

Höhere algorithmische Algebra II
Priv.Doz. Dr. Viktor Levandovskyy
Information
The first lecture will take place on Wed, 13.4 at 16:15 in RS 5, Rochusstrasse.
From 20.4 it will take place in SeMath, Pontdriesch 1016.
The first example class will take place on Thu, 14.4 at 16:15 in Semath. It will run
in the mixed lecture/example class mode.
From 28.4, the example classes take place on Thu at 12:00 in the seminar room of LBfM (405).
Matherials
Lectures and Lecture/Example Class 1,2,3. The homework assignment for 28.4 is in Section 3, marked with HW 1, 2 and so on.
Lectures and HW 45
Slides on noncommutative Gröbner bases
Lectures and HW 67 Homework for 2.6: prove (i)=>(ii) of Theorem 6 (the proof has been added
to the file later). Prove by using it: if gr A is Noetherian (Artinian), then so is A.
Lectures and Lecture/Example Classes 812.
Slides from last lectures on dimension theory
Contents

Generalities on algebraic structures (from magmas to finitely presented associative rings).
Examples of important rings and algebras.


Grading on an algebraic structure. Properties and invariants of graded rings and their ideals.
Algorithmic aspects and noncommutative factorization.


Gröbner bases of ideals in free associative algebra.


Filtration on an algebraic structure and the associated grading. Filteredgraded transfer of
ringtheoretic properties (like theorems of Jacobson). Finite dimensional and good
filtrations. HilbertPoincare series of rings and modules.


Dimension functions: definition, properties (incl. exactness), formalism.
Examples and nonexamples of dimensions, their exactness. The holonomic number
of a module category. Equidimensional and coquidimensional modules.


Krull dimension over commutative rings, its properties and the connection to Hilbert series.


Gel'fandKirillov dimension (GKdim) over Kalgebras. Its properties and behaviour on subalgebras,
submodules and its extension beyond finitely presented case. GKdim on commutative algebras and
comparison with the Krull dimension.


Limited exactness of GKdim and examples of categories where exactess is proven. Computation of GKdim: several approaches. The connection of GKdim to Hilbert series; more on GKdimholonomic modules and BernsteinHilbert multiplicity.

