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R-Mod/. f / ! R-Mod/. This means that we need to argue that the composition A ! f / ! A is homotopic to idA . If we call this composition h then we need to argue that idA h Š 0. f / CC CC CC CC C!  C / {{ {{ { {{ {} { A and A idA ?? ?? A ! idA h/ D 0. A ! C / D B. So we can think of idA h as a morphism k W A ! B. A/; B/ D 0. So 0 ! B ! k/ ! A/ ! 0 is split exact and K D idA h (as a morphism into B) is homotopic to 0. So then easily it is homotopic to 0 as a morphism into C . This completes our proof.

1 of Volume I). In fact it is such that if 0 W X 0 ! Y is another morphism of D with X 0 in C , then there is a unique morphism f W X 0 ! X so that ı f D 0 . So this W X ! Y will be a C -cover with this unique factorization property. 2. 4. Let T W D ! C be a functor. Suppose that for every object X of C there exists an object Y of D and a morphism W X ! Y / in C with the following universal property: if Y 0 is any object of D and if 0 W X ! Y 0 / is another morphism in C , then there is a unique morphism g W Y !

5. Let f W C ! D, g W D ! R-Mod/. f / ! g ı f / ! g/. R-Mod/ In this chapter we give the basic results concerning cotorsion pairs of classes of complexes of left R-modules. 1. If A is a class of complexes of left R-modules we let A? A; C / D 0 for all A 2 A. We let ? B; A/ D 0 for all A 2 A. 2. R-Mod/ if B ? D B and ? B D A. R-Mod/ are basically the same as those for R-Mod (see Chapter 7 of the Volume I). R-Mod/ with little modification. 3. R-Mod/ there are exact sequences 0 ! B ! A ! C ! 0 and 0 !

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