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Associativity and k-fold tensor products

April 16, 2012

Here I will use the universal property of tensor products regarding bilinear maps to prove that the construction of tensor products is associative. For simplicity, I will take the ring R to be commutative so that we can view R-modules as (R,R)-bimodules.

Theorem: Let R be a commutative and unitary ring, M, N and L be R-modules. Then there is a unique isomorphism

(M \otimes N) \otimes L \simeq M \otimes (N \otimes L).

Proof: For a fixed l, it is easy to see that the map \varphi_l : M \times N \to M \otimes (N \otimes L) such that \varphi_l(m,n) = m \otimes (n \otimes l) is bilinear. For example we have \varphi_l (m, n_1 + n_2) = m \otimes (n_1 \otimes l + n_2 \otimes l) = m \otimes (n_1 \otimes l) + m \otimes (n_2 \otimes l) = \varphi_l(m, n_1) + \varphi_l(m, n_2). By the universal property of tensor products, for every such fixed l \in L, we have a unique R-module homomorphism \phi_l : M \otimes N \to M \otimes ( N \otimes L) such that \phi_l (m,n) = \varphi_l (m,n) = m \otimes (n \otimes l). Thus, the map \psi : (M \otimes N) \times L \to M \otimes (N \otimes L) such that \psi(m \otimes n, l) = \phi_l(m,n) = m \otimes (n \otimes l) (and then extended by linearity on the first argument) is well-defined. Moreover, \psi is easily seen to be bilinear and thus (again by the universal property) induces a unique R-module homomorphism taking (m \otimes n) \otimes l to m \otimes (n \otimes l). Similarly, we can construct a homomorphism the other way around, inverse to this one, hence the isomorphism.

QED

We thus see that for a commutative ring R, we can unambiguously construct the k-fold tensor product M_1 \otimes M_2 \otimes \ldots \otimes M_k of k R-modules M_1, \ldots M_k. Moreover, from the universal property of tensor product and the last remark in the last post, we deduce that this k-fold tensor product is the universal object with respect to k-multilinear maps :

Theorem (Universal property): Let R be a commutative ring and let M_1, \ldots M_k, L be R-modules. Let \iota : M_1 \times \ldots \times M_k \to M_1 \otimes \ldots \otimes M_k be such that \iota(m_1, \ldots, m_k) = (m_1 \otimes \ldots \otimes m_k). Let \varphi : M_1 \times \ldots M_k \to L be a k-multilinear function. Then there exists a unique \phi : M_1 \otimes \ldots \otimes M_k \to L such that \varphi = \phi \circ \iota.

Once again, we deduce the existence of a bijection between the set of k-multilinear functions from M_1 \times \ldots M_k to L and the set of R-module homomorphisms from M_1 \otimes \ldots \otimes M_k to L.

Reference: D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.

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