If R is right hereditary, prove that Torj^R(A, B)={0} for all j ≥2 and for all right R-modules A

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If R is right hereditary, prove that Torj^R(A, B)={0} for...

If $R$ is right hereditary, prove that $\operatorname{Tor}_{j}^{R}(A, B)=\{0\}$ for all $j \geq 2$ and for all right $R$-modules $A$ and $B$.
Hint. Every submodule of a projective module is projective.

Key Concepts

Hereditary Rings

A hereditary ring is a ring in which every submodule of a projective module is itself projective. This property has strong implications in homological algebra because it ensures that every module has a projective resolution of length at most one. In the problem at hand, this property is used to prove that the Tor groups vanish for degrees j ? 2.

Tor Functor

The Tor functor is a homological tool used to measure the extent to which the tensor product fails to be exact. It is defined as the derived functor of the tensor product and is computed via projective resolutions. In essence, if you replace one of the modules with a projective resolution, you can calculate the Tor groups as the homology of the resulting complex after tensoring.

Projective Modules

A projective module is a module with the lifting property, meaning that every surjective module homomorphism onto it can be lifted to a map from a projective module. Projective modules are essential in homological algebra because they lead to exact sequences and make the computation of Tor functors more manageable.

Projective Resolutions

A projective resolution of a module is an exact sequence where each module, except possibly the module at the end, is projective. This resolution is used to compute derived functors such as the Tor functor. In the context of our problem, using a projective resolution allows one to compute higher Tor groups, and if projective resolutions have a certain bounded length, this can lead to vanishing higher Tor groups.

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