Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MML1003	Categories and Homological Algebra	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	3	2+2+0	speciality	optional	

Teaching Staff in Charge	Lect. MODOI Gheorghe Ciprian, Ph.D., cmodoi@math.ubbcluj.ro	
Aims	The course should teach the students the notions of (co)homology, module, algebra, category, functor, natural transformation, as well as the basic properties relative to these notions. The students should be also able to calculate the (co)homology of a triangulated space and to operate with the notions from module theory (kernel, cokernel, direct sum or product, projective or injective module, functors Hom, Ext, tensor product or Tor) using the categorial language.  	
Content	1. Simplicial sets and triangulated spaces. (Co)chain complexes and the long exact sequence of (co)homology. 
2. Modules and module homomorphisms. Direct sums and products. Tensor product. Graded algebras. Examples of algebras: tensor algebra, symmetric algebra and exterior algebra. 
3. Basic definitions and properties concerning categories and functors. Natural transformations and equivalences of categories. Kernels, cokernels, products and coproducts. Abelian categories. Exact funtors, projective and injective objects. 
4. Projective and injective resolutions. Derived functors. The functor Ext and the functor Tor. 	
Bibliography	1. H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, 1956.
2. S.I. Gelfand, Y.I. Manin, Methods of Homological Algebra, Springer Verlag, 1998.
3. P. Hilton, U. Stambach, A Course in Homological Algebra, Springer Verlag, 1971.
4. I.D. Ion, n. Radu, Algebra, Ed. Didactica si Pedagogica, Bucuresti, 1981.
5. I.D. Ion, C. Nita, N. Radu, D. Popescu, Probleme de algebra, Ed. Didactica si Pedagogica, Bucuresti, 1981.
6. S. MacLane, Homology, Springer Verlag, 1963.
7. C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994. 
	
Assessment	The students will get the grade as follows: The students will get points (from 0.5 to 5) for homeworks (exercices given during the course). The points for an exercise will be awarded to only one student. 10 points are equal to the grade 10 and so on. Additionally there is a writen exam for those students which are not satisfied by the grade obtained by homeworks. The subjects of the exam consists from basic definitions from the course and some exercises chosen from the homeworks.