Student seminar

## Goal

The goal of this seminar is to introduce Brauer groups of rings and schemes, and prove Gabber's theorem (in the version of de Jong) that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ provided $X$ has an ample line bundle.

The Brauer group has a long history: over a field $k$ one can consider central simple algebras, which are nice noncommutative algebras that are closely related to the arithmetic properties of $k$. Under an appropriate equivalence relation these form an abelian group, and this group is an important invariant of a field coinciding with the Galois cohomology of the absolute Galois group of $k$.

On a scheme the correct analogue of Galois cohomology is given by étale cohomology with coefficients in $\mathbb{G}_{\mathrm{m}}$, whereas central simple algebras become Azumaya algebras, and one can show that there is a chain of inclusions
\begin{equation}
\mathrm{Br}(X)\hookrightarrow\mathrm{Br}'(X)=\mathrm{H}_{\text{et}}^2(X,\mathbb{G}_{\mathrm{m}})_{\mathrm{tors}}\hookrightarrow\mathrm{H}_{\text{et}}^2(X,\mathbb{G}_{\mathrm{m}})
\end{equation}
Knowing that the first inclusion is an equality is an important property of a scheme $X$: on one hand it allows to study geometric objects (Azumaya algebras) using purely cohomological techniques, whilst on the other hand one has an explicit geometric description of abstractly defined cohomology classes.

This seminar can also serve as the basis for a follow-up seminar on orders next semester, for which a good understanding of Brauer groups of varieties and their function fields is important.

## Schedule

- October 21
- Grothendieck topologies and the étale site
- by Julia Ramos González, in G.016, at 11h00
- November 4
- Azumaya algebras (notes)
- Dennis Presotto, in G.016, at 11h00
- November 18
- Étale cohomology
- Jens Hemelaer, in G.016, at 11h00
- November 25
- The Brauer group of a ring
- Nikolaas Verhulst, in G.016, at 11h00
- December 2
- Stacks, gerbes and twisted sheaves
- Jens Hemelaer (notes) and Pieter Belmans, in G.016, at 11h00
- December 9
- $\mathrm{Br}=\mathrm{Br}'$ for affine schemes
- Jens Hemelaer, in G.016, at 11h00
- December 16
- $\mathrm{Br}=\mathrm{Br}'$ for affine schemes
- Jens Hemelaer, in G.016, at 11h00
- January 8
- $\mathrm{Br}=\mathrm{Br}'$ for quasiprojective schemes
- by Pieter Belmans, in G.015, at 10h00

## References

In due time I might add these.

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