Noncommutative Polynomial Algebras of Solvable Type and Their Modules

Huishi Li is an emeritus Professor at the Hainan University (China). He received his PhD degree from the University of Antwerp (Belgium) under the supervision of Professor, Doctor Fred Van Oystaeyen in 1990. His research interests include noncommutative rings and algebras, ltered and graded rings, noncommutative Gröbner basis theory and applications to noncommutative algebras. He has authored or co-authored six research books (five of them are written in English and one of them is written in Chinese). Before working at the Hainan University (China), he worked at the Shaanxi Normal Universty (China), the Bilkent University (Turkey), and the Jiaying University (China) respectively. He was also a visiting scholar at the Bielefeld University (Germany), the Antwerp University (Belgium), and the University of Reims (France) respectively. After retiring from the Hainan University, he worked at the Kashgar University (China) as a volunteer teacher of mathematics for one year, and he is now a volunteer teacher of mathematics at the Xinjiang Institute of Technology (China).

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.