# When does the Picard group embeds inside the divisor class group?

Let $$(X, \mathcal O_X)$$ be a Noetherian, separated, integral scheme that is locally regular in codimension $$1$$ (i.e. if $$\dim \mathcal O_{X,x}=1$$ then $$\mathcal O_{X,x}$$ is regular).

Then, is it true that $$Pic(X)$$ embeds inside $$Cl(X)$$ (the Weil divisor class group) ? Is it at least true if we also assume $$X$$ is normal ?

I know that if I assumed $$X$$ is locally factorial, then $$Pic(X)\cong Cl(X)$$. But otherwise, I'm not sure.

• Related, with discussion from Vakil. Jun 30 '20 at 20:33

1. If $$X$$ is normal it is true that the canonical group morphism of divisor classes $$\operatorname {CaCl}(X)=\operatorname {Pic}(X)\to \operatorname {WCl}(X)$$ is injective.
Indeed, it suffices to prove that there is already an injection between the groups of divisors themselves $$\operatorname {CaDiv}(X)\to \operatorname {WDiv}(X)$$ (before passing to the classes) and for that we can reduce to the case when $$X=\operatorname {Spec}A$$, where $$A$$ is an integrally closed domain.
The result then follows from the equality for an integrally closed domain $$A$$:$$A=\bigcap_{\operatorname {ht }\mathfrak(p)=1} A_\mathfrak p$$ Details in Görtz-Wedhorn,Theorem 11.38(1), page 307.
1. Injectivity need not hold when $$X$$ is not assumed normal.
For example consider the curve $$X\subset \mathbb P^2_\mathbb C\;$$ given by the equation $$y^2z=x^3$$.
Its Picard group is isomorphic to $$\mathbb Z\oplus \mathbb C$$ (Hartshorne, Exercise 6.9 (b), page 148)
Its Weil class group however is just $$\mathbb Z\cdot[P]$$, where $$[P]$$ is the class of any smooth point $$P$$ of $$X$$.