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0 indirectly. Let H denote A ® B'/im cp.. cp. = 0. ) =,O,, (x). We thus have a commutative diagram II A(9 B=>A(9 B' II A®B-+ H )0 to V). It suffices to show that 0 is an isomorphism, since then, for example, ker 0. = ker -7r (while ker 7r = im W. by definition of 7r).

But now on(x + In) = gn(x) = xgn(1) for x V In implies that gn(1) # 0 for all n. Thus, g(1) does not take values in DE,,, a contradiction. 4 Projectives, Injectives, and Flats 35 Exercises 1. Suppose only that A is a coproduct of Al and A2 in RM, that is, Al wz 4A< `°? A2 makes A into a coproduct of Al and A2 in RM. Show that there are unique 7ri : A --, Ai making A into a biproduct, using only the properties of a coproduct. 1, can be carried out. When assembled, this shows that "A® B A x B" holds in any additive category, since the left hand side is a biproduct.

Hence B is projective. 7 does not assert an equivalence of conditions. Further, we need to derive long exact sequences involving the variable we have called B. These issues will be addressed in Section 4.

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