Advanced Tilting Seminar

Winter semester 2024/2025

• October 14th - An overview of Morita theory, derived Morita theory, and the direction of the seminar.
Relevant literature:
• D. Murfet, Triangulated Categories, Link
• D. Murfet, Derived Categories 1, Link
• D. Murfet, Derived Categories 2, Link
• B. Keller, Derived Categories and Tilting, Link
• October 21th - No seminar.
• October 28th - No seminar.
• November 4th - Derived category of a hereditary abelian category, tilting modules over hereditary rings, example over the quiver A3.
Relevant literature:
• H. Krause, Derived categories, resolutions, and Brown representability, Link - Subsecton 1.6.
• A. Ergus, Morita Theory for Derived Categories of Algebras, Link
• November 11th - An example of non-classical tilting module over the integers, proof that the representable functor is fully faithful.
Relevant literature:
• L. Positselski and J. Šťovíček, The tilting-cotilting correspondence, Link - Example 8.4.
• S. Bazzoni, Equivalences induced by infinitely generated tilting modules, Link
• November 18th - T-structures in triangulated categories, the heart of a t-structure is abelian and induces a cohomological functor.
Relevant literature:
• A.A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Link
• D. Arapura, Triangulated categories and t-structures, Link - short but incomplete
• D. Miličić, Lectures on derived categories, Link - Section 4
• November 25th - Realization functors, effacability, basics on abstract silting objects.
Relevant literature:
• C. Psaroudakis, J. Vitória, Realisation functors in tilting theory, Link
• J. Lurie, Higher Algebra, Link - Sections 1.2, 1.3
• P. Nicolás, M. Saorín, A. Zvonareva, Silting Theory in triangulated categories with coproducts, Link
• S. Virili, Morita theory for stable derivators, Link
• December 2nd - Abstract tilting theorem, compact tilting objects.
Relevant literature:
• C. Psaroudakis, J. Vitória, Realisation functors in tilting theory, Link - Section 5
• December 9th - I'm not here!
• December 16th - Abstract Morita theorems for large and bounded tilting and cotilting complexes of R-modules.
Relevant literature:
• C. Psaroudakis, J. Vitória, Realisation functors in tilting theory, Link - Theorem 5.3
• D. Murfet, Derived Categories 1, Link - Section 4 for homotopy resolutions
• S. Breaz, -, G. C. Modoi, Silting, cosilting, and extensions of commutative ring, Link - Theorem 3.11
• January 6th - Example computed: The divisible tilting module over the integers, foreshadowing of contramodules.