Tensor product of modules: Difference between revisions

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=== Modules over commutative rings ===

Let ''R'' be a commutative ring, and ''M'', ''N'' and ''P'' be ''R''-modules. Then (in the below, "=" denotes [[canonical isomorphism]]s; this attitude is permissible since a tensor product is defined only up to unique isomorphisms)

; Identity : <math display=block>R \otimes_R M = M.</math>

; Associativity : <math display=block>(M \otimes_R N) \otimes_R P = M \otimes_R (N \otimes_R P).</math>

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*; Distribution over finite products : For any finitely many <math>N_i</math>, <math display="block">M \otimes_R \prod_{i = 1}^n N_i = \prod_{i = 1}^nM \otimes_R N_i.</math>

; Base extension : If ''S'' is an ''R''-algebra, writing <math>-_{S} = S \otimes_R -</math>, <math display="block">(M \otimes_R N)_S = M_S \otimes_S N_S;</math><ref>Proof: (using associativity in a general form) <math>(M \otimes_R N)_S = (S \otimes_R M) \otimes_R N = M_S \otimes_R N = M_S \otimes_S S \otimes_R N = M_S \otimes_S N_S</math></ref> cf. {{section link||Extension of scalars}}. A corollary is:

*; Distribution over [[localization of a module|localization]] : For any multiplicatively closed subset ''S'' of ''R'', <math display="block">S^{-1}(M \otimes_R N) = S^{-1}M \otimes_{S^{-1}R} S^{-1}N</math> as an <math>S^{-1} R</math>-module., Sincesince <math>S^{-1} R</math> is an ''R''-algebra and <math>S^{-1} - = S^{-1} R \otimes_R -</math>, this is a special case of:.

; Commutativity with [[direct limit]]s : For any direct system of ''R''-modules ''M''_''i'', <math display="block">(\varinjlim M_i) \otimes_R N = \varinjlim (M_i \otimes_R N).</math>

; [[tensor-hom adjunction|Adjunction]] : <math display=block>\operatorname{Hom}_R(M \otimes_R N, P) = \operatorname{Hom}_R(M, \operatorname{Hom}_R(N, P))\text{.}</math> A corollary is:

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<math display="block">\operatorname{Hom}_S (M \otimes_R S, P) = \operatorname{Hom}_R (M, \operatorname{Res}_R(P)).</math>

This says that the functor <math>-\otimes_R S</math> is a [[left adjoint]] to the forgetful functor {{tmath|1= \operatorname{Res}_R }}, which restricts an ''S''-action to an ''R''-action. Because of this, <math>- \otimes_R S</math> is often called the [[extension of scalars]] from ''R'' to ''S''. In the [[representation theory]], when ''R'', ''S'' are group algebras, the above relation becomes the [[Frobenius reciprocity]].

==== Examples ====