Commutative algebra 60: Regular local rings
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We define regular local rings as the local rings whose dimension is equal to the dimension of their cotangent space. We give s
From playlist Commutative algebra
Commutative algebra 61: Examples of regular local rings
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give some examples of regular local rings. We first give an example of a regular local ring that is not geometrically regul
From playlist Commutative algebra
This lecture is part of an online course on rings and modules. We discuss the operation of inverting the elements of a subset S of a ring R, called localization. We describe the localization in detail for commutative rings, and briefly discuss the non-commutative case. For the other lec
From playlist Rings and modules
Commutative algebra 16 Localization
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. In this lecture we construct the localization R[S^-1] of a ring with respect to a multiplicative subset S, and give some examp
From playlist Commutative algebra
Localization of Rings as Localizations of Categories
We show what it means to localize a category at a set of morphisms and show that usual localization of rings is an instance of this definition.
From playlist Category Theory
Commutative algebra 55: Dimension of local rings
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give 4 definitions of the dimension of a Noetherian local ring: Brouwer-Menger-Urysohn dimension, Krull dimension, degree o
From playlist Commutative algebra
Commutative algebra 56: Hilbert polynomial versus system of parameters
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. In this lecture we show that the dimension of a local ring, defined using Hilbert polynomials, is at most the dimension define
From playlist Commutative algebra
Visual Group Theory, Lecture 7.1: Basic ring theory
Visual Group Theory, Lecture 7.1: Basic ring theory A ring is an abelian group (R,+) with a second binary operation, multiplication and the distributive law. Multiplication need not commute, nor need there be multiplicative inverses, so a ring is like a field but without these properties.
From playlist Visual Group Theory
Ring Theory: We define rings and give many examples. Items under consideration include commutativity and multiplicative inverses. Example include modular integers, square matrices, polynomial rings, quaternions, and adjoins of algebraic and transcendental numbers.
From playlist Abstract Algebra
Commutative algebra 66: Local complete intersection rings
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We define local complete intersection rings as regular local rings divided by a regular sequence. We give a few examples to il
From playlist Commutative algebra
Schemes 5: Definition of a scheme
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We give some historical background, then give the definition of a scheme and some simple examples, and finish by explaining the origin of the word "spectrum".
From playlist Algebraic geometry II: Schemes
Schemes 10: Morphisms of affine schemes
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We try to define morphisms of schemes. The obvious definition as morphisms of ringed spaces fails as we show in an example. Instead we have to use the more su
From playlist Algebraic geometry II: Schemes
Algebraic geometry 41: Completions
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It reviews completions of rings and Hensel's lemma, and gives an example of two analytically isomorphic singularities.
From playlist Algebraic geometry I: Varieties
Commutative algebra 62: Cohen Macaulay local rings
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We define Cohen-Macaulay local rings, and give some examples of local rings that are Cohen-Macaualy and some examples that are
From playlist Commutative algebra
Benjamin Böhme: The Dress splitting and equivariant commutative multiplications
The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: Workshop "Fusion systems and equivariant algebraic topology"
From playlist HIM Lectures: Junior Trimester Program "Topology"
algebraic geometry 25 Morphisms of varieties
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of a morphism of varieties and compares algebraic varieties with other types of locally ringed spaces.
From playlist Algebraic geometry I: Varieties
Schemes 15: Quasicompact, Noetherian
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We define quasi-compact, Noetherian, and locally Noetherian schemes, give a few examples, and show that "locally Noetherian" is a local property.
From playlist Algebraic geometry II: Schemes
Commutative algebra 64: Gorenstein rings
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give an informal introduction to Gorenstein local rings, which are informally those with good duality properties. We give s
From playlist Commutative algebra