Since the last update we have added a new chapter. This chapter explains the Popescu-Ogoma-Andre-Swan proof of general Neron desingularization (GND). As explained here there is a way to reduce to the case of a base field. This does simplify the rest of the arguments somewhat, but not as much as I’d have liked.
The heart of the proof of GND is in the proof of Lemma Tag 07FJ. For some reason working through this proof made me think of playing chess, in that you have to think ahead several moves and the steps you take early in the proof almost don’t seem to make sense. I have a hard time explaining it, even to myself. But then, I was never any good at chess.