I try do the exercise 1.21 d (page 69) of Hartshorne's algebraic geometry, which is the following:
Let $X= \mathbb{P}^1 $, $\mathcal{O}$ its sheaf of regular functions and $\mathcal{O}_{p}$ the sheaf of regular functions of a point $p \in \mathbb{P}^1 $. Denote by $\mathcal{K}$ the constant sheaf associated to the function field $K$ of $X$. Show that the quotient sheaf $\mathcal{K}/\mathcal{O}$ is isomorphic to the direct sum of sheaves $\sum_{x \in X} i_{p}(I_{p})$, where $I_p$ is the group $K/\mathcal{O}_{p}$ and $i_{p}(I_{p})$ denotes the skyscraper sheaf given by $I_p$ at the point $p$
What is meant by the group $K/\mathcal{O}_{p}$?
Since $K$ is a group (even a field), but $\mathcal{O}_{p}$ is a sheaf, I don't know how this should be understand.
Only to be sure: The constant sheaf $\mathcal{K}$ on $X$ associated to $K$ is given by $\mathcal{K}(U)=K$ for any open $U$ of $X$. Is that correct?