# Picard group modulo codimension 2

Let $$X$$ be a normal (possibly singular) projective surface over $$\mathbb{C}$$. Consider the set $$M_X$$ of all coherent sheaves $$F$$ on $$X$$ such that there exists a finite subset $$Y\subset X$$ such that $$F$$ restricted to $$X\setminus Y$$ is a line bundle. $$M_X$$ becomes a monoid via the tensor product. Now let $$G_X$$ be the set of equivalence classes of $$M_X$$ where two sheaves $$F_1,F_2\in M_X$$ are equivalent if there is a finite subset $$Y\subset X$$ such that $$F_1$$ and $$F_2$$ are isomorphic on $$X\setminus Y$$. This equivalence relation is compatible with the tensor product and so $$G_X$$ is a group.

In general, one has the group homomorphism $$\textrm{Pic}(X)\to G_X$$ that sends a line bundle to its equivalence class. If $$X$$ is smooth, then this is an isomorphism and thus $$G_X$$ is just the usual Picard group.

My hope is that in general we can understand $$G$$ in terms of a desingularisation $$f: X'\to X$$. Because $$X$$ is normal, it has only finitely many singularities. Away from these singularities $$f$$ is an isomorphism and the group $$G_{X'}$$ is just the Picard group of $$X'$$. So my hope is that we can identify $$G_X$$ with $$\textrm{Pic}(X')$$. Is something like that true?

• I am confused. Why is $Pic(X) \to G_X$ an isomorphism? Doesn't $G_X$ parametrize coherent sheaves? What is the inverse of $[F]$ in $G_X$? – Ariyan Javanpeykar Apr 2 at 14:51
• Not any coherent sheaves, but such $F$ that are a line bundle on $U=X\setminus Y$ for a finite set $Y$. Then the inverse of $[F]$ is any extension of the inverse of $F|_U$ to $X$. – Hans Apr 2 at 16:20
• Ok, thank you. I misread the definition, sorry. – Ariyan Javanpeykar Apr 2 at 19:04

The group $$G_X$$ can be identified with the group of rank 1 reflexive sheaves on $$X$$ ($$F$$ is reflexive if the canonical morphism $$F \to F^{\vee\vee}$$ is an isomorphism) by taking a sheaf $$F$$ to the reflexive sheaf $$F^{\vee\vee}$$. The monoidal structure on the set of all reflexive sheaves is given by $$(F,G) \mapsto (F \otimes G)^{\vee\vee}.$$ Furthermore, the group $$G_X$$ can be identified with the class group $$\operatorname{Cl}(X)$$ of Weil divisors on $$X$$.
The relation of $$\operatorname{Pic}(X')$$ to $$\operatorname{Cl}(X)$$ is given by the following exact sequence $$0 \to \bigoplus \mathbb{Z}[E_i] \to \operatorname{Pic}(X') \to \operatorname{Cl}(X) \to 0,$$ where $$E_i$$ are the components of the exceptional divisor of $$X' \to X$$.