 Solution

The sheaf of nonzero rational functions on an irreducible variety ${\mathcal{K}}_X^{\ast }$ is a flasque sheaf. Indeed, its sections on any nonempty open subset are $K^{\ast }$: irreducibility makes nonempty open subsets connected and makes any two of them meet, so locally constant rational-function values must agree. Restrictions between such opens are identities; restriction to the empty set is also surjective. The preliminary argument therefore givesApply the long exact sequence in sheaf cohomology to the short exact sequence of sheaves of multiplicative abelian groupsIts relevant part isPart (b) identifies the first map with the principal divisor map $K^{\ast }\to \mathrm{Div}(X)$. Exactness therefore identifies its cokernel with $H^1(X,{\mathcal{O}}_X^{\ast })$. Combining this with part (a) givesThe connecting map sends a divisor to the class of the unit ratios of its local rational representatives; these are precisely the transition data of the associated line bundle, up to the usual choice of inverse transition convention.