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Then s = pis = − pjt = 0 and −t = −q jt = qis = 0. Therefore, ϕ is an isomorphism, and its inverse is m → ( pm, qm). (v) ⇒ (i). Obvious. 21. ance, then • If T : R Mod → Ab is an additive functor of either variT (A B) ∼ = T (A) T (B). In particular, if T is covariant, then x → (T ( p)x, T (q)x) is an isomorphism, where p : A B → A and q : A B → B are the projections. Proof. 20(iv), and the displayed isomorphism is that given in the proof of (iv) ⇒ (i) of the proposition. • Internal direct sum is the most important instance of a module isomorphic to a direct sum.

Let ϕ : S T → M be an isomorphism. Define σ : S → S T by s → (s, 0) and τ : T → S T by t → (0, t). Clearly, σ and τ are injective R-maps, and so their composites i = ϕσ : S → M and j = ϕτ : T → M are also injections. If m ∈ M, then ϕ surjective implies that there exist s ∈ S and t ∈ T with m = ϕ(s, t) = ϕ(s, 0) + ϕ(0, t) = is + jt ∈ im i + im j. Finally, if x ∈ im i ∩ im j, then x = ϕσ (s) = ϕ(s, 0) and x = ϕτ (t) = ϕ(0, t). Since ϕ is injective, (s, 0) = (0, t), so that s = 0 and x = ϕ(s, 0) = 0.

En ] is not a singular (n − 1)-simplex. We remedy this by introducing face maps. 10 Simplicial homology H is also functorial, but defining H ( f ) for a simplicial map f n n is more complicated, needing the Simplicial Approximation Theorem. 30 Introduction Ch. 1 Definition. Define the ith face map in : n−1 → n , where 0 ≤ i ≤ n, by putting 0 in the ith coordinate and preserving the ordering of the other coordinates: the points of [e0 , . . , en−1 ] are convex combinations (t0 , . . , tn ) = t0 e0 + · · · + tn−1 en−1 , and so n i : (t0 , .

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