General information 
Course unit name: Local Algebra
Course unit code: 568184
Academic year: 20182019
Coordinator: Santiago Zarzuela Armengou
Department: Department of Mathematics and Computer Science
Credits: 6
Single program: S
Estimated learning time 
Total number of hours 150 
Facetoface learning activities 
70 
 Lecture 
30 

 Problemsolving class 
30 

 Seminar 
10 
Supervised project 
20 
Independent learning 
60 
Recommendations 

Competences to be gained during study 

Learning objectives 
Referring to knowledge
Referring to abilities, skills

Teaching blocks 
1. Introduction
1.1. Elements of commutative rings, ideals and modules
1.2. Graded rings and modules
2. Tor and Ext functors
2.1. Module categories; Limits and colimits
2.2. Tensor product and Hom functors
2.3. Projective and injective resolutions
2.4. Tor and Ext functors
2.5. Applications for commutative ring theory
3. Homological dimension; Syzygy theorem
3.1. Global dimension of a ring
3.2. Regular local rings; Characterization: The AuslanderBuchsbaum and Serre theorem
3.3. Syzygy theorem
3.4. Factoriality of regular local rings: AuslanderBuchsbaum theorem
4. Completion
4.1. Completion of abelian topological groups
4.2. Adic completion of a ring
4.3. Noetherian completion
4.4. Hensel’s lemma and some applications of complete rings
5. Koszul complex
5.1. Koszul complex associated to a linear form
5.2. Koszul complex associated to a family of elements
5.3. Regular sequences and the Koszul complex
6. Grade theory
6.1. Module dimensions
6.2. The grade of a module
6.3. Ideals generated by regular sequences
6.4. Complete intersection ideals
7. CohenMacaulay rings and modules
7.1. CohenMacaulay modules
7.2. CohenMacaulay rings
7.3. Transfer of CohenMacaulay properties for morphisms
8. Computation and applications of the Hilbert function
8.1. Hilbert functions
8.2. HilbertSamuel functions
8.3. Multiplicity of local rings
8.4. Examples and geometrical interpretation
Teaching methods and general organization 
The teaching methodology for the subject includes:

Official assessment of learning outcomes 
Examinationbased assessment

Reading and study resources 
Consulteu la disponibilitat a CERCABIB
Book
Bourbaki, N. Commutative algebra. Paris : Hermann ; Reading (Mass) : AddisonWesley, 1972.
Chapters 17 
Bourbaki, N. Algèbre commutative : chapitres 8 et 9. Berlin [etc.] : Springer, 2006.
Bourbaki, N. Algèbre. Chapitre 10, Àlgebre homologique. Paris [etc.] : Masson, 1980.
Bourbaki, N. Algèbre commutative, Chapitre 10,/ SpringerVerlag, Berlin, 2007.
Bruns, W. ; Herzog, J. CohenMacaulay rings. Cambridge University Press, 2005.
Eisenbud, D. Commutative algebra with a view toward Algebraic Geometry. Springer, 1996.
Kaplansky, I. Commutative rings. Chicago [etc.] : University of Chicago Press, 1974.
Kunz, E. Introduction to commutative algebra and algebraic geometry. Boston : Birkhäuser, 1991.
Lafon, J. P. Les formalismes foundamentaoux de l’algèbre commutative. Paris : Herrmann, 1974.
Lang, S. Algebra. New York : Springer, 2005.
Matsumura, H. Commutative ring theory. New York : Cambridge University Press, 2008.
Osborne, M. S. Basic homological algebra. New York [etc.] : Springer, 2000.
Sally, J. D. Numbers of generators of ideals in local rings. New York : Marcel Dekker, 1978.
Serre, J. P. Local algebra. Berlin : Springer, 2000.
Zariski, O. ; Samuel, P. Commutative algebra. New York : Springer, [19751976]