Serre's conjecture on projective modules over polynomial rings
| Thesis title in Czech: | Serreho hypotéza o projektivních modulech nad okruhy polynomů |
|---|---|
| Thesis title in English: | Serre's conjecture on projective modules over polynomial rings |
| Key words: | projektivní|modul|komutativní algebra|tenzor|okruhy polynomů |
| English key words: | projective|module|commutative algebra|tensor product|polynomial ring |
| Academic year of topic announcement: | 2022/2023 |
| Thesis type: | Bachelor's thesis |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | Liran Shaul, Ph.D. |
| Author: | Bc. Edward Young - assigned and confirmed by the Study Dept. |
| Date of registration: | 29.09.2022 |
| Date of assignment: | 05.10.2022 |
| Confirmed by Study dept. on: | 05.12.2022 |
| Date and time of defence: | 28.06.2023 08:00 |
| Date of electronic submission: | 11.05.2023 |
| Date of submission of printed version: | 15.05.2023 |
| Date of proceeded defence: | 28.06.2023 |
| Opponents: | Isaac Bird, Ph.D. |
| Guidelines |
| Serre conjectured that any finitely generated projective module over a polynomial ring in finitely many variables is free.
This is now known as the Quillen–Suslin theorem, and was proven independently by Quillen and Suslin (see [1], [2]). The aim of this thesis is to write down a modern proof of this conjecture, and to study generalizations of it related to the Bass–Quillen conjecture. |
| References |
| [1] Quillen, Daniel (1976), "Projective modules over polynomial rings", Inventiones Mathematicae, 36 (1): 167–171.
[2] Suslin, Andrei A. (1976), Проективные модули над кольцами многочленов свободны [Projective modules over polynomial rings are free], Doklady Akademii Nauk SSSR (in Russian), 229 (5): 1063–1066. [3] Lam, T. Y. (2006), Serre's problem on projective modules, Springer Monographs in Mathematics, Berlin; New York: Springer Science+Business Media, pp. 300pp., ISBN 978-3-540-23317-6. [4] Rotman, J. J. (2008). An introduction to homological algebra. Springer Science & Business Media. |
| Preliminary scope of work |
| Serre conjectured that any finitely generated projective module over a polynomial ring in finitely many variables is free.
This is now known as the Quillen–Suslin theorem, and was proven independently by Quillen and Suslin (see [1], [2]). The aim of this thesis is to write down a modern proof of this conjecture, and to study generalizations of it related to the Bass–Quillen conjecture. |