I try to read Orlov's papers on Landau-Ginzburg model, but I am quite puzzled，there are several questions：

1 the method of truncation is used frequently，(that is: using a bounded above complex $Q$ of locally free sheaves and quasi-isomorphism $Q^.\to E^.$ and consider a good truncation $τ^{≥−k}Q$.)

I am quite unfamaliar with this, are there any reference? And the language of derived category of coherent sheaves in the paper is far beyond what I learned in orinary homological algebra， are there any reference?

2 What is the meaning for a "morphism" between a scheme X and a ring A(not spec(A))? Just a map?

3 The object of $DB_{w0}(W)$ is defined to be a pair of module: $P_0 \mapsto P_1\mapsto P_0 $ where $p_0p_1=(W- w_0)$. However， I cannot understand, is the module a single module, or a sheaf of module. Either case, the relation $(W- w_0)\in A$ is difficult to understand. So it is not understanded for me the exact sequence relation $$ 0\mapsto Coker p_1\mapsto P_1/W \mapsto P_0/W \mapsto 0 $$ in the proof of Lemma 3.6.