Overview

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COURSE OUTLINE: Algebraic geometry played a central role in 19th century math. The deepest results of Abel, Riemann, Weierstrass, and the most important works of Klein and Poincar/’e were part of this subject. In the middle of the 20th century algebraic geometry had been through a large reconstruction. The domain of application of its ideas had grown tremendously, in the direction of algebraic varieties over arbitrary fields and more general complex manifolds. Many of the best achievements of algebraic geometry could be cleared of the accusation of incomprehensibility or lack of rigor. The foundation for this reconstruction was (commutative) algebra. In the 1950s and 60s have brought substantial simplifications to the foundation of algebraic geometry, which significantly came closer to the ideal combination of logical transparency and geometric intuition. 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.

Syllabus

Intro Introduction to Algebraic Geometry and Commutative Algebra.
noc20 ma20 lec01 Motivation for K algebraic sets.
noc20 ma20 lec02 Definitions and examples of Affine Algebraic Set.
noc20 ma20 lec03 Rings and Ideals.
noc20 ma20 lec04 Operation on Ideals.
noc20 ma20 lec05 Prime Ideals and Maximal Ideals.
noc20 ma20 lec06 Krull's Theorem and consequences.
noc20 ma20 lec07 Module, submodules and quotient modules.
noc20 ma20 lec08 Algebras and polynomial algebras.
noc20 ma20 lec09 Universal property of polynomial algebra and examples.
noc20 ma20 lec10 Finite and Finite type algebras.
noc20 ma20 lec11 K Spectrum K rational points.
noc20 ma20 lec12 Identity theorem for Polynomial functions.
noc20 ma20 lec13 Basic properties of K algebraic sets.
noc20 ma20 lec14 Examples of K algebraic sets.
noc20 ma20 lec15 K Zariski Topology.
noc20 ma20 lec16 The map VL.
noc20 ma20 lec17 Noetherian and Artinian Ordered sets.
noc20 ma20 lec18 Noetherian induction and Transfinite induction.
noc20 ma20 lec19 Modules with Chain Conditions.
noc20 ma20 lec20 Properties of Noetherian and Artinian Modules.
noc20 ma20 lec21 Examples of Artinian and Noetherian Modules.
noc20 ma20 lec22 Finite modules over Noetherian Rings.
noc20 ma20 lec23 Hilbert’s Basis TheoremHBT.
noc20 ma20 lec24 Consequences of HBT.
noc20 ma20 lec25 Free Modules and rank.
noc20 ma20 lec26 More on Noetherian and Artinian modules.
noc20 ma20 lec27 Ring of FractionsLocalization.
noc20 ma20 lec28 Nil radical, contraction of ideals.
noc20 ma20 lec29 Universal property of S 1A.
noc20 ma20 lec30 Ideal structure in S 1A.
noc20 ma20 lec31 Consequences of the Correspondence of Ideals.
noc20 ma20 lec32 Consequences of the Correspondence of IdealsContd.
noc20 ma20 lec33 Modules of Fraction and universal properties.
noc20 ma20 lec34 Exactness of the functor S 1.
noc20 ma20 lec35 Universal property of Modules of Fractions.
noc20 ma20 lec36 Further properties of Modules and Module of Fractions.
noc20 ma20 lec37 Local Global Principle.
noc20 ma20 lec38 Consequences of Local Global Principle.
noc20 ma20 lec39 Properties of Artinian Rings.
noc20 ma20 lec40 Krull Nakayama Lemma.
noc20 ma20 lec41 Properties of IK and VL maps.
noc20 ma20 lec42 Hilbert’s Nullstelensatz.
noc20 ma20 lec43 Hilbert’s NullstelensatzContd.
noc20 ma20 lec44 Proof of Zariski’s LemmaHNS 3.
noc20 ma20 lec45 Consequences of HNS.
noc20 ma20 lec46 Consequences of HNSContd.
noc20 ma20 lec47 Jacobson Ring and examples.
noc20 ma20 lec48 Irreducible subsets of Zariski TopologyFinite type K algebra.
noc20 ma20 lec49 Spec functor on Finite type K algebras.
noc20 ma20 lec51 Zariski Topology on arbitrary commutative rings.
noc20 ma20 lec52 Spec functor on arbitrary commutative rings.
noc20 ma20 lec53 Topological properties of Spec A.
noc20 ma20 lec54 Example to support the term “Spectrum”.
noc20 ma20 lec55 Integral Extensions.
noc20 ma20 lec56 Elementwise characterization of Integral extensions.
noc20 ma20 lec57 Properties and examples of Integral extensions.
noc20 ma20 lec58 Prime and Maximal ideals in integral extensions.
noc20 ma20 lec59 Lying over Theorem.
noc20 ma20 lec60 Cohen Siedelberg Theorem.