Course Objective* The student knows an advanced topic in algebra, in this case part of
the theory of Brauer groups. This will include most of the following:
+ the theory of central simple (finite dimensional) algebras over a
field, including the Brauer group;
+ Galois cohomology, and its relation to the Brauer group;
+ cyclic division algebras;
+ norms on algebras;
+ division algebras over local fields;
+ division algebras over number fields.
* The student can independently learn part of this material, and give a
talk about it.
* The student can independently set an assignment on that part, provide
model solutions, and mark the work handed in.
Course ContentOver the real numbers, the only finite dimensional division algebras are
the real numbers, the complex numbers, and the Hamiltonians. Over any
field k, the problem can be studied by means of the Brauer group of k.
For this one considers finite dimensional k-algebras with only the
obvious two ideals, and as centre k. Any such algebra is isomorphic to a
matrix ring over a division ring over k. Under the tensor product over
k, such algebras do not form a group as there is no inverse. But on a
set of equivalence classes defined using matrix sizes, one can put a
group structure, which gives the Brauer group. By studying the structure
of the algebras involved, this group can be linked with group cohomology
of finite Galois extensions of k (Galois cohomology). This makes the
group more computable for certain types of fields, in particular for
finite extensions of the p-adic numbers or the rationals.
The non-commutative case has several surprising properties. For example,
any automorphism of the type of algebra considered is induced by
conjugation with a unit of the algebra. The techniques involved are
therefore also often quite different from, e.g., the ones used in field
theory or Galois theory.
Brauer groups found an application in diophantine geometry, where they
can sometimes be used to show a variety has no rational point over a
given number field (the Brauer-Manin obstruction).
Teaching MethodsA seminar, for 2 hours per week during each week of periods 1 and 2.
Each student gives one or two talks. The student also creates a
corresponding assignment for the other students (which has to be
approved by the coordinators), and grades the corresponding solutions of
the other students based on (approved) etailed model solutions. This
grading is checked at random by the coordinators.
Method of AssessmentThe grade is determined as follows:
* 30% by the talk(s) of the student;
* 30% by the understanding of the material by the student (both for the
talk and the preparation of the assignment and the model solution);
* 40% by the grades of the work handed in.
There will be a minimum attendance requirement in order to get a grade
for this course. The exact requirement will be determined at the
beginning of the seminar.
Due to the nature of the seminar, there will not be a resit.
LiteratureWe shall work mostly from `Associative algebras', Graduate texts in
mathematics volume 88, Springer Verlag. (There is a 2012 reprint of the
original 1982 edition.) It contains most of the material we want to
treat, e.g., central simple algebras, Brauer groups, group cohomology,
relation between Brauer groups and Galois cohomology, completions of
finite dimensional algebras, the Brauer groups of local fields and of
number fields, which corresponds to Chapters 12 through 18. Several of
those topics are covered in other texts as well, and we may use one or
more of those during the seminar.
Additional InformationThe seminar will only run if there are at least ten participating
Recommended background knowledgeThe VU courses Linear Aagebra, Group theory, and Rings and fields, as
well as basic knowledge of module theory (as in the intensive course of
mastermath), or equivalent. Specifically, the relevant material (and
much more) can be found in Chapters 1 through 4, 7 through 11, and
Chapter 13, of Dummit and Foote, Abstract Algebra, third edition, but
also in many other places.
At some stage the course Galois theory will be necessary as well (the
material contained in Section 14.1 and 14.2 of the book by Dummit and
Foote).Knowledge of p-adic numbers can help provide background for
Brauer groups of p-adic fields, but as completion in the non-commutative
case has its own peculiarities, we have to treat it anyway. Algebraic
number theory can be useful but is not required.
|Language of Tuition||English|
|Faculty||Faculty of Science|
|Course Coordinator||prof. dr. R.M.H. de Jeu|
|Examiner||dr. S.R. Dahmen|
prof. dr. R.M.H. de Jeu
dr. S.R. Dahmen
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