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The main theorem, I.a, is the existence for excellent Deligne-Mumford champof characteristic zero of a resolution functor independent of the resolutionprocess itself. Perceived wisdom was that this was impossible, but thecounterexamples overlooked the possibility of using weighted blow ups. Thefundamental local calculations take place in complete local rings, and areelementary in nature, while being self contained and wholly independent ofHironaka's methods and all derivatives thereof, i.e. existing technology.Nevertheless Abramovich, Temkin, and Wlodarczyk, [ATW], have varied existingtechnology to obtain an even shorter proof of principalisation, I.f, in thegeometric case. Excellent patching is more technical than varieties over afield, and whence easier geometric arguments are pointed out when they exist.

Very fast, very functorial, and very easy resolution of singularities