Let, $X$ be a scheme and $L$ be a line bundle on $X$,then we know that at any point $x \in X$ we have $L_x \cong \mathcal O_x$. At this point my question is the following :
Under this isomorphism is it also true that $m_x.L_x$ is isomorphically mapped to $m_x$?
(I started with an element from $\mathcal O_x -m_x$ ,then it's a unit but I don't see why its image does not belong to $m_x.L_x$)
Any help from anyone is welcome.
If $\varphi: M\rightarrow N$ is an isomorphism of $A$-modules and $I\subseteq A$ is an ideal, then by restriction you have an isomorphism $\varphi: IM\rightarrow IN$ (you can just check this set theoretically).
Now, if $L$ is a line bundle, by definition it is locally isomorphic to $\mathcal{O}_X$. Hence, there is an open neighborhood $U$ of $x$, and an isomorphism $\varphi:\mathcal{O}_X|_U\rightarrow L|_U$. Looking at the stalk we get an isomorphism $$\varphi_x:\mathcal{O}_x\rightarrow L_x$$ And now we are in the previous case with $I=m_x$, $A=M=\mathcal{O}_x$ and $N=L_x$