What is the motivation for excellent rings? First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and applications are for excellent rings and by extension quasi-excellent rings conceptually. I know they were first described by Grothendieck in effort to describe phenomena with the resolutions of singularities, but I can't seem to find any specific conceptual applications they have. Every time I've tried researching this I get the definition of them without any further elaboration of why they are defined the way they are.  If someone could describe their specific uses or provide a source to do so that would be very much appreciated.
 A: (As the name suggests,) (quasi-)excellent rings are rings that are very well-behaved under various natural operations: localisation, finite type extension, formal completion (some hypothesis here), henselisation. One may well imagine that Grothendieck was looking for a property permanent under these operations. Excellent integral domains are also universally Japanese, which may be useful to know.
The first non-trivial example of an excellent ring is that of a complete noetherian local ring. The formal completion $\widehat{A}$ of a local ring $A$ may not inherit some properties of $A$, like reduceness or normality; it does when $A$ is excellent.
Now, my familiarity with excellent rings is rather superficial and I can never remember all the things that go into defining them. My go to place is Exposé I in Travaux de Gabber. There, in 2.11, Raynaud briefly mentions the link with the resolution of singularities: every characteristic zero quasi-excellent scheme $X$ admits a desingularisation à la Hironaka; conversely, if every integral scheme $Y$ finite over $X$ admits a resolution of singularities, then $X$ is quasi-excellent.
