Tensor product of a sheaf of module and a residue field Let $S$ be a scheme over a finite field $\mathbb{F}_q$ and $\mathcal{L}$ an invertible sheaf on $S$. 
In Pink's article (here, see section 1.4 on page 13), he states that:
For any section $\ell\in \mathcal{L}(S)$ and any point $s\in S$ we let $\ell(s)\in \mathcal{L}\otimes_{\mathcal{O}_S}k(s)$ denote the value of $\ell$ over the residue field $k(s)$ of $s$.
My question is: what is the definition of $\mathcal{L}\otimes_{\mathcal{O}_S}k(s)$?
We know we can define the tensor product of two $A$-modules or of two $\mathcal{O}_X$-modules, but what is the tensor product of an $\mathcal{O}_X$-module and a field?  

The notation $\ell(s)\in \mathcal{L}\otimes_{\mathcal{O}_S}k(s)$ does not make sense.
It is an abuse of language for the correct description $$\ell(s)\in \mathcal{L}_s\otimes_{\mathcal{O}_{S,s}}k(s)=\mathcal{L}_s/\mathfrak m_{S,s} \mathcal{L}_s$$  (Where of course $\mathfrak m_{S,s}$ is the maximal ideal of the local ring $\mathcal O_{S,s}$)
