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I have just started to learn about derived categories, I am now trying to understand what morphisms look like in some easy examples. Let me describe one for you.

Let $D(\mathcal A)$ be the derived category of an Abelian category $\mathcal A$. Pick two objects $E,F$ in $\mathcal A$ and look at them in $D(\mathcal A)$. Now the $i$th shift $F[i]$ of $F$ is not an object of $\mathcal A$ anymore (I guess).

Question. Is there an easy or standard description of $\hom_{D(\mathcal A)}(E,F[i])$?

Essentially I have $E$, which I look at as a complex $$\dots\to 0\to E\to 0\to \dots\,\,\,\,\,\,\,\,\,\,\,\,\,E\,\textrm{in degree }0$$ concentrated in degree $0$, and $F[i]$ is the complex $$\dots \to 0\to F\to 0\to \dots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F\,\textrm{in degree }-i.$$

How to construct roofs $E\leftarrow R\to F[i]$ and equivalence classes of them?

I would be very glad to have some explicit example, e.g. for $\mathcal A$ the category of $\mathcal O_X$-modules, or the category of (quasi)coherent $\mathcal O_X$-modules on a scheme $X$.

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Yes!! $Hom_{D(\mathcal{A})}(E, F[i])=Ext^i_A (E, F)$ – Matt Oct 31 '13 at 1:04
... where $\mathrm{Ext}^i$ is understood to be zero when $i<0$. – Martin Brandenburg Oct 31 '13 at 8:17

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