Docente: Marco Fontana
Dipartimento di Matematica e Fisica,
Matematica: Stanza 204 tel. [+39] 06 5733 8232 e-mail: fontana(at)mat.uniroma3.it Orario di ricevimento (I Semestre 2013/14): Martedì ore 10-11 e Giovedì ore 13-14 oppure per appuntamento da fissare via email |
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Appunti on-line di corsi di introduzione all’algebra commutativa ed altri links utili
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This course is an introduction to modules over rings, Noetherian modules, unique factorization domain and polynomial rings over them, modules over principal ideal domains, localization [pdf file 202K].
Chapter 0 Ring Theory Background (7 pp.); Chapter 1 Primary Decomposition and Associated Primes (15 pp.); Chapter 2 Integral Extensions (9 pp.); Chapter 3 Valuation Rings (9 pp.); Chapter 4 Completion (10 pp.); Chapter 5 Dimension Theory (15 pp.); Chapter 6 Depth (4 pp.); Chapter 7 Homological Methods (8 pp.); Chapter 8 Regular Local Rings (3 pp.; ) Exercises (7 pp.); Solutions (8 pp.); List of Symbols; Index
1 Rings and Modules 3
1.1 Ideals and Radicals 3
1.2 Polynomial rings and Localization of rings 8
1.3 Modules 11
1.4 Zariski Topology 12
Exercises 14
2 Noetherian Rings 17
2.1 Noetherian Rings and Modules 17
2.2 Primary Decomposition of Ideals 19
2.3 Artinian Rings and Modules 23
2.4 Krull's Principal Ideal Theorem 27
Exercises 14
3 Integral Extensions 32
3.1 Integral Extensions 32
3.2 Noether Normalization 35
3.3 Finiteness of Integral Closure 38
Exercises 42
4 Dedekind Domains 44
4.1 Dedekind Domains 45
4.2 Extensions of Primes 50
Exercises 42
A Appendix: Primary Decomposition of Modules 55
A.1 Associated Primes of Modules 55
A.2 Primary Decomposition of Modules 58
Exercises 62
References
S. Kleiman, Commutative Algebra, MIT OpenCourseWare >>
A. Altman and S. Kleiman, A Term of Commutative Algebra (2012) .pdf
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . iii
1. Rings and Ideals . . . . . . . . . . . . . . . . . . . 1
2. Prime Ideals . . . . . . . . . . . . . . . . . . . . 6
3. Radicals . . . . . . . . . . . . . . . . . . . . . . 10
4. Modules . . . . . . . . . . . . . . . . . . . . . . 14
5. Exact Sequences . . . . . . . . . . . . . . . . . . . 20
6. Direct Limits . . . . . . . . . . . . . . . . . . . . 26
7. Filtered Direct Limits . . . . . . . . . . . . . . . . . 33
8. Tensor Products . . . . . . . . . . . . . . . . . . . 37
9. Flatness . . . . . . . . . . . . . . . . . . . . . . 43
10. Cayley–Hamilton Theorem . . . . . . . . . . . . . . . 49
11. Localization of Rings . . . . . . . . . . . . . . . . . 55
12. Localization of Modules . . . . . . . . . . . . . . . . 61
13. Support . . . . . . . . . . . . . . . . . . . . . 66
14. Krull–Cohen–Seidenberg Theory . . . . . . . . . . . . . 71
15. Noether Normalization . . . . . . . . . . . . . . . . 75
Appendix: Jacobson Rings . . . . . . . . . . . . . . . 80
16. Chain Conditions . . . . . . . . . . . . . . . . . . 82
17. Associated Primes . . . . . . . . . . . . . . . . . . 87
18. Primary Decomposition . . . . . . . . . . . . . . . . 91
19. Length . . . . . . . . . . . . . . . . . . . . . . 97
20. Hilbert Functions . . . . . . . . . . . . . . . . . . 101
Appendix: Homogeneity . . . . . . . . . . . . . . . . 107
21. Dimension . . . . . . . . . . . . . . . . . . . . 109
22. Completion . . . . . . . . . . . . . . . . . . . . 115
23. Discrete Valuation Rings . . . . . . . . . . . . . . . 122
24. Dedekind Domains . . . . . . . . . . . . . . . . . 127
25. Fractional Ideals . . . . . . . . . . . . . . . . . . 131
26. Arbitrary Valuation Rings . . . . . . . . . . . . . . . 136
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . iii
1. Rings and Ideals . . . . . . . . . . . . . . . . . . . 1
2. Prime Ideals . . . . . . . . . . . . . . . . . . . . 6
3. Radicals . . . . . . . . . . . . . . . . . . . . . . 10
4. Modules . . . . . . . . . . . . . . . . . . . . . . 14
5. Exact Sequences . . . . . . . . . . . . . . . . . . . 20
6. Direct Limits . . . . . . . . . . . . . . . . . . . . 26
7. Filtered Direct Limits . . . . . . . . . . . . . . . . . 33
8. Tensor Products . . . . . . . . . . . . . . . . . . . 37
9. Flatness . . . . . . . . . . . . . . . . . . . . . . 43
10. Cayley–Hamilton Theorem . . . . . . . . . . . . . . . 49
11. Localization of Rings . . . . . . . . . . . . . . . . . 55
12. Localization of Modules . . . . . . . . . . . . . . . . 61
13. Support . . . . . . . . . . . . . . . . . . . . . 66
14. Krull–Cohen–Seidenberg Theory . . . . . . . . . . . . . 71
15. Noether Normalization . . . . . . . . . . . . . . . . 75
Appendix: Jacobson Rings . . . . . . . . . . . . . . . 80
16. Chain Conditions . . . . . . . . . . . . . . . . . . 82
17. Associated Primes . . . . . . . . . . . . . . . . . . 87
18. Primary Decomposition . . . . . . . . . . . . . . . . 91
19. Length . . . . . . . . . . . . . . . . . . . . . . 97
20. Hilbert Functions . . . . . . . . . . . . . . . . . . 101
Appendix: Homogeneity . . . . . . . . . . . . . . . . 107
21. Dimension . . . . . . . . . . . . . . . . . . . . 109
22. Completion . . . . . . . . . . . . . . . . . . . . 115
23. Discrete Valuation Rings . . . . . . . . . . . . . . . 122
24. Dedekind Domains . . . . . . . . . . . . . . . . . 127
25. Fractional Ideals . . . . . . . . . . . . . . . . . . 131
26. Arbitrary Valuation Rings . . . . . . . . . . . . . . . 136
Contenuto: Teoria della dimensione; Grado di Trascendenza; Polinomi di Hilbert; Dimensione di Krull; Anelli regolari e singolari; Molteplicita. Successioni regolari ad anelli di Cohen-Macaulay; Completamenti. Metodi omologici; Introduzione agli schemi; Topologa di Zariski; Fasci di anelli e moduli; Il linguaggio dei funtori; Discesa fedelmente piatta. Morfismi étale, piatti e lisci; Differenziali algebrici; Algebre étale; Anelli Henseliani; Introduzione alle topologie di Grothendieck; La topologia étale.
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- Grothendieck Circle >>
- The long-term goal of the Grothendieck Circle is to make publicly available (and in some cases translate) the material written by and about Alexandre Grothendieck as well as to provide biographical material on Grothendieck's life and his origins. For the present, we have posted several of his writings appearing on the web in complete form for the first time, together with many links to other online sources of his work. Since many of these texts are unpublished or are out-of-print we hope this site will serve as a valuable resource, expanding over time
- Emmy Noether. Un articolo di Aldo Brigaglia apparso su Lettera Pristem .pdf
N.B. Le indicazioni bibliografiche sono minimali, da integrare con altre fonti bibliografiche (sia tradizionali che on-line) indicate nella bibliografia del corso.
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