Advanced Tilting Seminar
Summer semester 2024/2025
- February 24th - Linear topological rings, finite topology on endomorphism rings.
Relevant literature:
- Stacks Project tag 07E7, Link
- nLab page and references within, Link
- Tibor Szelem, On a topology in endomorphism rings of Abelian groups. Publ. Math. Debrecen 5 (1957): 1-4. Link - Seems to be the first appearance of "finite topology" in the literature.
- March 3rd - Complete and separated linear topological rings.
Relevant literature:
- Stacks Project tag 07E7, Link
- Tibor Szelem, On a topology in endomorphism rings of Abelian groups. Publ. Math. Debrecen 5 (1957): 1-4. Link
- March 10th - Endomorphism ring is complete, "Complete free modules" over complete linear rings, monadic definition of modules.
Relevant literature:
- Leonid Positselski, Contramodules, Link - Subsection 2.1.
- March 17th - Definition of a contramodule, some basic properties of the category of contramodules.
Relevant literature:
- Leonid Positselski, Contramodules, Link - Subsection 2.1.
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Sections 6.1 & 6.2
- March 24th - Abstract properties of the category of contramodules.
Relevant literature:
- Leonid Positselski, Contramodules, Link - Subsection 2.1.
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Sections 6.2 & 6.3
- Leonid Positselsky and Jiří Rosická, Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories, Link - Section 5.
- Leonid Positselsky, Answer to the following question at Mathoverflow, Link.
- Locally presentable categories on nLab, Link.
- Adjoint functor theorem on nLab, Link.
- March 31th - Cocomplete categories a with projective generator, equivalence between Add(M) and Proj(Ctra-End(M)).
Relevant literature:
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Sections 6.3 & 7.1
- April 7th - category of discrete modules, contratensor product.
Relevant literature:
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Section 7.2
- Grothendieck categories on nLab, Link.
- April 14th - fullness of the forgetful functor from contramodules to modules.
Relevant literature:
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Theorem 6.10, Section 7.3
- Jan Šťovíček and Leonid Positselski, Topologically semisimple and topologically perfect topological rings, Link - Corollary 4.4, Section 5
- Leonid Positselski, Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures, Link - Subsection A.1.2--- A counterexample to fullness of the forgetful functor
- Leonid Positselski, Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors, Link - Example 7.2 --- Another counterexample to fullness of the forgetful functor
- April 28th - The heart of a tilting module as a contramodule category, counterexample for a tilting complex
Relevant literature:
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Section 8
- -, Topological endomorphism rings of tilting complexes, Link - Sections 2, 3 and 7
- May 5th - Abelianization, restricted Yoneda embedding
Relevant literature:
- Jan Šťovíček and Leonid Positselski, The tilting-cotilting correspondence, Link - Theorem 7.1
- -, Topological endomorphism rings of tilting complexes, Link - Section 2
- Daniel Murfet, Rings with Several Objects, Link
- Henning Krause, Smashing subcategories and the telescope conjecture – an algebraic approach, Link
- Amnon Neeman, Triangulated Categories, Chapters 5 and 6
- May 12th and last - definable subcategories, derived topological Morita theory
Relevant literature:
- -, Topological endomorphism rings of tilting complexes, Link - Section 4,5,6/li>
- R. Göbel and J. Trlfaj, Approximations and Endomorphism Algebras of Modules, Section 13 and 15