Tensor product of modules

The tensor product of modules over an (arbitrary) ring with 1 is a generalization of the tensor product of vector spaces over a field . It has meaning in abstract algebra and is used in homological algebra , in algebraic topology and in algebraic geometry .

definition

Be a ring (with , but not necessarily commutative). Be one - right module and a -Linksmodul. The tensor product over is defined by an Abelian group ${\displaystyle R}$ ${\displaystyle 1}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle (M\otimes _{R}N,\otimes _{R})}$ ${\displaystyle R}$

${\displaystyle M\otimes _{R}N}$

and one - bilinear mapping ${\displaystyle \mathbb {Z} }$

${\displaystyle {\begin{matrix}\otimes _{R}:&M\times N&\to &M\otimes _{R}N\\&(m,n)&\mapsto &m\otimes _{R}n,\end{matrix}}}$

so by a figure with
	${\displaystyle \forall m\in M;\,n_{1},n_{2}\in N:}$	${\displaystyle \otimes _{R}(m,n_{1}+n_{2})\;\;=\;\otimes _{R}(m,n_{1})+\otimes _{R}(m,n_{2})}$	(Dl ⊗ )
	${\displaystyle \forall m_{1},m_{2}\in M;\,n\in N:}$	${\displaystyle \otimes _{R}(m_{1}+m_{2},n)\;=\;\otimes _{R}(m_{1},n)+\otimes _{R}(m_{2},n)}$	(Dr ⊗ ),

that also

${\displaystyle \forall m\in M;\,n\in N;\,r\in R:}$

${\displaystyle \otimes _{R}(mr,n)=\otimes _{R}(m,rn)}$

(A ⊗ )

which together have the following universal property :

For every Abelian group and every bilinear map ${\displaystyle G}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle g\,\colon M\times N\to G\,}$

with the additional property

${\displaystyle \forall m\in M;\,n\in N;\,r\in R:}$

${\displaystyle g(mr,n)=g(m,rn)}$

(A g )

there is a group homomorphism

${\displaystyle g_{\otimes }:M\otimes _{R}N\to G}$     With     ${\displaystyle g_{\otimes }\circ \otimes _{R}=g}$

and this is clearly determined.

This universal property defines a tensor product that is uniquely determined except for isomorphism, and is called the canonical (mediating) bilinear mapping of the tensor product. ${\displaystyle \otimes _{R}}$

The abbreviated spellings and are common for. ${\displaystyle \otimes _{R}(m,n)}$ ${\displaystyle m\otimes _{R}n}$ ${\displaystyle m\otimes n}$

Remarks

1. The requirement (Dl) means the left distributivity of over the module addition and (Dr) the legal distributivity. ${\displaystyle \otimes _{R}}$
2. Requirement (A) is reminiscent of the associative law of ring multiplication.
3. From (Dl _g that each follows) due to the neutral element is mapped; accordingly from (Dr _g ). ${\displaystyle (m,0)}$ ${\displaystyle g(m,0)=g(m,0-0)=g(m,0)-g(m,0)=0_{G}}$ ${\displaystyle 0_{G}\in G}$ ${\displaystyle g(0,n)=0_{G}}$

Basic construction

The existence of the tensor product is shown by the following construction.

Consider the free modulus generated by all pairs , which is isomorphic to ( direct sum ). Since one contains, the pairs can be understood as the basis of . The sub-module is formed , which is defined by the linear combinations of basic elements in ${\displaystyle (m,n)\in M\times N}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle F}$ ${\displaystyle {\mathbb {Z} }^{(M\times N)}=\bigoplus _{M\times N}\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle 1}$ ${\displaystyle (m,n)\in M\times N}$ ${\displaystyle F}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle Q}$ ${\displaystyle F}$

	${\displaystyle \forall m\in M;\,n_{1},n_{2}\in N:}$	${\displaystyle (m,n_{1}+n_{2})-(m,n_{1})-(m,n_{2})}$	(Dl Z )
	${\displaystyle \forall m_{1},m_{2}\in M;\,n\in N:}$	${\displaystyle (m_{1}+m_{2},n)-(m_{1},n)-(m_{2},n)}$	(Dr Z )
	${\displaystyle \forall m\in M;\,n\in N;\,r\in R:}$	${\displaystyle (mr,n)-(m,rn)}$	(A Z )

is produced.

The Abelian group is defined as the quotient of to , in signs ${\displaystyle M\otimes _{R}N}$ ${\displaystyle F}$ ${\displaystyle Q}$

${\displaystyle M\otimes _{R}N:=F/Q}$,

and the image of under the bilinear mapping as the subclass of , in characters ${\displaystyle (m,n)}$ ${\displaystyle \otimes _{R}}$ ${\displaystyle (m,n)}$

${\displaystyle \otimes _{R}(m,n):=(m,n)+Q}$.

Objects defined by universal properties are always uniquely determined (except for isomorphism). ■

Remarks

1. For follows from (Dl _Z ) and from (Dr _Z ) analogously , together . Therefore it is sufficient to establish the conditions (Dl _Z ) and (Dr _Z ) for Abelian groups ( modules) - the condition (A _Z ) is then automatically established. ${\displaystyle r\in \mathbb {N} }$
     ${\displaystyle r\cdot (m,n)=\underbrace {(m,n)+\dots +(m,n)} _{r{\text{ Summanden}}}\equiv (m,\underbrace {n+\dots +n} _{r{\text{ Summanden}}})=(m,rn)\mod Q}$
   ${\displaystyle r\;(m\otimes n)=(mr)\otimes n}$ ${\displaystyle r\;(m\otimes n)=m\otimes (rn)=(mr)\otimes n}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle M,N}$
2. If one denotes with the resp. underlying -modules, then can modulus are canonically identified with the quotient of the modulus after the -submodule formed by elements of the form with is generated. ${\displaystyle M',N'}$ ${\displaystyle M,N}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle M\otimes _{R}N}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle M'\otimes _{\mathbb {Z} }N'}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle (mr)\otimes n-m\otimes (rn)}$ ${\displaystyle m\in M,\,n\in N,\,r\in R}$

Construction as R module

If the ring is commutative (in this case a right module can be given a left module structure and vice versa), then the tensor product is not just an Abelian group, but a module and a bilinear map, and not just a bilinear one. The scalar multiplication can be done with the help of the definition (for the sake of clarity the suffix is omitted from the figure ) ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle M\otimes _{R}N}$ ${\displaystyle R}$ ${\displaystyle \otimes _{R}}$ ${\displaystyle R}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle _{R}}$ ${\displaystyle \otimes _{R}}$

${\displaystyle \forall m\in M;\,n\in N;\,r\in R:}$

${\displaystyle r\;(m\otimes n):=(mr)\otimes n}$

(S R )

To be defined. This link is well defined , since for each the independence of Representatives or the minor class of ${\displaystyle s\in R}$ ${\displaystyle (ms,n)}$ ${\displaystyle (m,sn)}$ ${\displaystyle (ms)\otimes n=m\otimes (sn)}$

${\displaystyle r\;((ms)\otimes n)=((ms)r)\otimes n=(m(sr))\otimes n=(m(rs))\otimes n=((mr)s)\otimes n=(mr)\otimes (sn)=r\;(m\otimes (sn))}$

follows. Note that the third equality needs the commutativity of . ${\displaystyle R}$

Alternatively, the tensor product can be constructed directly as a module. In the basic construction, instead of the free Abelian group, the free module generated by is used . When generating (which in this case becomes not just a subgroup, but a sub-module) one also takes the linear combinations ${\displaystyle M\times N}$ ${\displaystyle R}$ ${\displaystyle Q}$

${\displaystyle \forall m\in M;\,n\in N;\,r\in R:}$

${\displaystyle r\cdot (m,n)-(mr,n)}$

(S ′ R )

added. The commutativity of ensures the associativity of the scalar multiplication, for it is ${\displaystyle R}$

${\displaystyle r\;(s\;(m\otimes n))=r\;(ms\otimes n)=((ms)r)\otimes n=(m(sr))\otimes n=(m(rs))\otimes n=(rs)\;(m\otimes n)}$

For ${\displaystyle m\in M;\ n\in N;\ r,s\in R.}$

The R module constructed in these two ways has a corresponding universal property:

For every R module and every R bilinear mapping ${\displaystyle G}$

${\displaystyle g\,\colon M\times N\to G\,}$

there is an R -module homomorphism

${\displaystyle g_{\otimes }:M\otimes _{R}N\to G}$     With     ${\displaystyle g_{\otimes }\circ \otimes _{R}=g}$

and this is clearly determined.

Remarks

1. Specialization: If there is a field, then the -modules and the tensor product are -vector spaces, and the latter agrees with from the article Tensor product of vector spaces . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle M,N}$ ${\displaystyle M\otimes _{R}N}$ ${\displaystyle R}$ ${\displaystyle M\otimes N}$
2. Generalization: It is the non-commutativity of permit and as the name for the center of the ring with both constructions in this section by replacing, to the uniquely determined module and the -bilinearen figure to arrive. To fulfill (A _⊗ ), linear combinations (A _Z ) with scalars are generated from the original ring as before . It is this ring that characterizes the tensor product . To avoid confusion, it is best to start with the definition of a module , which is provided a posteriori with (S _R ) with a (left or right) scalar multiplication as a subring of ${\displaystyle R}$ ${\displaystyle Z(R)}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle Z(R)}$ ${\displaystyle Z(R)}$ ${\displaystyle M\otimes _{R}N}$ ${\displaystyle Z(R)}$ ${\displaystyle \otimes _{R}}$ ${\displaystyle Q}$ ${\displaystyle R}$ ${\displaystyle M\otimes _{R}N}$
   ${\displaystyle \mathbb {Z} }$ ${\displaystyle M\otimes _{R}N}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle Z(R).}$
3. The ring at the operator can have a big impact, as the examples and show. ${\displaystyle R}$ ${\displaystyle \otimes _{R}}$ ${\displaystyle \mathbb {C} \otimes _{\mathbb {C} }\mathbb {C} =\mathbb {C} }$ ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} =\mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{2}=\mathbb {R} ^{4}}$

Change of ring

• ${\displaystyle R}$and let be rings, be a ring homomorphism and be a right module, be a left module. Then there is - in the names of module (mathematics) #change of the ring  - exactly one linear mapping ${\displaystyle S}$ ${\displaystyle \rho \colon S\to R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle \varphi \;\colon \;M_{[S]}\otimes _{S}N_{[S]}\;\to \;M\otimes _{R}N}$

such that for everyone ${\displaystyle m\in M,n\in N}$

${\displaystyle \varphi (m\otimes _{S}n)=m\otimes _{R}n.}$

This mapping is surjective and is called canonical .

• Is there , then is ${\displaystyle S\subseteq R}$

${\displaystyle M\otimes _{R}N=(M\otimes _{S}N)/Q,}$

where is generated by with . ${\displaystyle Q}$ ${\displaystyle mr\otimes _{S}n-m\otimes _{S}rn}$ ${\displaystyle m\in M;\,n\in N;\,r\in R}$

• Let be a two-sided ideal in which is contained in both the annihilator of and of . Then resp. a canonical right resp. left module structure, and the canonical homomorphism ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R/I}$

${\displaystyle \varphi \;\colon \;M\otimes _{R}N\;\to \;M\otimes _{R/I}N,}$

which corresponds to canonical homomorphism is identity. ${\displaystyle \rho \colon R\to R/I}$

Special cases

Let R , R ₁ , R ₂ , R _{3 be} (not necessarily commutative) rings.

• Is M ₁₂ a R ₁ - R ₂ - bimodule and M ₂₀ , a left R ₂ module, then the tensor

${\displaystyle M_{12}\otimes _{R_{2}}M_{20}}$

a left R ₁ module.

• If M _{02 is} a right R ₂ module and M _{23 is} an R ₂ - R ₃ bimodule, then is the tensor product

${\displaystyle M_{02}\otimes _{R_{2}}M_{23}}$

a right R ₃ module.

• If M _{01 is} a right R ₁ module, M _{12 is} an R ₁ - R ₂ bimodule and M _{20 is} a left R ₂ module, then the associativity law applies

${\displaystyle (M_{01}\otimes _{R_{1}}M_{12})\otimes _{R_{2}}M_{20}=M_{01}\otimes _{R_{1}}(M_{12}\otimes _{R_{2}}M_{20})}$.

Hence leads in the notation without brackets

${\displaystyle M_{01}\otimes _{R_{1}}M_{12}\otimes _{R_{2}}M_{20}}$

any order of execution of ⊗ to the same result.

• Each ring is a - -Bimodule. So is ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle R\otimes _{R}R=R}$

with the ring multiplication

${\displaystyle mn=:m\otimes _{R}n}$

than the canonical- bilinear mapping. ${\displaystyle \mathbb {Z} }$

• For all R modules M and N is

${\displaystyle M\otimes _{R}\{0\}=\{0\}=\{0\}\otimes _{R}N.}$

• If commutative, then are the modules ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle M\otimes _{R}N}$ and ${\displaystyle N\otimes _{R}M}$

canonically isomorphic.

• Is a - algebra is so ${\displaystyle A}$ ${\displaystyle R}$

${\displaystyle A\otimes _{R}N}$

a left module; the module operation is given by ${\displaystyle A}$

${\displaystyle b(a\otimes n)=(ba)\otimes n}$for , in . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle A}$

• Each ring with is a - -Bimodul. So is ${\displaystyle R}$ ${\displaystyle 1}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle R\otimes _{R}R=R}$

with the ring multiplication

${\displaystyle m\otimes _{R}n=mn}$

than the canonical- bilinear mapping. ${\displaystyle \mathbb {Z} }$

• Is a commutative ring, and are and associative - algebras , so is ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle R}$

${\displaystyle A\otimes _{R}B}$

again an associative algebra; the multiplication is given by ${\displaystyle R}$

${\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(a_{1}a_{2})\otimes (b_{1}b_{2}).}$

Categorical properties

Different variants of the tensor product have right adjoint functors:

• If there is a ring, a right module, a left module and an Abelian group, then: ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle G}$

${\displaystyle \mathrm {Hom} _{\mathbb {Z} }(M\otimes _{R}N,G)=\mathrm {Hom} _{R}(M,\mathrm {Hom} _{\mathbb {Z} }(N,G));}$

there is a legal module fortune ${\displaystyle \mathrm {Hom} _{\mathbb {Z} }(N,G)}$ ${\displaystyle R}$

${\displaystyle (f\cdot r)(n)=f(rn)\quad {\text{für }}f\in \mathrm {Hom} _{\mathbb {Z} }(N,G),\,r\in R,\,n\in N.}$

• If there is a ring, an -algebra, a -link module and a -link module, then: ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N}$ ${\displaystyle A}$

${\displaystyle \mathrm {Hom} _{A}(A\otimes _{R}M,N)=\mathrm {Hom} _{R}(M,N)}$.

• Is a commutative ring with identity and , , three -modules, then: ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle R}$

${\displaystyle \mathrm {Hom} _{R}(M\otimes _{R}N,G)=\mathrm {Hom} _{R}(M,\mathrm {Hom} _{R}(N,G))}$.

In particular, the tensor product is a right exact functor.

The tensor product is the pushout in the category of commutative rings with one element; in particular, for a commutative ring with one, the tensor product over the coproduct (for finitely many objects) is in the category of -algebras. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

Examples

• ${\displaystyle \mathbb {Z} /n\mathbb {Z} \otimes _{\mathbb {Z} }\mathbb {Z} /m\mathbb {Z} =\mathbb {Z} /\mathrm {ggT} (m,n)\mathbb {Z} }$
• ${\displaystyle \mathbb {Q} \otimes _{\mathbb {Z} }\mathbb {Z} /n\mathbb {Z} =\{0\}}$
• ${\displaystyle \mathbb {Q} \otimes _{\mathbb {Z} }\mathbb {R} =\mathbb {R} }$
• Localizations of modules are tensor products with the localized rings, so is for example

${\displaystyle \mathbb {Q} \otimes _{\mathbb {Z} }\mathbb {Q} =\mathbb {Q} .}$

• If a ring, a two-sided ideal and a left module, then is ${\displaystyle R}$ ${\displaystyle I}$ ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle M/IM=M\otimes _{R}R/I.}$

• If a commutative ring with one element is, then ${\displaystyle R}$

${\displaystyle R[X]\otimes _{R}R[Y]=R[X,Y].}$

• ${\displaystyle R[X]=R\otimes _{\mathbb {Z} }\mathbb {Z} [X]}$

Structure of the elements

Elementary tensors

An elementary tensor or pure tensor in the tensor product is an element of the form with . ${\displaystyle M\otimes _{R}N}$ ${\displaystyle m\otimes n}$ ${\displaystyle m\in M,\,\,n\in N}$

General shape

Every element of the tensor product is a finite sum ${\displaystyle x}$ ${\displaystyle M\otimes _{R}N}$

${\displaystyle x=\sum _{i}m_{i}\otimes n_{i},\,m_{i}\in M,n_{i}\in N}$

of elementary tensors. This representation is not clear. Furthermore, in general not every tensor can be written as an elementary tensor.

For example, the tensor is not an elementary tensor in the tensor product , where the standard basis vectors are im ; on the other hand, absolutely. ${\displaystyle e_{1}\otimes e_{2}\pm e_{2}\otimes e_{1}}$ ${\displaystyle \mathbb {R} ^{2}\otimes _{\mathbb {R} }\mathbb {R} ^{2}}$ ${\displaystyle e_{i}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle e_{1}\otimes e_{1}\pm e_{1}\otimes e_{2}\pm e_{2}\otimes e_{1}+e_{2}\otimes e_{2}=(e_{1}\pm e_{2})\otimes (e_{1}\pm e_{2})}$

If R is a commutative ring and an R -module generated by an element , then every tensor of the tensor product is an elementary tensor for any R -module . ${\displaystyle M}$ ${\displaystyle M\otimes _{R}N}$ ${\displaystyle N}$

Additional terms

In algebra :

• Flatness
• Brewers group

In differential geometry :

• Tensor field
• Differential form
• Vector bundle

In functional analysis

• Tensor product for Von Neumann algebras
• Spatial tensor product ( C * -algebras )
• Maximum tensor product (C * -algebras)

literature

• Siegfried Bosch : Algebra. 7th edition. Springer-Verlag, 2009, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 . “Tensor product over rings” : Section 7.2 p. 299.

Individual evidence

1. ↑ read as " tensor product of with over ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R}$ " or also as " tensored over with " ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle N}$
2. ↑ so also with N. Bourbaki: Elements of Mathematics, Algebra I, Chapters 1–3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products ,, p. 243 ( Internet Archive ).
3. ↑ (.: Balanced multiplying dt approximately) over for a picture with these three properties sometimes the term "balanced product" is found in English literature R .
4. ↑ N. Bourbaki: Elements of Mathematics, Algebra I, Chapters 1-3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products ,, p. 244 ( Internet Archive ).
5. ↑ N. Bourbaki: Elements of Mathematics, Algebra I, Chapters 1-3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products ,, p. 244 ( Internet Archive ).
6. ↑ The R -Bilinearity has the property (A _g ).
7. ↑ N. Bourbaki: Elements of Mathematics, Algebra I, Chapters 1-3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products ,, p. 246 ( Internet Archive ).
8. ↑ N. Bourbaki: Elements of Mathematics, Algebra I, Chapters 1-3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products ,, p. 258 ( Internet Archive ).