Tensor product of a sheaf of module and a residue field

Question

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?

Answer 1

Georges' answer (the other one) is more general and more common than the following, but this does still show up sometimes:

Here $k(s)$ can refer to the skyscraper sheaf with stalk $k(s)$ at the point $s$ and stalk $0$ everywhere else, especially when $s$ is a closed point. It is easy to check that this is an $\mathcal{O}_X$-module: we define the action by $f\in \mathcal{O}_X(U)$ on $k(s)(U)$ to be the $0$ action on the $0$ module if $s\notin U$ and to be multiplication by the image of $f$ under the composite map $\mathcal{O}_X(U)\to\mathcal{O}_{X,s}\to \mathcal{O}_{X,s}/\mathfrak{m}_s \cong k(s)$ otherwise.

Answer 2

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 $\mathcal O_{S,s}$)