updat

date: 2022-03-17 19:50 modification date: Thursday 17th March 2022 19:50:58 title: koszul duality aliases: [koszul dual]

• Tags
  • #todo/untagged
• Refs:
  • Brantner’s Lecture notes: #resources
    • Lecture 1\
    • Lecture 2\
    • Lecture 3\
    • Lecture 4\
    • Lecture 5\
    • Lecture 6\
    • Lecture 7 (Lecture 8 missing)
• Links:
  • Morita equivalence
  • compact
  • thick subcategory
  • Morita equivalence
  • perfect complexes
  • operad
  • deformation
  • reflexive pair
  • Barr-Beck
  • simplicial set
  • infinity category

koszul duality

Koszul duality patterns have influenced several recent developments in algebraic geometry, ranging from the classification of formal deformations by dg Lie algebras and the unobstructedness of Calabi-Yau varieties to purely inseparable Galois theory and derived Galois deformation rings.

Definition

For augmented k-algebras

For modules over associative algebras

Applications

Setup

Deformation Problems

#AG/deformation-theory of CYs are unobstructed:

Notes

Koszul algebras

Defining the Adams grading:

Every Koszul algebra is a quadratic algebra:

Dualizing a Koszul algebra:

Examples

Checking if something is a Koszul algebra using the Poincare series. See PBW basis.

Examples

The BGG correspondence:

In topology