Abstract
Introducing the notions of quasi-excellent rings and more generally of P-rings, GrothendieckGrothendieck, A. left some unsolved problems. Among them, two questions seem to be the most important ones: the localization problem for P-morphisms and the lifting problem for P-rings. Probably the most spectacular result is Corollary 4.6.21 stating that power series ring over quasi-excellent rings remains quasi-excellent. The results presented in this chapter need a serious geometric preparation, because the proofs use an important result of GabberGabber, O., known as the weak local uniformization theorem. Because FlennerFlenner, H.’s form of the second Theorem of Bertini is used in the proof, we decided to present this result that is important in many places in Commutative Algebra and Algebraic Geometry. This chapter uses notions and results from Algebraic Geometry. They are collected in Sect. 4.1.
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