Noncommutative Polynomial Algebras of Solvable Type and Their Modules Basic Constructive-Computational Theory and Methods
Noncommutative Polynomial Algebras of Solvable Type and Their Modules is the ﬁrst book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. In doing so, this book covers:
- A constructive introduction to solvable polynomial algebras and Gröbner basis theory for left ideals of solvable polynomial algebras and submodules of free modules
- The new ﬁltered-graded techniques combined with the determination of the existence of graded monomial orderings
- The elimination theory and methods (for left ideals and submodules of free modules) combining the Gröbner basis techniques with the use of Gelfand-Kirillov dimension, and the construction of diﬀerent kinds of elimination orderings
- The computational construction of ﬁnite free resolutions (including computation of syzygies, construction of diﬀerent kinds of ﬁnite minimal free resolutions based on computation of diﬀerent kinds of minimal generating sets), etc.
This book is perfectly suited to researchers and postgraduates researching noncommutative computational algebra and would also be an ideal resource for teaching an advanced lecture course.
1. Solvable Polynomial Algebras. 1.1. Definition, Examples, Basic Properties. 1.2. A Constructive Characterization. 1.3. The Solvable Polynomial Algebras H(A). 1.4. Gröbner Bases of Left Ideals. 1.5. Finite Gröbner Bases → The Noetherianess. 1.6. Elimination in Left Ideals. 2. Gröbner Basis Theory of Free Modules. 2.1. Monomial Orderings on Free Modules. 2.2. Gröbner Bases of Submodules. 2.3. The Noncommutative Buchberger Algorithm. 2.4. Elimination in Submodules. 2.5. Application to Module Homomorphisms. 3. Computation of Finite Free Resolutions and Projective Dimension. 3.1. Computation of Syzygies. 3.2. Computation of Finite Free Resolutions. 3.3. Global Dimension and Stability. 3.4. Computation of p.dimAM. 4. Computation of Minimal Finite Graded Free Resolutions. 4.1. N-graded Solvable Polynomial Algebras of (B; d( ))-type. 4.2. N-Graded Free Modules. 4.3. Computation of Minimal Homogeneous Generating Sets. 5. Computation of Minimal Finite Filtered Free Resolutions. 5.1. N-Filtered Solvable Polynomial Algebras of (B; d( ))-Type. 5.2. N-Filtered Free Modules. 5.3. Filtered-Graded Transfer of Gröbner Bases for Modules. 5.4. F-Bases and Standard Bases with Respect to Good Filtration. 5.5. Computation of Minimal F-Bases and Minimal Standard Bases. 5.6. Minimal Filtered Free Resolutions and Their Uniqueness. 5.7. Computation of Minimal Finite Filtered Free Resolutions. Appendix.