Resolution of Singularities
Quasi-excellent rings are closely related to the problem of resolution of singularities, and this seems to have been Grothendieck's motivation for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi-excellent rings. Hironaka (1964) proved this for all complete integral Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral finite algebras over a Noetherian ring R then the ring R is quasi-excellent.
Read more about this topic: Excellent Ring
Famous quotes containing the word resolution:
“We often see malefactors, when they are led to execution, put on resolution and a contempt of death which, in truth, is nothing else but fearing to look it in the face—so that this pretended bravery may very truly be said to do the same good office to their mind that the blindfold does to their eyes.”
—François, Duc De La Rochefoucauld (1613–1680)