By Garrett P.

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F(B) ~ 11 0 51 = F(go), ltugF(go) = F(go), nglug = F(go)nug = F(go), ltglUg = F(go)nug = F(go). Hence nugF(go) = nglug and ltugF(go) = ltgl ug. We have nugF(go) This means that the following diagram is commutative F(A) -i F(go) ~ F 2(A) -i F(B) ~ F 2(B) and therefore sgF(go) = F(gl)Sg. Assuming the commutativity of F(gk) with the degeneracy morphisms has been proved for k = 0,1, . , n - 1 we prove now that siF(gn) = F(gn+dsi, 0 < i < n. ~l{)j F(gn) when 0 j < i, lj+lU? 9) 52 CHAPTER 2. DERIVED FUNCTORS On the other hand, Ij+Ign+IUi = F(gnW;+Iui .

32, Theorem 2. 34 and Quillen's spectral sequence. 35. 36. Let T : C --+ Gr be a cosheaf and let (X*, a, A) be a contractible P -resolution such that L~T(Xm) = 0 for all n > 0, m 2:: 0 and L5T(Xm) = T(Xm) , m 2:: O. Then ·there is an isomorphism L~T(A) ~ 1l"nT(X*), n 2:: O. Proof. 35. 37. If the projective class P is induced by a cotriple F then there is an isomorphism for any T : C --+ Gr. 38. Let X* be the Cech complex of a P-epimorphism f : X --+ A of the category C. Let T : C --+ Gr be a covariant v functor .

DERIVED FUNCTORS On the other hand, Ij+Ign+IUi = F(gnW;+Iui . 9) only with the difference that the factor F(gn) appears at the beginning, not the end, of each product. Since 8j F(gn) = F(gn+I)8j and by the induction assumption Sf-l F(gn-l) = F(gn)sf- 1 , one has for 0 ~ j ~ n+ 1, 0 ~ i ~ n. Hence uiF(gn) = gn+lui, 0 ~ i ~ n. Thus the diagram F(Ln(A)) F 2(Ln(A)) F(u~) F(Ln+I(A)) 1- F2(gn) 1- F(gn+I) F(Ln(B)) F 2(Ln(B)) ~ F(Ln+I(B)) 1- is commutative. It follows that sfF(gn) = F(gn+dsi and therefore for 0 ~ i ~ n, n> O.

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