Tensor product of modules: Difference between revisions

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; 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. Since <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:

; CommutationCommutativity 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:

*; Right-exactionexactness : If <math display="block">0 \to N' \overset{f}\to N \overset{g}\to N'' \to 0</math> is an exact sequence of ''R''-modules, then <math display="block">M \otimes_R N' \overset{1 \otimes f}\to M \otimes_R N \overset{1 \otimes g}\to M \otimes_R N'' \to 0</math> is an exact sequence of ''R''-modules, where <math>(1 \otimes f)(x \otimes y) = x \otimes f(y).</math>

; Tensor-hom relationadjunction : There is a canonical ''R''-linear map: <math display="block">\operatorname{Hom}_R(M, N) \otimes P \to \operatorname{Hom}_R(M, N \otimes P),</math> which is an isomorphism if either ''M'' or ''P'' is a [[finitely generated projective module]] (see {{section link||As linearity-preserving maps}} for the non-commutative case);<ref>{{harvnb|Bourbaki|loc=ch. II §4.4}}</ref> more generally, there is a canonical ''R''-linear map: <math display="block">\operatorname{Hom}_R(M, N) \otimes \operatorname{Hom}_R(M', N') \to \operatorname{Hom}_R(M \otimes M', N \otimes N')</math> which is an isomorphism if either <math>(M, N)</math> or <math>(M, M')</math> is a pair of finitely generated projective modules.

To give a practical example, suppose ''M'', ''N'' are free modules with bases <math>e_i, i \in I</math> and <math>f_j, j \in J</math>. Then ''M'' is the [[direct sum of modules|direct sum]] <math>M = \bigoplus_{i \in I} R e_i</math>