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Descent in Motivic Triangulated Categories

There will be a minicourse by Denis-Charles Cisinski from Toulouse.

Schedule:

All lectures and tutorials will take place in room 69/118. Please register by email to mgausman@uos.de. 

Abstract: For all three lectures: we will recall various constructions of triangulated categories of motivic sheaves, after Morel and Voevodsky. We will see how Lurie's sheaves of higher groupoids allows to work with fairly general schemes (in particular, without any noetherian assumptions). We will also see why the smooth Nisnevich site plays a central role, by examining the proof as well as the consequences of the localization theorem. The second part of these lectures will focus on various aspects of proper descent. First, we will see why descent by blow-ups shows up naturally (and easily) from the proper base change formula. Descent by blow-ups is fundamental in the study of Bloch's higher Chow groups (via moving lemmas) as well as in the study of motives of singular algebraic varieties with integral coefficients. Second, we will study the relationship between proper descent and étale descent for motivic sheaves. In the case of finite coefficients, this will lead us to the rigidity theorem of Suslin and Voevodsky, which describes usual étale sheaves as mixed motives, and thus gives a natural interpretation (and construction) of theétale realization functor

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