Крис Брав (Chris Brav)

DG categories and non-commutative geometry

(учебно-исследовательский семинар)

Summary:

A fair amount of the geometry of a finite type scheme is encoded in its differential graded bounded derived category of coherent sheaves. In particular, various (co)homological invariants of a scheme can be recovered from its bounded derived category, for example algebraic K-theory and Z/2-graded algebraic de Rham cohomology. With this in mind, one can develop a fairly robust non-commutative geometry by thinking of sufficiently nice differential graded categories as being analogues of bounded derived categories of coherent sheaves, but for some putative non-commutative spaces, allowing for a common framework in which to treat problems from classical algebraic geometry, algebraic topology, and symplectic topology.

Students taking the course for credit will give talks based on assigned reading and provide detailed notes in TeX, which we shall collaboratively assemble into a coherent whole. Prerequisites: Familiarity with basic homological and homotopical algebra. (For the latter, one could concurrently read the first chapter of Hovey's Model Categories.)

Outline:

Basics of dg categories, dg functors, and dg modules. Standard adjunctions. Yoneda embedding. Basic properties of dg categories (properness, smoothness, perfection). Generalised Serre duality.
Homotopy theory and Morita theory of dg categories. Mapping spaces and bimodules. Finite type dg categories.
Cyclic homology of dg categories and its variations. The degeneration conjecture for the non-commutative Hodge-to-de-Rham spectral sequence.
Left and right Calabi-Yau structures and their relative versions.
Calabi-Yau completion and Calabi-Yau deformations Further topics (depending on time and interest):

Moduli of objects after Toen-Vaquie., non-commutative motives after Tabuada and Robalo, the Waldhausen S-construction via quiver representations after Dyckerhoff-Kapranov.

Literature:

Keller, On differential graded categories

Toen, Lectures on dg categories

Toen, The homotopy theory of dg-categories and derived Morita theory

Toen-Vaquie, Moduli of objects in dg categories

Keller, Deformed Calabi-Yau completions

Brav-Dyckerhoff, Notes on Calabi-Yau structures on dg categories