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| | I am Oscar and I completely dig that name. North Dakota is where me and my spouse live. Bookkeeping is what I do. To collect badges is what her family and her enjoy.<br><br>Here is my web blog :: [http://www.neweracinema.com/tube/blog/74189 http://www.neweracinema.com] |
| In [[abstract algebra]], '''restriction of scalars''' is a procedure of creating a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] <math>R</math> from a module over another ring <math>S</math>, given a [[ring homomorphism|homomorphism]] <math>f : R \to S</math> between them. Intuitively speaking, the resulting module "remembers" less information than the initial one, hence the name.
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| In [[algebraic geometry]], the term "restriction of scalars" is often used as a synonym for [[Weil restriction]].
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| == Definition ==
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| Let <math>R</math> and <math>S</math> be two rings (they may or may not be [[commutative ring|commutative]], or contain an [[identity element|identity]]), and let <math>f:R \to S</math> be a homomorphism. Suppose that <math>M</math> is a module over <math>S</math>. Then it can be regarded as a module over <math>R</math>, if the action of <math>R</math> is given via <math>r \cdot m = f(r) \cdot m</math> for <math>r \in R</math> and <math>m \in M</math>.
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| == Interpretation as a functor ==
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| Restriction of scalars can be viewed as a [[functor]] from <math>S</math>-modules to <math>R</math>-modules. An <math>S</math>-homomorphism <math>u : M \to N</math> automatically becomes an <math>R</math>-homomorphism between the restrictions of <math>M</math> and <math>N</math>. Indeed, if <math>m \in M</math> and <math>r \in R</math>, then
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| : <math>u(r \cdot m) = u(f(r) \cdot m) = f(r) \cdot u(m) = r\cdot u(m)\,</math>.
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| As a functor, restriction of scalars is the [[right adjoint]] of the [[extension of scalars]] functor.
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| If <math>R</math> is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
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| == The case of fields ==
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| When both <math>R</math> and <math>S</math> are [[field (mathematics)|fields]], <math>f\ </math> is necessarily a [[monomorphism]], and so identifies <math>R</math> with a [[subfield]] of <math>S</math>. In such a case an <math>S</math>-module is simply a [[vector space]] over <math>S</math>, and naturally over any subfield thereof. The module obtained by restriction is then simply a vector space over the subfield <math>R \subset S</math>.
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| <!-- #REDIRECT[[Weil_restriction]] -->
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| [[Category:Abstract algebra]]
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| {{algebra-stub}}
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I am Oscar and I completely dig that name. North Dakota is where me and my spouse live. Bookkeeping is what I do. To collect badges is what her family and her enjoy.
Here is my web blog :: http://www.neweracinema.com