$K_X^*/O_X^*$ is a flasque sheaf for smooth variety over $\mathbb{C}$? Suppose $X$ is a smooth variety over $\mathbb{C}$, why do we have $K_X^*/O_X^*$ is a flasque sheaf? (Beauville "Complex Algebraic Surface" p.28)
(To show the surjection $K_X^*/O_X^*(X)\to K_X^*/O_X^*(U)$, given an element of the latter group, a collection of $(U_I,f_i)$ with $f_i/f_j\in O^*_{U_i\cap U_j}$.  How to find the corresponding data for a global section? )
 A: a) We have-completely tautologically- an exact sequence of sheaves in the Zariski topology $$  1\to   O_X^*       \to   K_X^*  \to  K_X^*/O_X^*   \to 0      \quad (\bigstar)            $$
It is clear that $K_X$ is flasque by definition of "rational function" (= regular function defined on some open subset of $X$ ), and thus that $K_X^*$ is flasque too.
But why is $K_X^*/O_X^*$ flasque?
The secret is to reinterpret that sheaf as the sheaf of Cartier divisors on $X$: $$(K_X^*/O_X^*)(U)=CaDiv(U)$$
And then to use that on a smooth variety Cartier divisors coincide with Weil divisors:  $$CaDiv(U)=WDiv(U)$$
So finally we are reduced to show that $WDiv$ is a flasque sheaf on $X$.
But nothing is easier: given a prime divisor $Y\subset U$ we obtain a prime divisor $\bar Y\subset X$ on $X$ just by  taking the Zariski closure $\bar Y$ of $Y$ in $X $.   
b) Since flasque sheaves are acyclic, the exact sequence $(\bigstar)$ (extended by zeros) provides us with an acyclic  resolution of $O_X^*$, which permits us to compute the cohomology of that sheaf.
Since all sheaves of index $\geq 2$ are zero in that resolution we conclude:  $$   H^i(X, O_X^*)=0 \quad \operatorname {for all}   i\geq 2  .                   $$
