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15 A-module homomorphism M ⊗A A[tα ] → M ⊗A B defined by mapping x ⊗A g(tα ) to x ⊗A g(fα ) for all x ∈ M and g(tα ) ∈ A[t], the element α∈J eα ⊗A tα in M ⊗A A[t], where J is the subset of I where να = 0, maps to z. It follows that every polynomial law U ∈ P n (M, A) is uniquely determined by the elements UA[t] ( α∈J eα ⊗A tα ) for all finite subsets J of I. Moreover, it is clear that we obtain a polynomial law in P n (M, N ) by choosing an arbitrary family (fν )ν∈N(I) of elements fν in A and defining U by UA[t] ( α∈J eα ⊗A tα ) = µ∈N(J ) ,|µ|=n fµ ⊗A tµ .

12) Corollary. Let B be an A-algebra, and let M be a B-module that is free as an A-module of rank m. Moreover let C be a B-algebra, and let N be a C-module that is free as a B-module of rank n. We consider M ⊗B N as a C-module via the action of C on N . 1) A is commutative → → Proof. χ! 1) a natural B-algebra homomorphism ϕC : C → EndB (N ). 1) that we have a composite homomorphism υ given by Γm B (ϕC ) Γm B (C) −−−−−→ Γm (χ) Γm −B−−→ B (EndB (N )) − Γm B (EndB Γm (id) ϕ M (N )) −−− −−→ Γm B (EndB (N )).

2) ξ∈Nµ,ν → n → for all x in E (I) and y in G(I) , and all µ and ν in N(I) with |µ| = |ν| = n, ξ where we write ! ξG (xy) = α,β∈I γGα,β (xα yβ )!. 2) that the multiplication of E defined by vE is associative, and that it is commutative when E is commutative. 3) that n ν vG (γE (x) ⊗A γE (y)) = νβ β∈I γE (xyβ ) n for all x in E and y in E I and n ∈ N. In particular we have that γE (1) is a unit for the multiplication, and it is clear that the homomorphism A → ΓnA (E) → n that maps f to f γE (1) gives ΓnA (E) a structure of an A-algebra.

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