# Babytop Seminar

## Spring 2021

In Spring 2021, Babytop was about "yet another Bloch-Kato seminar".-
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### rwb

In this talk I will give an overview of Voevodsky's resolution of the Bloch-Kato and Beilinson-Lichtenbaum conjectures. This talk will follow Ch.1 of HW.

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### rwb

I will set out the key properties which characterize the category of motives. The goal is to provide a self-contained point of reference for the remainder of the seminar.

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### rwb

Following Geisser–Levine as well as Kahn's 'Geisser–Levine revisited' and HW Ch.2 I will present the proof that H90(n) implies BL(n). At its core the argument relies on an analysis of the effective slice filtration and Gabber's rigidity theorem.

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### rwb

The second half of the argument that H90(n) implies BL(n).

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### Elden Elmanto

Voevodsky's "motivic Eilenberg-Maclane spaces" has always been, to me at least, a daunting ingredient in the proof of the Bloch-Kato conjectures. I present some (attempts at) simplification.

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### Elden Elmanto

Cotinuation of the previous talk.

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### rwb

I will present the proof of H90(n) in the case where Milnor K-theory vanishes. The proof relies on a direct analysis of how etale cohomology changes under a cyclic degree ell Galois extension.

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### rwb

I will explain how motivic cohomology provides a rich collection of obstructions to the existence of rational points on a variety, then introduce Rost varieties as a family of varieties which exhibit a surprising mixture of chromatic and galois theoretic obstructions simultaneously.

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### rwb

I will explain how assuming the existence of a Rost varieties associated with each symbol we can prove Bloch-Kato.

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### rwb

I will explain the construction of a Rost variety associated to each symbol and sketch the proof that such a variety has all the desired properties.

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This seminar was organized by Robert Burklund.