Lecture Notes on Topology and Algebraic Geometry

Table of Contents

I developed these notes for the course Advanced Geometry, Spring 2018, in ShanghaiTech University. The course covers Point Set Topology, and gives students some tastes of Algbraic Geometry. (Lecture 27-30 is absent)

1 Point Set Topology

Lecture 01
Introduction to this course, some set theory.
Lecture 02
Metric space, open ball, neighborhood, and continuity.
Lecture 03
Limit of a sequence in metric space, continuous functions, and characterization of continuity in terms of open set.
Lecture 04
Closed set and limit point.
Lecture 05
Open set and closed set on the real line.
Lecture 06
Topological equivalence.
Lecture 07
Re-work of what we learned in metric space in the context of topological space, Hausdorff Space, irreducible space, and density.
Lecture 08
Irreducibility, basis and product topology.
lecture 09-10
Lecture 11 (unfinished)

2 Commutative Algebra and Algebraic Geometry

Lecture 12-13
Introduction to group, ring, and field. A first taste of algebraic geometry, polynomial, ideal and linear form, and some theories related to dimension.
Lecture 14-15
Operations between ideals, prime ideals, radical, Zariski Topology, and some theorems.
Lecture 16-18
Hilbert Basis Theorem, for which modules, Noetherian modules, etc. are introduced.
Lecture 19
Quotients of modules and isomorphism theorems.
Lecture 20-21
Hilbert's Nullstellensatz and its consequences, for which maximal ideals and localization are introduced. Irreducible spaces are discussed.
Lecture 22-23
Irreducible Decomposition Theorem, Projective Spaces and Varieties.
Lecture 24-26
Dimension, prime ideals in a ring, in quotient spaces and in their localization, integral things.
Lecture 27
A proof of of the fact that \(\dim \mathbb{F}[x_1,\dots,x_n]=n\).
Lecture 28
Noether Normalization and its geometric interpretation.
Lecture 29
Noether projection for Noether Normalization: the induced map is surjective and has finite fibers.
Lecture 30
Hyperplane intersection characterization of dimension.