Koszul Duality

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The classical Koszul duality is the following statment:

For a graded algebra $R$ we dentoe by $grmod-R$ denote the category of finitely generated graded $R$ modules.

Theorem (Bernstein--Gelfand--Gelfand) Let $V$ be a vector space, then there is an exact equivalence \[ \KD^b(grmod-S^* V ) \; \isom \; \KD^b( grmod-\Lambda^* D V ), \] where $DV$ is the dual vector space and $S^*$ (resp. $\Lambda^*$) denote the symmetric (resp. exterior) algebra.

The goal of this note is to understand this statement from the viewpoint of derived Morita theory. See also Ben Websters answer on MathOverflow - What is Koszul duality?.

References

Expositions

Plan

  1. Morita theory in Abelian Categories
  2. Morita theory in Derived Categories
  3. Example: Derived Categories of Projective Spaces
  4. Koszul Dulaity Revisited

Video

The video stops after 30 minutes and therefore covers only the first part on abelian categories.

Slides

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