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
- Compactness.
- Lecture 11 (unfinished)
- Connectedness.
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.