Commutative Algebra

Noetherian rings and modules: primary decomposition, Krull’s dimension theory. Integral extensions. Finite type algebras over a field: Noether’s normalization lemma; Hilbert’s Nullstellensatz; integral closure of an algebra of finite type. Local algebra: Systems of parameters and depth; regular local rings and Cohen-Macaulay rings; Hilbert’s “Syzygies” theorem; homological characterization of regular rings (Serre-Auslander-Buchsbaum). Characteristic polynomial: Hilbert-Serre characteristic polynomial; Samuel’s characteristic polynomial; graded rings and multiplicity; application: invariants in algebraic geometry.

References:
ATIYAH, M. F. e 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. – Comutative Algebra. Vols. 1 e 2, New York, Van- Nostrand, 1960.

 

* Standard program. The teacher has the autonomy to make any changes