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Concept
Change of rings
Résumé
In algebra, a change of rings is an operation of changing a coefficient ring to another.
Given a ring homomorphism , there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N, one can form
the induced module, formed by extension of scalars,
the coinduced module, formed by co-extension of scalars, and
formed by restriction of scalars.
They are related as adjoint functors:
and
This is related to Shapiro's lemma.
Throughout this section, let and be two rings (they may or may not be commutative, or contain an identity), and let be a homomorphism. Restriction of scalars changes S-modules into R-modules. In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction.
Suppose that is a module over . Then it can be regarded as a module over where the action of is given via
where denotes the action defined by the -module structure on .
Restriction of scalars can be viewed as a functor from -modules to -modules. An -homomorphism automatically becomes an -homomorphism between the restrictions of and . Indeed, if and , then
As a functor, restriction of scalars is the right adjoint of the extension of scalars functor.
If is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
Tensor product of modules and Tensor product of modules#Extension of scalars
Extension of scalars changes R-modules into S-modules.
Let be a homomorphism between two rings, and let be a module over . Consider the tensor product , where is regarded as a left -module via . Since is also a right module over itself, and the two actions commute, that is for , (in a more formal language, is a -bimodule), inherits a right action of . It is given by for , . This module is said to be obtained from through extension of scalars.
Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an R-module with an -bimodule is an S-module.
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