We know that the morphisms between objects of derived category are roofs. But how to understand them,and how to compute them. For example, we consider the derived category $D(X)$ of a projective variety $X$, then $\Hom(O_X, E^.)=?$ for a complex $E^.$ and why $\Hom(A, B[1])=\Ext^1(A, B)$ for sheaves $A$ and $B$. Can we understand them only using the roofs.
closed as offtopic by abx, Joonas Ilmavirta, Alex Degtyarev, Hugh Thomas, Ryan Budney May 4 at 4:52This question appears to be offtopic. The users who voted to close gave this specific reason:



First, the derived category (say of an abelian category) should be defined as the localisation of the category of complexes at the class of quasiisomorphisms. In particular, 'the' derived category is really defined up to equivalence. Ignoring settheoretic subtleties, you can use roofs/a calculus of fractions to show that this localisation exists, but it gives only one among many models for the derived category (and a not very useful model at that). For the relation $Hom(A,B[1]) \simeq Ext^{1}(A,B)$, you simply need to replace $B$ with an injective resolution $B \rightarrow I^{\bullet}_{B}$. Then, in terms of roofs, you can represent an element of $Hom(A,B[1])$ in terms of a map of complexes $A \rightarrow I^{\bullet}_{B}[1]$, up to homotopy, which you might recognize as nothing but an element of $Ext^{1}$ in some more classical construction (like CartanEilenberg or Grothendieck's Tohoku paper). All of this and more is explained in great detail in Chapter 3 of Gelfand and Manin's Methods of Homological Algebra. 


Just an addition to the above answer, since $I^.$ is a complex of injectives, in computing $Hom_{D(A)}(A, I[i])$ one case use a lemma (if $A^.$ and $I^.$ are in $K^+(A)$ such that all $I^i$ are injective then $Hom_{D(A)}(A^., I^.)=Hom_{K(A)}(A^., I^.)$) and compute the $Hom$ in homotopy category itself without having to worry about roofs. 

