Teaching

October 6h - Triangulated categories, (co)homological functors, representable functors.

Relevant literature:

• A. Neeman, Triangulated Categories.

• D. Murfet, Triangulated Categories, Link

October 13th - Examples triangulated categories: Stable module category, stable homotopy category, derived category.

Relevant literature:

• Stable module category: Wiki.

• Stable homotopy category, Wiki, nLab

• Derived categories: D. Murfet, Derived Categories, Link

October 20th - Products of triangles via pre-triangles, detection of iso/mono/epi-morphisms/non-functoriality of mapping cones.

Relevant literature:

• D. Murfet, Triangulated Categories, Link

• G. Stevenson, On the failure of functorial cones in triangulated categories, Link

October 27th - Homotopy colimits.

Relevant literature:

• D. Murfet, Triangulated Categories, Link

November 3rd - Functor category, restricted Yoneda embedding

Relevant literature:

• D. Murfet, Triangulated Categories 3, Link

• H. Krause, Smashing subcategories and the telescope conjecture – an algebraic approach, Link

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

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.