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Крис Брав (Chris Brav)
DG categories and non-commutative geometry
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.)
Moduli of objects after Toen-Vaquie., non-commutative motives after
Tabuada and Robalo, the Waldhausen S-construction via quiver
representations after Dyckerhoff-Kapranov.
- 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
- Left and right Calabi-Yau structures and their relative versions.
- Calabi-Yau completion and Calabi-Yau deformations
Further topics (depending on time and interest):
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