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Semi-orthogonal Decompositions of Derived Categories

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 240)

Abstract

In this survey, we are mostly interested in:
  1. 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. 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.

Keywords

Semi-orthogonal Decomposition Exceptional Collection Exceptional Objects Smooth Projective Variety Admissible Subcategory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author thanks the University of Alberta and the organizers for the hospitality and excellent opportunity to participate in the superschool. For detailed treatments of this survey, see [4, 6, 8, 9, 13, 15, 17].

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.McGill UniversityMontrealCanada

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