Let $X$ be a either a projective scheme or a compact complex space. Then one has an exact sequence $$ (1) \quad 0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{x \in \textrm{Sing}(X)} \textrm{Cl}(\mathcal{O}_{X,x}),$$ where $\textrm{Pic}(X)$ and $\textrm{Cl}(X)$ denote the groups of Cartier (resp. Weil) divisors on $X$ up to linear equivalence.

The last arrow is in general not surjective, as can be shown by simple examples.

Now, in the paper by J.Birgener and U. Storch

*Zur Berechnung der Divisorenklassengruppen kompletter lokaler Ringe*, Nova Acta Leopoldina **52** Nr. 240, 7-63 (1981)

page 11, it is claimed that *in the algebraic case* one actually has the sequence
$$ (2) \quad 0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{x \in \textrm{Sing}(X)} \textrm{Cl}(\mathcal{O}_{X,x}) \to H^2(X, \mathcal{O}_X^{\ast}) \to 0.$$

My problem is that $(2)$ seems to me not true. For instance, take a *smooth* cubic threefold $X \subset \mathbb{P}^4$. Then the inclusion $\textrm{Pic}(X) \to \textrm{Cl}(X)$ is an isomorphism, so $(2)$ would imply $ H^2(X, \mathcal{O}_X^{\ast})=0$. However, by the exponential sequence one finds $H^2(X, \mathcal{O}_X^{\ast})=H^3(X, \mathbb{Z})= \mathbb{Z}^{10}$, and this is a contradiction.

So my question is:

am I missing something? If not, can one correct $(2)$ in some way?

Any answer or reference to the existing literature will be appreciated. Thank you!