ABOUT THE COURSE:Commutative algebra and Algebraic geometry are twin topics. They are very much interrelated. Giving a course in algebraic geometry requires good grasp of commutative algebra.. This course will motivate students to learn more advanced topics in commutative algebra and algebraic geometry. I am hoping person who are interested to learn commutative algebra and algebraic geometry will see a beautiful mixture of these two important topics in this course.PREREQUISITES:1. Basic rings and modules, commutative algebra2. Abstract Algebra (Groups, Rings, ideals Modules, Linear Algebra)INDUSTRY SUPPORT: Industry related to algebraic statistics, coding, artificial intellegence, mathematical modeling

Syllabus

Week 1: Some basics of commutative algebra (with a brief sketch of proof) : Integral Extension, going up and going down, Noether normalization, valuation, Krull dimension theorem. Week 2:Affine (quasi) varieties and Projective (quasi) varieties, Hilbert Nullstelensatz.Irreducibility, dimension . Week 3:Regular functions , local rings, function field. category of affine varieties and it’s equivalence with reduced finitely generated k-algebrasWeek 4:Sheaves and affine varieties. Definition of variety (=prevariety) , projective variety, product of variety.Week 5:Morphism of varieties. Various types of morphism. Fiber of a morphism, Chevalley’s theorem.Week 6:Gluing of morphism, Morphism between projective varieties. Week 7:Function fields, birationality of algebraic varieties, blow upsWeek 8:Nonsingular varieties (various definitions) examples. C nonsingular curves and abstract non-singular curves and their equivalence. Week 9:Intersection theory of curves. Bezouts theoremWeek 10:Introduction to schemes and it’s necessity. Morphism of schemes. The category of schemesWeek 11:Properties of schemes and morphism. (1) Open, closed, finite, affine, separatedWeek 12:Properties of schemes and morphism (2) proper , flat, etale, projective, smooth , valuative criterion of properness, separated-ness (without proof)