# Koszul Duality

From Notes Wiki

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**

- Beilinson, Ginzburg, Soegel - Koszul Duality Patterns in Representation Theory
- Beilinson - Coherent sheaves on $\IP^n$ and Problems in Linear Algebra (MSN)
- Bernstein, Gelfand, Gelfand - Algebraic Bundles over $\IP^n$ and Problems in Linear Algebra (pdf)
- Bondal - Representations of Associative Algebras and Coherent Sheaves pdf
- Rickard - Morita Theory for derived Categories
- Keller - Deriving DG categories
- Ginzburg, Kapranov - Koszul Duality for Operads

**Expositions**

- Schwede - Morita theory in abelian, derived and stable model categories
- MathOverflow - What is Koszul duality?

## Plan

- Morita theory in Abelian Categories
- Morita theory in Derived Categories
- Example: Derived Categories of Projective Spaces
- Koszul Dulaity Revisited

## Video

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

## Slides