Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
Video Lectures
Displaying all 30 video lectures.
I. The Zariski Topology | |
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Lecture 1 Play Video |
What is Algebraic Geometry? In this beginning lecture, we introduce Algebraic Geometry as the study of the geometry of the set of common zeros of a collection of polynomials. We indicate that this would involve setting up a dictionary of sorts between the Geometric side and the Commutative Algebra side. |
Lecture 2 Play Video |
The Zariski Topology and Affine Space In this lecture, we describe how the Zariski topology is defined using sets of common zeros of collections of polynomials as closed sets, also called algebraic sets. We explain that it is more natural to study ideals generated by subsets of polynomials and indicate how the Hilbert Basis Theorem (or Emmy Noether's Theorem) makes sure that we deal only with finitely many polynomials at a time |
Lecture 3 Play Video |
Going back and forth between subsets and ideals In this lecture, we describe how to pass from a subset of affine space to its associated ideal. We explain how this can be undone by taking zero sets. We also indicate how these procedures are inclusion-reversing and would eventually lead to a correspondence |
II. Irreducibility in the Zariski Topology | |
Lecture 4 Play Video |
Irreducibility in the Zariski Topology In this lecture we describe how closed subsets of affine space correspond to radical ideals, and in particular how points correspond to maximal ideals. We introduce the notion of topological irreducibility and explain how it is a stronger form of the usual connectedness. We outline properties of irreducible spaces. We show how irreducible closed subsets correspond to prime ideals |
Lecture 5 Play Video |
Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime In this lecture we explain in detail why irreducible closed subsets correspond to ideals whose radicals are prime. This is an extension of the correspondence between points and maximal ideals. It in turn extends to the correspondence between closed (algebraic) subsets and radical ideals |
III. Noetherianness in the Zariski Topology | |
Lecture 6 Play Video |
Understanding the Zariski Topology on the Affine Line |
Lecture 7 Play Video |
The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties |
IV. Dimension and Rings of Polynomial Functions | |
Lecture 8 Play Video |
Topological Dimension, Krull Dimension and Heights of Prime Ideals |
Lecture 9 Play Video |
The Ring of Polynomial Functions on an Affine Variety |
Lecture 10 Play Video |
Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces |
V. The Affine Coordinate Ring of an Affine Variety | |
Lecture 11 Play Video |
Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ? |
Lecture 12 Play Video |
Capturing an Affine Variety Topologically |
VI. Open sets in the Zariski Topology and Functions on such sets | |
Lecture 13 Play Video |
Analyzing Open Sets and Basic Open Sets for the Zariski Topology |
Lecture 14 Play Video |
The Ring of Functions on a Basic Open Set in the Zariski Topology |
VII. Regular Functions in Affine Geometry | |
Lecture 15 Play Video |
Quasi-Compactness in the Zariski Topology |
Lecture 16 Play Video |
What is a Global Regular Function on a Quasi-Affine Variety? |
VIII. Morphisms in Affine Geometry | |
Lecture 17 Play Video |
Characterizing Affine Varieties |
Lecture 18 Play Video |
Translating Morphisms into Affines as k-Algebra maps |
Lecture 19 Play Video |
Morphisms into an Affine Correspond to k-Algebra Homomorphisms |
Lecture 20 Play Video |
The Coordinate Ring of an Affine Variety |
Lecture 21 Play Video |
Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture |
IX. The Zariski Topology on Projective Space and Projective Varieties | |
Lecture 22 Play Video |
The Various Avatars of Projective n-space |
Lecture 23 Play Video |
Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology |
X. Graded Rings, Homogeneous Ideals and Homogeneous Localisation | |
Lecture 24 Play Video |
Translating Projective Geometry into Graded Rings and Homogeneous Ideals |
Lecture 25 Play Video |
Expanding the Category of Varieties |
Lecture 26 Play Video |
Translating Homogeneous Localisation into Geometry and Back |
Lecture 27 Play Video |
Adding a Variable is Undone by Homogenous Localization |
XI. The Local Ring of Germs of Functions at a Point | |
Lecture 28 Play Video |
Doing Calculus Without Limits in Geometry |
Lecture 29 Play Video |
The Birth of Local Rings in Geometry and in Algebra |
Lecture 30 Play Video |
The Formula for the Local Ring at a Point of a Projective Variety In the previous lecture, we showed that the local ring of regular functions at a point of an affine variety is given by the localisation of its affine coordinate ring at the maximal ideal corresponding to that point; in the present lecture we prove an analogous result for a projective variety. The proof of the formula involves interplay between localisations at single elements, at prime and at maximal ideals, and taking quotients, and homogenisation and dehomogenisation of polynomials |