# Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity

## Video Lectures

Displaying all 30 video lectures.
I. The Zariski Topology
Lecture 1

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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

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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

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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

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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

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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

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Understanding the Zariski Topology on the Affine Line
Lecture 7

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The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties
IV. Dimension and Rings of Polynomial Functions
Lecture 8

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Topological Dimension, Krull Dimension and Heights of Prime Ideals
Lecture 9

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The Ring of Polynomial Functions on an Affine Variety
Lecture 10

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Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces
V. The Affine Coordinate Ring of an Affine Variety
Lecture 11

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Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?
Lecture 12

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Capturing an Affine Variety Topologically
VI. Open sets in the Zariski Topology and Functions on such sets
Lecture 13

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Analyzing Open Sets and Basic Open Sets for the Zariski Topology
Lecture 14

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The Ring of Functions on a Basic Open Set in the Zariski Topology
VII. Regular Functions in Affine Geometry
Lecture 15

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Quasi-Compactness in the Zariski Topology
Lecture 16

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What is a Global Regular Function on a Quasi-Affine Variety?
VIII. Morphisms in Affine Geometry
Lecture 17

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Characterizing Affine Varieties
Lecture 18

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Translating Morphisms into Affines as k-Algebra maps
Lecture 19

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Morphisms into an Affine Correspond to k-Algebra Homomorphisms
Lecture 20

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The Coordinate Ring of an Affine Variety
Lecture 21

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Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture
IX. The Zariski Topology on Projective Space and Projective Varieties
Lecture 22

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The Various Avatars of Projective n-space
Lecture 23

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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

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Translating Projective Geometry into Graded Rings and Homogeneous Ideals
Lecture 25

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Expanding the Category of Varieties
Lecture 26

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Translating Homogeneous Localisation into Geometry and Back
Lecture 27

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Adding a Variable is Undone by Homogenous Localization
XI. The Local Ring of Germs of Functions at a Point
Lecture 28

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Doing Calculus Without Limits in Geometry
Lecture 29

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The Birth of Local Rings in Geometry and in Algebra
Lecture 30

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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