Homological algebra | Algebraic geometry | Module theory
In algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by Jean-Pierre Serre in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism. (Wikipedia).
Commutative algebra 44 Flat modules
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 summarize some of the properties of flat modules. In particular we show that for finitely presented modules over local ring
From playlist Commutative algebra
What is a Module? (Abstract Algebra)
A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we introduce the module, give a variety of examples, and talk about the ways in which modules and vector spaces are different from one an
From playlist Abstract Algebra
Drinfeld Module Basics - part 01
This is a very elementary introduction to Drinfeld Modules. We just give the definitions. My wife helped me with this. Any mistakes I make are my fault.
From playlist Drinfeld Modules
Building Drone Rotors - PART 1
We present a multi-part series covering the construction and testing of large multi-rotor propellers.
From playlist Drones
Modular Forms | Modular Forms; Section 1 2
We define modular forms, and borrow an idea from representation theory to construct some examples. My Twitter: https://twitter.com/KristapsBalodi3 Fourier Theory (0:00) Definition of Modular Forms (8:02) In Search of Modularity (11:38) The Eisenstein Series (18:25)
From playlist Modular Forms
Commutative algebra 22 Flatness, tensor products, 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 introduce flat modules, show that R[1/S] is flat, and show that vanishing, flatness, and exactness are all
From playlist Commutative algebra
Commutative algebra 45: Torsion free modules
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 finish the survey of types of modules by briefly discussing torsion-free and coprimary modules. We show that flat modules a
From playlist Commutative algebra
Commutative algebra 38 Survey of module properties
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 a short survey of some of the properties of modules, in particular free, stably free, Zariski locally free, projectiv
From playlist Commutative algebra
Commutative algebra 47: Colimits and exactness
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 discuss the question of when a colimit of exact sequences is exact. We first show that a colimit of right exact sequences i
From playlist Commutative algebra
The top-heavy conjecture for vectors and matroids - Tom Braden
Members’ Seminar Topic: The top-heavy conjecture for vectors and matroids Speaker: Tom Braden SPEAKER AFFILIATION Affiliation: University of Massachusetts, Amherst; Member, School of Mathematics Date: February 08, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. Given a morphism of schemes, we define two maps f^* and f_* taking sheaves of modules on one space to sheaves of modules on the other, and discuss their exac
From playlist Algebraic geometry II: Schemes
Peter SCHOLZE (oct 2011) - 3/6 Perfectoid Spaces and the Weight-Monodromy Conjecture
We will introduce the notion of perfectoid spaces. The theory can be seen as a kind of rigid geometry of infinite type, and the most important feature is that the theories over (deeply ramified extensions of) Q_p and over F_p((t)) are equivalent, generalizing to the relative situation a th
From playlist Peter SCHOLZE (oct 2011) - Perfectoid Spaces and the Weight-Monodromy Conjecture
This lecture is part of an online course on rings and modules. We mainly discuss the problem of whether free modules over a ring have a well defined ran, generalizing the dimension of a vector space. We show that they do over many rings, including all non-zero commutative rings, but give
From playlist Rings and modules