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.


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.
| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.