The main task is to give an introduction to modern commutative algebra with a special regard to commutative ring theory, arithmetic, homological methods and algebraic geometry.
Expected learning outcomes
(first part) Theory and computations of primary decompositions, integral extensions, regular rings & a first step in dimension theory. (9 credits) The additional 3 credits course is providing the next key step in dimension theory.
Lesson period: First semester
(In case of multiple editions, please check the period, as it may vary)
We assume known the basic language of categories and functors up to the Yoneda Lemma. We also assume the standard notions: ideals, polynomial rings, multiplicatively closed subsets and localizations, tensor products of modules, Noetherian rings & modules. For example, with respect to M. Reid Undergraduate Commutative Algebra LMS student text series C.U.P. 1995 we will be dealing fast with Chapters 4, 5 & 7, 8 in the commutative algebra course assuming the other Chapters.
Assessment methods and Criteria
The final examination consists of three parts: a written exam, an oral exam and some homework.
Commutative Algebra (first part)
Substitution principle, prime spectrum & points. Hilbert's Nullstellensatz Primary decomposition & regular rings. Integral ring extensions & valuations Noether's normalization. A first step in dimension theory. Derivations & Zariski tangent space.
Lectures and exercises.
Course notes are available at the official ARIEL page. Moreover, the following are classical references: - S. Bosch: Algebraic Geometry and Commutative Algebra. Universitext, Springer, 2013, 504 p. - M.F. Atiyah & I.G. MacDonald: Introduction to Commutative Algebra. Addison-Wesley 1969 (ed. Feltrinelli, 1981)
Commutative Algebra mod/2
Next step in dimension theory. Primary decomposition of modules, support & associated primes. Filtered/graded modules & Artin-Rees. Hilbert-Samuel polynomial & the dimension theorem.
S. Raghavan: Balwant Singh & R. Sridharan Homological Methods in Commutative Algebra Oxford Univ. Press/TIFR, 1975