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> The first three properties (plus identities on morphisms) say that the category of ''R''-modules, with ''R'' commutative, forms a [[symmetric monoidal category]]. Thus <math>M \otimes_R N \otimes_R P</math> is well-defined.

; Symmetry : <math display=block>M \otimes_R N = N \otimes_R M.</math> In fact, for any permutation ''σ'' of the set {1, ..., ''n''}, there is a unique isomorphism: <math display="block">\begin{cases} M_1 \otimes_R \cdots \otimes_R M_n \longrightarrow M_{\sigma(1)} \otimes_R \cdots \otimes_R M_{\sigma(n)} \\ x_1 \otimes \cdots \otimes x_n \longmapsto x_{\sigma(1)} \otimes \cdots \otimes x_{\sigma(n)} \end{cases}</math>

: The first three properties (plus identities on morphisms) say that the category of ''R''-modules, with ''R'' commutative, forms a [[symmetric monoidal category]].

; Distribution over [[direct sum]]s : <math display=block>M \otimes_R (N \oplus P) = (M \otimes_R N) \oplus (M \otimes_R P).</math> In fact, <math display="block">M \otimes_R \left (\bigoplus\nolimits_{i \in I} N_i \right ) = \bigoplus\nolimits_{i \in I} \left ( M \otimes_R N_i \right ),</math> for an [[index set]] ''I'' of arbitrary [[cardinality]]. Since finite products coincide with finite direct sums, this imples:

*; 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:

*; Right-exactness : 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 adjunctionrelation : 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>

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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 ====