Commutative Algebra
Noetherian rings and modules: primary decomposition, Krull dimension theory. Integer extensions. Finite type algebras over a body: Noether’s normalization lemma; Hilbert’s zeros theorem; integer closure of a finite type algebra. Local algebra: system of parameters and depth; regular and Cohen-Macaulay local rings; “Syzygies” theorem (Hilbert); homological characterization of regular rings (Serre-Auslander-Buchsbaum). Characteristic polynomial: Hilbert-Serre characteristic polynomial; Samuel characteristic polynomial; graded rings and multiplicity: application: invariants of algebraic geometry.
References:
ATIYAH, M. F. and MACDONALD, I. G. – Introduction to Commutative Algebra. Reading, Mass., Addison-Wesley, 1969.
MATSUMURA, H. – Commutative Algebra. Reading, Mass., Benjamin- Commings, 1980.
SERRE, J. P. – Algebre Locale – Multiplicités. Berlin. Springer-Verlag, 1965
ZARISKI, O., SAMUEL, P. – Commutative Algebra. Vols. 1 and 2, New York, Van-Nostrand, 1960.
ATIYAH, M. F. and MACDONALD, I. G. – Introduction to Commutative Algebra. Reading, Mass., Addison-Wesley, 1969.
MATSUMURA, H. – Commutative Algebra. Reading, Mass., Benjamin- Commings, 1980.
SERRE, J. P. – Algebre Locale – Multiplicités. Berlin. Springer-Verlag, 1965
ZARISKI, O., SAMUEL, P. – Commutative Algebra. Vols. 1 and 2, New York, Van-Nostrand, 1960.
* Basic syllabus. The teacher has the autonomy to make any changes.