Complete residue field of a point of type 5

Question

Let $(F,|.|)$ be a complete algebraically closed field. Let $x$ be the point of type 5 corresponding to the unit open disc of the adic affine line over $F$. Can we obtain a concrete description of the complete residue field of $x$ (as for point of type 2 or 3)? Does the local ring of $x$ coincide with the Robba ring?

Answer 1

A point of type 5 corresponds to a valuation of rank 2, so I am not sure if the meaning of completion is completely clear here. I think that the usual thing is to complete with respect to the associated rank 1 valuation. Then, you would get back the complete residue field at the associated type 2 point (the Gauss point in your case).

Your point of type 5 is closed and is in the closure of the Gauss point. As a result, any neighborhood of the type 5 point will also contain the Gauss point, so you will get something strictly smaller than the Robba ring. It is contained in the bounded Robba ring, but still strictly smaller I guess. For a rather concrete description, you can have a look at the notes by Florent Martin here: https://florentguymartin.github.io/pdf/intro_adic.pdf.