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.

  • $\begingroup$ Related, with discussion from Vakil. $\endgroup$
    – KReiser
    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 $X$.

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