The Tensor Product
Having considered bilinear maps, we now come to multilinear maps and basic theorems concerning their structure. There is a universal module representing multilinear maps, called the tensor product. We derive its basic properties, and postpone to Chapter X
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XVI
The Tensor Product
Having considered bilinear maps , we now come to multilinear maps and basic theorems concerning their structure . There is a universal module representing multilinear maps, called the tensor product. We derive its basic properties, and postpone to Chapter XIX the special case of alternating products . The tensor product derives its name from the use made in differential geometry, when this product is applied to the tangent space or cotangent space of a manifold . The tensor product can be viewed also as providing a mechanism for "extending the base" ; that is, passing from a module over a ring to a module over some algebra over the ring . This "extension" can also involve reduction modulo an ideal, becau se what matters is that we are given a ring homomorphism f : A ~ B, and we pass from modules over A to module s over B . The homomorphism f can be of both types , an inclu sion or a canonical map with B = AIJ for some ideal J, or a composition of the two . I have tried to provide the basic material which is immediately used in a variety of applications to many fields (topology, algebra , differential geometry, algebraic geometry, etc .) .
§1.
TENSOR PRODUCT
Let R be a commutative ring. If E 1,
• •• ,
En , F are modules, we denote by
the module of n-multilinear maps
f : E 1 x '" x En
-+
F. 601
S. Lang, Algebra © Springer Science+Business Media LLC 2002
602
THE TENSOR PRODUCT
XVI, §1
We recall that a multilinear map is a map which is linear (i.e., R-linear) in each variable. We use the words linear and homomorphism interchangeably. Unless otherwisespecified, modules, homomorphisms, linear, multilinear referto the ringR. One may view the multilinear maps ofa fixed set of modules Ej , .. . , En as the objects of a category. Indeed, if
f : E1
X • •.
x En
--+
F and
g: E 1
X •..
x En
--+
G
are multilinear, we define a morphism f --+ g to be a homomorphism h : F which makes the following diagram commutative:
E. x . . . x
»:
--+
G
F
E.~ j. G
A universal object in this category is called a tensor product of E 1 , • •• , En (over R). We shallnow provethat a tensor product exists, and in fact construct one in a natural way. By abstract nonsense, we know of course that a tensor product is uniquely determined, up to a unique isomorphism. Let M be the free module generated by the set of all n-tuples (Xl"' " X n) , (Xi E Ei ) , i.e. generated by the set E I x . .. x En . Let N be the submodule generated by all the elements of the following type:
for all
Xi E
Ej ,
X; E
E j , a E R. We have the canonical injection
of our set into the free module generated by it. We compose this map with the canonical map M --+ MIN on the factor module, to get a map
We contend that ip is multilinear and is a tensor product. It is obvious that qJ is multilinear-our definition was adjusted to this purpose. Let
f: E 1 x ... x En
--+
G
be a multilinear map . By the definition of free module generated by
TENSOR PRODUCT
XVI, §1
we have an induced linear map M commutative:
-+
603
G
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