Wall Finiteness Obstruction for DG Categories I

Explore a comprehensive lecture on the Wall finiteness obstruction theorem for DG categories over a field. Delve into the analogue of this classical result for DG categories, presented by Alexander Efimov from the Steklov Mathematical Institute of Russian Academy of Sciences. Learn about the criteria for homotopically finitely presented DG categories to be Morita equivalent to finite cell DG categories, and discover the characterization of finite cell DG categories as quotients of proper DG categories with full exceptional collections. Examine applications of this theory to smooth and proper phantom DG categories, the derived category of the Barlow surface, and varieties with stratifications by affine spaces. Gain insights into how these findings disprove two conjectures by Orlov and explore the implications for smooth proper algebraic varieties with specific stratifications.

Intro
Thomason's classification of dense subcategories
DG reformulation of Thomason's theorem
Classical Wall obstruction
Preliminaries on DG categories
Homotopically finitely presented DG categories
Summarizing table
Generating cofibrations
Main theorem
Speial case: smooth and proper DG categories
Application: phantom DG categories
Application: Barlow and Dolgachev surfaces
Application: varieties with a nice stratification
Application proper schemes with a nice stratification