Isomorphism between morphisms of derived category and homotopy category

Question

This is exercise 5.1 of chapter3 of Gelfand's methods of homological algebra.

I want to show $Hom_{K(A)} (X^*,Y^*)$ and $Hom_{D(A)} (X^*,Y^*)$ is isomorphic if $Y^*$ is in $ ObKom^+(I)$, the set of bounded below chain complex consisting of injective objects.

I am stuck in constructing a inverse map. Any suggestion?

Answer 1

Here is a more general result, which shows where the above isomorphism comes from. Let $H$ be the homotopy category and $D$ the derived category. Call a complex K homotopical injective if $Hom_H(N,K)=0$ for every acyclic complex N. An example of such a K is a left bounded complex of injectives, as you have. Let $H_i$ denote the subcategory of $H$ consisting of homotpically injective complexes. Then the quotient functor $q:H \rightarrow D$ has a fully faithful right adjoint $i$. Now restricting gives an equivalence $q_i : H_i \rightarrow D$ and an isomorphism $Hom_H(L,iM) \cong Hom_D(qL,M)$ is clear in this case. What is left is the construction of $i$ (which is a little complicated) and you can find this in the book "Derived equivalences for group rings" by Steffen König and Alexander Zimmermann as Chapter 8, written by Bernhard Keller.