On understanding Orlov's LG B model

Question

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.

Answer 1

First of all, I do not think these are research-level questions nor they clarify something that is hard to understand with enough background.

1. I suggest you start with learning some derived category language. In particular, you will learn about truncations, t-structures and much more. Without knowing the language it's hard to read Orlov's papers. The canonical source in your case would be Daniel Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry.
2. This should be jargon for a morphism $X\to \mathrm{Spec}\ A$.
3. I assume you are trying to read Triangulated Categories of Singularities… As far as I remember, it deals with matrix factorisations for affine schemes only. In particular, there is no distinction between modules and quasi-coherent sheaves. As for the exact sequence, it should actally look like $$0\to \mathrm{Coker}\ p_1\to P_1\mid_W\to P_2\mid_W\to \mathrm{Coker}\ p_1\to 0,$$ and it doesn't go beyond Hartshorne. I suggest you figure it out yourself as a good exercise.