal domain if and only if $A$ is a normal domain in codimension at most $1$. (ii) Suppose that $A$ is a normal domain and a complete intersection. Then the divisor class group of $A$ is a subgroup of the projective limit of the divisor class groups of the localizations $A_p$, where $p$ runs through all prime ideals of height at most $3$ in $A$. We use this fact to describe for an integral Noetherian locally complete intersection scheme $X$ the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if $X$ is locally a UFD.