Student seminar


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.


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


In due time I might add these.