Resolution For Schemes and Status of The Problem
It is easy to extend the definition of resolution to all schemes. Not all schemes have resolutions of their singularities: Grothendieck (1965, section 7.9) showed that if a locally Noetherian scheme X has the property that one can resolve the singularities of any finite integral scheme over X, then X must be quasi-excellent. Grothendieck also suggested that the converse might hold: in other words, if a locally Noetherian scheme X is reduced and quasi excellent, then it is possible to resolve its singularities. When X is defined over a field of characteristic 0, this follows from Hironaka's theorem, and when X has dimension at most 2 it was prove by Lipman. In general it would follow if it is possible to resolve the singularities of all integral complete local rings.
Hauser (2010) gave a survey of work on the unsolved characteristic p resolution problem.
Read more about this topic: Resolution Of Singularities
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