Proposition 1

Let A be a Krull domain. If every divisorial ideal of A is invertible, then, for every maximal ideal m of A, $A_m$ is factorial. The converse is true if it is also assumed that every divisorial ideal of A is finitely generated (in particular if A is Noetherian).

Suppose that every divisorial ideal of $A$ is invertible; as $A_{,,}$ is a Krull domain ($§ 1$, no. 4, Proposition 6), every divisorial ideal $a$ of $A_{,,}$ is the intersection of two principal fractional ideals ($§ 1$, no. 5, Corollary 2 to Proposition 9); hence $a = bA_m$, where $b$ is a divisorial ideal of $A$ (Chapter II, $§ 2$, no. 4); as $b$ is invertible by hypothesis, we deduce from Chapter II, $§ 5$, no. 6, Theorem 4 that $a$ is principal and hence $A_{,,}$ is a factorial domain (no. 1, Definition 1). Conversely, if all the $A_{,,}$ are factorial and $c$ is a finitely generated divisorial ideal of $A$, $cA_m$ is a divisorial ideal of $A_{,,}$, as follows from $§ 1$, no. 5, Corollary 2 to Proposition 9 and Chapter II, $§ 2$, no. 4; by hypothesis $cA_m$ is principal and hence it follows from Chapter 11, $§ 5$, no. 6, Theorem 4 that $c$ is invertible.