An equivariant version of Gabber’s lemma
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Salle Pierre Grisvard
IHP
Gabber’s preparation lemma is a fundamental result in algebraic geometry, originally established by Gabber in 1994 in his work on the exactness of certain Gersten complexes. Following the work of Morel, Gabber’s lemma has become a cornerstone of motivic homotopy theory over fields, the homotopy theory of algebraic varieties. When algebraic varieties are equipped with an action of an algebraic group G, one can similarly define an equivariant motivic homotopy category, providing a framework for studying the homotopy theory of varieties with G-action. In this talk, I will first recall Gabber’s preparation lemma and discuss some of its main consequences. I will then propose and study an equivariant version of Gabber’s lemma for a finite constant abelian group G, based on the C2-equivariant version introduced by Bachmann. If time permits, I will also discuss some applications of this equivariant version of Gabber’s lemma.