Modules-Multilinear Maps and Tensor Products
المؤلف:
David R. Wilkins
المصدر:
Algebraic Topology
الجزء والصفحة:
96-97
4-7-2017
1781
Let M1, M2, . . . , Mn be modules over a unital commutative ring R, and let P be an R-module. A function f: M1 × M2 × · · · × Mn → P is said to be R-multilinear if
f(x1, . . . , xk−1, x`k + x``k, xk+1, . . . , xn)
= f(x1, . . . , xk−1, x`k, xk+1, . . . , xn)
+ f(x1, . . . , xk−1, x``k, xk+1, . . . , xn)
and
f(x1, . . . , xk−1, rxk, xk+1, . . . , xn) = rf(x1, . . . , xk−1, xk, xk+1, . . . , xn)
for k = 1, 2, . . . , n, for all xl, x`l, x``l ∈ Ml (l = 1, 2, . . . , n), and for all r ∈ R.
(When k = 1 the list x1, . . . , xk−1 should be interpreted as the empty list in the formulae above; when k = n the list xk+1, . . . , xn should be interpreted as the empty list.) One can construct a module M1 ⊗R M2 ⊗R · · · ⊗R Mn, referred to as the tensor product of the modules M1, M2, . . . , Mn over the ring R, and an R-multilinear mapping
jM1×M2×···×Mn: M1 × M2 × · · · × Mn → M1 ⊗R M2 ⊗R · · · ⊗R Mn
where the tensor product and multilinear mapping jM1×M2×···×Mn satisfy the following universal property:
given any R-module P, and given any R-multilinear function f: M1 × M2 × · · · × Mn → P, there exists a unique R-module homomorphism θ: M1 ⊗R M2 ⊗R · · · ⊗R Mn → P such that f = θ ◦ jM1×M2×···×Mn
This tensor product is defined to be the quotient of the free module FR(M1×M2×· · ·×Mn) by the submodule K generated by elements of the free module that are of the form
iM1×M2×···×Mn (x1, . . . , xk−1, x`k + x``k, xk+1, . . . , xn)
− iM1×M2×···×Mn (x1, . . . , xk−1, x`k, xk+1, . . . , xn)
− iM1×M2×···×Mn (x1, . . . , xk−1, x``k, xk+1, . . . , xn),
or are of the form
iM1×M2×···×Mn (x1, . . . , xk−1, rxk, xk+1, . . . , xn)
− riM1×M2×···×Mn (x1, . . . , xk−1, xk, xk+1, . . . , xn),
where xl, x`l, x``l ∈ Ml for l = 1, 2, . . . , n, and r ∈ R. There is an R-multilinear function
jM1×M2×···×Mn: M1 × M2 × · · · × Mn → M1 ⊗R M2 ⊗R · · · ⊗R Mn, where jM1×M2×···×Mn
is the composition π ◦ iM1×M2×···×Mn of the natural embedding
iM1×M2×···×Mn: M1 × M2 × · · · × Mn → FR(M1 × M2 × · · · × Mn)
and the quotient homomorphism
π: FR(M1 × M2 × · · · × Mn) → M1 ⊗R M2 ⊗R · · · ⊗R Mn.
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