Viktor Levandovskyy HAA II, SS2016

### 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 10-16.
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 4-5
Slides on noncommutative Gröbner bases
Lectures and HW 6-7 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 8-12.
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. Filtered-graded transfer of ring-theoretic properties (like theorems of Jacobson). Finite dimensional and good filtrations. Hilbert-Poincare series of rings and modules. Dimension functions: definition, properties (incl. exactness), formalism. Examples and non-examples of dimensions, their exactness. The holonomic number of a module category. Equidimensional and co-quidimensional modules. Krull dimension over commutative rings, its properties and the connection to Hilbert series. Gel'fand-Kirillov dimension (GKdim) over K-algebras. 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 GKdim-holonomic modules and Bernstein-Hilbert multiplicity.

RWTH Aachen University Division of Mathematics Lehrstuhl D für Mathematik