Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers in number fields constitute an important class of commutative rings — the Dedekind domains. This has led to the notions of integral extensions and integrally closed domains.The notion of localization of a ring (in particular the localization with respect to a prime ideal leads to an important class of commutative rings — the local rings.The set of the prime ideals of a commutative ring is naturally equipped with a topology — the Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory — a generalization of algebraic geometry introduced by Grothendieck.The main purpose of this course is to provide important workhorses of commutative algebra assuming only basic course on commutative algebra. Special efforts are made to present the conceptsat the center of the field in a coherent, tightly knit way, streamlined proofs and a focus on the core results. Virtually all concepts and results of commutative algebra have natural interpretations. It is the geometric view point that brings out the true meaning of the theory. The main focus in the course are the folloing core results :• Noether’s Normalisation.• Dimension theory.• Homological characterization of Regular local rings.• Discrete Valuation rings and Dedekind Domains.Apart from deepening the knowledge in commutative algebra, participants of this course are prepared to continue their studies in different directions, for example, algebraic geometry. Another possible direction to go in computational aspects of commutative algebra.
PREREQUISITES: Linear Algebra ;Algebra – First Course ;Commutative Algebra – Basic Course ;Homological Algebra – Elementary Level
Week 1 : Noether’s Normalisation Lemma — Classical Version Week 2 : Noether’s Normalisation Lemma — Classical Version Week 3 : Dimension of Graded Rings and Modules Week 4 : Digression on Basic Concepts Week 5: Dimension Theorem Week 6 : Krull’s Principal Ideal Theorem and its Generalisation Week 7 : Digression on the Language of Algebraic Geometry Week 8: Regular Local Rings Week 9: Homological Dimension of Modules and Global Dimension of Rings Week 10: Homological Characterisation of Regular Local Rings Week 11: Discrete Valuation Rings Week 12: Dedekind Domains