Dimension theory of affine algebras: Principal ideal theorem, Noether normalization lemma, dimension and transcendence degree, catenary property of affine rings, dimension and degree of the Hilbert polynomial of a graded ring, Nagata's altitude formula, Hilbert's Nullstellensatz, finiteness of integral closure.
Hilbert-Samuel polynomials of modules: Associated primes of modules, degree of the Hilbert polynomial of a graded module, Hilbert series and dimension, Dimension theorem, Hilbert-Samuel multiplicity, associativity formula for multiplicity. Complete local rings: Basics of completions, Artin-Rees lemma, associated graded rings of filtrations, completions of modules, regular local rings. Basic Homological algebra: Categories and functors, derived functors, Hom and tensor products, long exact sequence of homology modules, free resolutions, Tor and Ext, Koszul complexes. Cohen-Macaulay rings: Regular sequences, quasi-regular sequences, Ext and depth, grade of a module, Ischebeck's theorem, basic properties of Cohen-Macaulay rings, Macaulay's unmixed theorem, Hilbert-Samuel multiplicity and Cohen-Macaulay rings, rings of invariants of finite groups.

To have basic ideas of commutative algebra.

Students will have very good knowledge of commutative algebra.

D. Eisenbud, Commutative Algebra (with a view toward algebraic geometry,), Springer-Verlag , 2003.

M. F. Atiyah and I. G. MacDonald, Introduction to commutative algebra, Taylor & Francis , 1994.

H. Matsumura, Commutative ring theory, Cambridge University press , 1980.

W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press , 1998.