Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

In his proof of the equal characteristic case of the Gersten conjecture for algebraic K-theory, Quillen used a geometric presentation lemma that may be viewed as a refinement of the Noether normalization theorem. Gabber, Gros--Suwa, and others have subsequently generalized Quillen's presentation lemma to establish Gersten-type phenomena for other cohomological functors. I will discuss presentation lemmas in mixed characteristics and present some of their consequences for the study of torsors under reductive groups.