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Let I • and J • be two injective resolutions of an object A in A . 1) of the right derived functor of F at A, we will prove 47 Derived Functors that Hj (F I • ) and Hj (F J • ) are isomorphic for j ≥ 0. Let us consider 0 bI 0 GA d0 G I1 ~ ~~ ~ ~ ~~ dd dd dd d2 d1 G I2 G ... 2) J0 d0 d1 G J1 G J2 G ... 2), the injectiveness of I 0 implies that there exists a morphism f 0 from J 0 to I 0 . Namely, we have the following commutative diagrams 0 0 0 I ~c y1 ~~ 1 f 0 ~ ~~ 1 ~~ GA G J0 GA G I0 dd 1 dd dd 1 g0 dd 1 1  J0 where the second diagram is obtained by the injectiveness of J 0 .

6). Then the contravariant functor HomA (·, A) is a left exact functor from A to Ab. 2) where, for instance, φ∗ := HomA (φ, A). 3) G 0. An injective object I in A is an object to guarantee the exactness of the functor HomA (·, I) : A Ab. 4) 44 Derived Functors any morphism f : C → I can be lifted to f : C → I satisfying f = f ◦ φ. 2) from φ becomes an epimorphism: also HomA (C, I) φ∗ G HomA (C , I) G0 is exact. That is, an object I is said to be an injective object if HomA (·, I) : A Ab becomes an exact functor.

1) 0 0 G R0 F A R F φ G R0 F A R F ψG R0 F A 0 ... ∂ j−1G Rj F A Rj F φ j G Rj F A R F ψG Rj F A ∂0 ∂j 1 G R1 F A R F φ G . . G Rj+1 F A G ... may be an exact sequence in B. 1) can be done as follows. , the initial terms of injective resolutions for A and A ). 2) 0 where ι0 : I 0 → I 0 = I 0 ⊕ I 0 is defined by ι0 (x ) = (x , 0) ∈ I 0 ⊕ I 0 and π 0 : I 0 = I 0 ⊕ I 0 → I 0 is the projection defined by π 0 (x , x ) = x . Then ι0 is a monomorphism and π 0 is an epimorphism satisfying ker π 0 = im ι0 .

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Algebra I. Lecture Notes by Thomas Keilen

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