Cartier-divisor description of the Picard group ($\mathrm{Pic}(X)\cong \Gamma (X,{\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast })/k(X{)}^{\ast }$)

On an irreducible variety, a Cartier divisor is a section of ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$: its local rational equations differ by regular units. Their ratios define an invertible sheaf, and a global rational equation gives the trivial class. Conversely a rational trivialization of an invertible sheaf gives local equations. The rational-function sheaf is flasque, so its long exact sequence in sheaf cohomology also identifies the quotient with $H^1(X,{\mathcal{O}}_X^{\ast })$.