Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Algebră şi Geometrie (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MO267	Special Topics in Modern Mathematics (2)	

Semester	Hours: C+S+L	Category	Status	
2	2+2+0		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
MODOI Gheorghe Ciprian	Ph.D.	Lect.	Algebra	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The introduction of the notions of (co)homology relative to different structures (triangulated spaces, groups, Lie algebras). 
The exposition of the main results concerning the homology of the complexes. 
The introduction of the notions: category, functor and natural transformation, as well as the presentation of some basic properties relative to these notions. 
The definition of the derived functors using (co)resolutions and the study of some homological dimensions.   	

IV. Course contents
1. Simplicial sets and triangulated spaces. The topological motivation of homological algebra. 
2. Homology  and cohomology. Complexes, homomorphisms of complexes and homotopy of complexes.
3. Examples of homology and cohomology.  
4. Categories and functors. 
5. Derived functors and homological dimensions. 	

V. Bibliography
[GM] S.I. Gelfand,Yu.I. Manin, Methods of Homological Algebra, Springer-Verlag 1998.
[HS] P. Hilton, U. Stammbach, A Course in Homological Algebra, Springer-Verlag 1971.
[S] B. Stenstrom, Rings of Quotients, Springer-Verlag, 1975.
[W] C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994.	

VI. Thematic of didactic activities per weeks
Week 1: Simplicial sets. Triangulated spaces. Maps of simplicial sets. [GM, 1.1, 1.2].
Week 2: Topological spaces. Sheaves. [GM, 1.3, 1.5].
Week 3: Complexes and complex homomorphisms. Homotopy of complexes. [GM, 1.7], [W, 1.1, 1.2].
Week 4: Coefficient systems. Long exact sequences. [GM, 1.6], [W, 1.3]. 
Week 5: Group (co)homology. Hochschild and cyclic homology. [HS, 6.1, 6.2], [W, 6.1, 6.2, 9.1]. 
Week 6: Cohomology of  Lie algebras.[HS, 7.1, 7.2], [W, 7.1, 7.2, 7.4]
Week 7: Categories. Limits and colimits. [HS, 2.1], [W, A.1, A.5], [S, 1.4, 4.8].
Week 8: Functors and natural transformations. [HS, 2.4], [S, 1.4], [W, A.2, A.3].
Week 9: Equivalences of categories. [GM, 2.2], [S, 4.1].
Week 10: Adjoint functors. [HS, 2.7], [S, 4.9], [W, A.6].
Week 11: Additive and abelian categories. [GM, 2.5], [HS, 2.9], [S, 4.4], [W, A.4].
Week 12: Resolutions and coresolutions. Derived functors. [W, 2.2, 2.3, 2.4, 2.5].
Week 13: The functors Ext and Tor. [W, 3.1, 3.2, 3.3, 3.4].
Week 14: Homological dimensions. [W, 4.1, 4.2].   	

VII. Didactic methods used
Expositions accompanied by proofs, heuristic conversations, exemplifications, oriented inductive and deductive discover, individual or group research projects, individual study, homework assignments.	

VIII. Assessment
The activity ends with a written final exam (grade E), a project (grade P) and the mean of the continuous grading of the homework (M). The exam subjects have theoretical and practical questions (exercises and problems) from all the studied topics. The subject of the project may be a theoretical one (additionally with respect to the studied topics) or a more complex exercise (by example some computations involving the use of the computer and of some mathematical software) and IT WILL BE established in common by teacher and each student. The final grade is the weighted mean of the three grades mentioned above, conditioned by all the grades being at least 5 from 10. Otherwise, the exam will not be passed. The final grade will be calculated by formula  N= 50%E + 25%L + 25%R.	

IX. Additional references
[CE] H. Cartan, S. Eilenberg - Homological Algebra, Princeton University Press, 1956.
[L] J. L. Loday - Cyclic Homology, Springer-Verlag 1992.
[M] S. MacLane - Homology, Springer-Verlag 1963.
[N] D.G. Northcott, An Introduction to Homological Algebra, Cambridge University Press, 1960.	

	Dean,							Chair head,
	Prof. TAMBULEA Leon, Ph.D			Conf. BREAZ Simion Sorin, Ph. D.

	Date,							Course responsible,
	4.03.2009						Lect. MODOI Gheorghe Ciprian, Ph.D.