The Alternating Product
The alternating product has applications throughout mathematics. In differential geometry, one takes the maximal alternating product of the tangent space to get a canonical line bundle over a manifold. Intermediate alternating products give rise to differ
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XIX
The Alternating Product
The alternating product has applications throughout mathematics . In differential geometry, one takes the maximal alternating product of the tangent space to get a canonical line bundle over a manifold . Intermediate alternating products give rise to differential forms (sections of these products over the manifold) . In this chapter, we give the algebraic background for these constructions . For a reasonably self-contained treatment of the action of various groups of automorphisms of bilinear forms on tensor and alternating algebras, together with numerous classical examples, I refer to: R. HOWE, Remarks on classical invariant theory, Trans . AMS 313 (1989), pp . 539-569
§1
DEFINITION AND BASIC PROPERTIES
Consider the category of modules over a commutative ring R. We recall that an r-multilinear map f: E(') --+ F is said to be alternating if !(XI, ... , x.) = 0 whenever Xi = Xj for some i =1= j . Let ar be the submodule of the tensor product F(E) generated by all elements of type Xl
where
Xi
= x j for some
@ . . . @x,
i =1= j. We define /\'(E) = T'(E)/a, .
Then we have an r-multilinear map E(') --+ /\'(E) (called canonical) obtained
731 S. Lang, Algebra © Springer Science+Business Media LLC 2002
732
XIX, §1
THE ALTERNATING PRODUCT
from the composition
E(r)
--+
Tr(E)
--+
T'(Ei]«, = /\ r(E).
It is clear that our map is alternating. Furthermore, it is universal with respect to r-multilinear alternating maps on E. In other words, if f : E(r) --+ F is such a map, there exists a unique linear map f* : /\r(E) --+ F such that the following diagram is commutative :
/\r(E)
E(r) / l
' . . , xr) ~
U(XI) 1\ • • • 1\
u(xr) E Ar(F).
This map is multilinear alternating, and therefore induces a homomorphism
A r(u): Ar(E) ~ Ar(F). The association u ~ Ar(u) is obviously functorial.
Example. Open any book on differential geometry (complex or real) and you will see an application of this construction when E is the tangent space of a point on a manifold , or the dual of the tangent space . When taking the dual, the con struction gives rise to differential forms . We let /\(E) be the direct sum 00
/\ (E) =
EB /\ r(E). r=O
DEFINITION AND BASIC PROPERTIES
XIX, §1
733
We shall make I\(E) into a graded R-algebra and call it the alternating algebra of E, or also the exterior algebra, or the Grassmann algebra. We shall first discuss the general situation, with arbitrary graded rings . A r be a G-graded Let G be an additive monoid again, and let A =
EB
reG
R-algebra. Suppose given for each A r a submodule an and let a =
EB o.. reG
Assume that a is an ideal of A . Then a is called a homogeneous ideal, and we can define a graded structure on A/a. Indeed, the biline ar map
sends o, x As into ar + s and similarly, sends Ar x as into ar + s' Thus using representatives in An As respectively, we can define a bilinear map
and thus a bilinear map A/a x A/a -+ A/a, which obviously makes A/a into a graded R -algebra . We apply this to r(E) and the modules u, defined previously. If
i
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