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Published in:
2018 | OriginalPaper | Chapter
Semi-orthogonal Decompositions of Derived Categories
Author : Yijia Liu
Published in: Superschool on Derived Categories and D-branes
Publisher: Springer International Publishing
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Abstract
In this survey, we are mostly interested in:
1.
\({\text {D}}^{{\text {b}}}(X):={\text {D}}^{{\text {b}}}(\mathop {\text{ Coh }}(X))\), where X is a smooth projective variety over a filed k and
2.
\({\text {D}}^{{\text {b}}}(Q,I):={\text {D}}^{{\text {b}}}(\mathop {\text {mod-}}kQ/I) \cong {\text {D}}^{{\text {b}}}(\mathop {\text{ rep }}(Q,I)^{op})\), where Q is a quiver and k is a field.
The second one it rather easy to understand while the first one is relatively hard to understand in general. We are going to see in some nice cases, they are equivalent. The theory of semi-orthogonal decompositions and exceptional collections will give you an idea how one might study the structure of derived categories: we decompose the category into smaller pieces. We hope these pieces are as simple as possible - as the simplest derived category \({\text {D}}^{{\text {b}}}(pt) \cong {\text {D}}^{{\text {b}}}(\text {Vec}_{k}^{\text{ f }.d.})\), which is given exactly by exceptional objects.