In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in). The name comes from a duality property of singular plane curves studied by Gorenstein (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ ⊃ complete intersection rings ⊃ regular local rings (Wikipedia).
Symphony of the Rings - Linking Rings
With some moves from Jeff McBride and Dan Harlan
From playlist My Magic
Ring Definition (expanded) - Abstract Algebra
A ring is a commutative group under addition that has a second operation: multiplication. These generalize a wide variety of mathematical objects like the integers, polynomials, matrices, modular arithmetic, and more. In this video we will take an in depth look at the definition of a rin
From playlist Abstract Algebra
The Mandelbrot set is a churning machine
Its job is to fling off the red pixels and hang onto the green ones. Audio by @Dorfmandesign
From playlist mandelstir
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
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
John Greenlees: The singularity category of C^*(BG) for a finite group G
SMRI Algebra and Geometry Online John Greenlees (Warwick University) Abstract: The cohomology ring H^*(BG) (with coefficients in a field k of characteristic p) is a very special graded commutative ring, but this comes out much more clearly if one uses the cochains C^*(BG), which can be vi
From playlist SMRI Algebra and Geometry Online
Gorenstein Rings In Local Algebra by Srikanth Iyengar
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
This lecture is part of an online course on rings and modules. We continue the previous lecture on complete rings by discussing Hensel's lemma for finding roots of polynomials over p-adic rings or over power series rings. We sketch two proofs, by slowly improving a root one digit at a tim
From playlist Rings and modules
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
Stefano Marseglia, Computing isomorphism classes of abelian varieties over finite fields
VaNTAGe Seminar, February 1, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk: Honda: https://doi.org/10.2969/jmsj/02010083 Tate: https://link.springer.com/article/10.1007/BF01404549 Deligne: https://eudml.org/doc/141987 Hofmann, Sircana: https://arxiv.org/ab
From playlist Curves and abelian varieties over finite fields
Abstract Algebra: The definition of a Ring
Learn the definition of a ring, one of the central objects in abstract algebra. We give several examples to illustrate this concept including matrices and polynomials. Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ We recommend th
From playlist Abstract Algebra
Ring Examples (Abstract Algebra)
Rings are one of the key structures in Abstract Algebra. In this video we give lots of examples of rings: infinite rings, finite rings, commutative rings, noncommutative rings and more! Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦
From playlist Abstract Algebra
Fallout 3 Soundtrack - Main Theme
Main Theme from Fallout 3, composed by Inon Zur. Enjoy the awesomeness :)
From playlist Brilliant Music
Yongnam Lee: Q-Gorenstein Deformations and their applications
In this talk we will discuss Q-Gorenstein schemes and Q-Gorenstein morphisms in a general setting. Based on the notion of Q-Gorenstein morphism, we define the notion of Q-Gorenstein deformations and discuss their properties. Versal properties of Q-Gorenstein deformations and their applicat
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
Alina Marian - On the tautological cohomology of the moduli space of curves
Worldwide Center of Mathematics. Research lecture
From playlist Center of Math Research: the Worldwide Lecture Seminar Series
Miles Reid: Graded rings and Fano 3-Folds - an introduction to Tom and Jerry
It is known that there are about 60000 possibilities for the Hilbert series of anticanonical rings of Q-Fano 3-folds in the Mori category. Graded ring methods allow the construction of Q-smooth Q-Fano 3-folds corresponding to many hundreds of these Hilbert series. One expects the methods t
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
Muon-g-2 ring on land for final transport
After a long water journey, a large crowd gathers to watch the muon-g-2 ring being placed onto the Emmert transporter for its journey to Fermilab.
From playlist Muon g-2
Rahul PANDHARIPANDE - Stable quotients and relations in the tautological ring
The topic concerns relations among the kappa classes in the tautological ring of the moduli space of genus g curves. After a discussion of classical constructions in Wick form, we derive an explicit set of relations obtained from the virtual geometry of the moduli space of stable quotients
From playlist École d’été 2011 - Modules de courbes et théorie de Gromov-Witten