Binary operations | Homological algebra

Tor functor

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group. In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Algebra. (Wikipedia).

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Torus Magic

Buy at http://www.shapeways.com/shops/GeometricToy Torus Magic is a transformable torus. This torus object is constructed with many rings,and transforms flat,spherical etc. Also you can turn inside out the torus. Copyright (c) 2014,AkiraNishihara

From playlist 3D printed toys

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Torus Magic 2

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From playlist Handmade geometric toys

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Torus Magic with Ring 1

Buy at http://www.shapeways.com/shops/GeometricToy "Torus Magic" can eat another torus.This torus object is constructed with 30 large rings(70mm diameter) and many small rings. Copyright (c) 2015,AkiraNishihara

From playlist 3D printed toys

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Torus Magic (50mm)

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From playlist 3D printed toys

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Buy at http://www.shapeways.com/shops/GeometricToy This object consists of two "Torus Magic".These torus objects are constructed with 30 large rings(70mm diameter) and many small rings. Copyright (c) 2015,Akira Nishihara

From playlist 3D printed toys

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Torus Magic with Ring 2

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From playlist 3D printed toys

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From playlist Commutative algebra

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From playlist Commutative algebra

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From playlist Tutorial-a-thon 2021 Fall

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From playlist AATRN 2020

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From playlist PHYSICS 16.6 TORSION

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Homological algebra 4: Properties of Tor over rings

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From playlist TUTORIALS & HOW TO's

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Hopf algebra | Group representation | Koszul complex | Category of abelian groups | Samuel Eilenberg | Direct sum of modules | Homological algebra | Zero divisor | Group cohomology | Algebraic topology | Torsion subgroup | Chain complex | Bimodule | Eilenberg–Moore spectral sequence | Regular sequence | Tensor product of modules | Algebra over a field | Polynomial ring | Derived tensor product | Free module | Projective module | Group ring | Künneth theorem | Flat module | Cokernel | Free abelian group | Henri Cartan | Integer | Mathematics | Lie algebra | Ring (mathematics) | Category theory | Exterior algebra | Hochschild homology | Derived functor | Universal coefficient theorem | Ext functor | Multiplicatively closed set | Flat morphism | Universal enveloping algebra | Finitely generated abelian group | Abelian group | Module (mathematics) | Commutative ring