Invertible ideals and locally free module

Question

I was reading Pete Clark beautiful notes and in particular the invertible ideals part. I'm really stuck at the point where he demonstrates that for every locally free module $M$ there is an ivertible ideals s.t $M \cong I$. In that part, he says $M \otimes K \cong K$ (where $K$ is the field of quotient). I'm not getting this:locally this is true,but the isomorphisms change ,so what is he really saying?