Separable extensions & topology vs inseparable extensions and algebra

Question

In the note Properties of fibers and applications, Osserman writes above Definition 1.5:

Intuitively, the point is that phenomena relating to topology can only change under separable extensions, while phenomena relating to algebra can only change under inseparable extensions.

What is the "yoga" behind this intuition, and where is it explained conceptually?

For instance, why are "geometrically unibranch" local rings 0BPZ "geometric", and why isn't this contradictory to the excerpt of Osserman (for geometrically unibranch we ask the extension of residue fields to be purely inseparable)?

Answer 1

I would guess that the intuition is that separable extensions extend uniquely over "infinitesimal thickenings", i.e. deformations over a nilpotent base. This makes their classification problem "rigid", as topological invariants like homotopy groups should be. On the other hand inseparable ones do not. For example let $L/K$ be the extension $k(y)/k(x)$ where $k$ has characteristic $p$ and $x^p = y.$ Then over the square-zero thickening $R = K[\epsilon]/(\epsilon^2 = 0)$ the two flat families of rings $R[y]/(y^p = x)$ and $R[y]/(y^p = x + \epsilon)$ are non-isomorphic deformations. Similarly and relatedly, nonseparable extensions can have nontrivial deformations of the identity automorphism over a thickening whereas separable ones cannot: for example, $R[y]/(y^p = x)$ above has the automorphism $y\mapsto y + \epsilon$. To be clear being "non-topological" is different from being "non-geometric". Topologists are (not really but in an archetypical sense) interested in finding invariants that do not change under deformation whereas geometers are interested in finding invariants that do. For example the automorphism group of an inseperable extension is a (non-discrete) group scheme: an object that has more geometry than a separable Galois group.