A seminar in the winter semester 2025/26 under the code NMAG496 (Geometry Elective 1). The aim is to learn about the derived Morita theory and the use of differential graded rings and categories in homological algebra.

Basic information

The seminar is scheduled on Fridays at 10:40-12:10, in the seminar room of the Department of Algebra.

The plan is to understand Keller's proof [Ke94] of Rickard's defived Morita theorem [Ri89] for rings, and then to continue with differential graded techniques in homological algebra:

  1. A possible short introduction to derived and triangulated categories (depending on who joins, not to leave people behind, but maybe Michal Hrbek's complementary seminar will save us from that).
  2. Rickard's derived Morita theorem characterizing when two rings have equivalent derived categories.
  3. A conceptual proof of Rickard's theorem was given by Keller using DG rings.
  4. DG categories, algebraic triangulated categories (as a solution to some known deficiencies of triangulated categories) and Keller's extension of the Morita theorem for them.
  5. Pretriangulated dg categories as an (in principle more flexible) enhancement for triangulated categories.

Program

A brief overview of what has been told in the seminar will be updated below.

DateTopicSource
Oct 17The classical Morita theorem and a version of Rickard’s derived Morita theorem. The technical issue with constructing derived equivalences and a sketch how DG rings will solve it.[Ja21], §1.1–1.3
Oct 24Conceptual point of view at DG k-algebras: as monoids in the closed symmetric monoidal category of DG k-modules. Recollections on abstract closed symmetric monoidal categories. The monoidal category of graded k-modules. More structure induced by the closed symmetric monoidal structure: enrichment and the enriched hom-⊗ adjunction.[Ja21], §2.1
Oct 31The monoidal category of DG k-modules and the definition of a DG k-algebra. The homotopy category of complexes in a category of ordinary modules (or more generally in an abelian category).[Ja21], §2.2, 2.3,
[We94], §10.1
Nov 7Axioms of triangulated categories and the triangulated structure on the homotopy category of complexes of ordinary modules.[Ja21], §5.2.1,
[We94], §10.1, 10.2,
[Ma]
Nov 21DG categories as "DG algebras with many objects". The DG category structure on the category of DG modules over a DG k-algebra A. The underlying and the homotopy categories of a general DG category and specializing this to the DG category of DG A-modules.[Ya25], §4.2,
[Ja21], §3.1, §3.3, §4.1
Nov 28Identification of Z0(A) and the abstract underlying category (in the sense of enriched category theory) of a DG category A. DG functors and DG natural transformations, the DG Yoneda lemma. The underlying and homotopy functors of DG functors. The shift DG functor on the category of DG k-modules. [Ya25], §4.3,
[Ja21], §3.1, §3.2,
[Ke05], §1.2, §2.2, §2.4
Dec 5Shift DG functors in general DG categories. Cones in general DG categories. Pretriangulated DG categories (in a strict form) and the standard triangulated structure on their homotopy categories.[Ya25], §4.4
Dec 12The DG category of DG functors between two DG categories, DG modules over a small DG category as a special case. Compactly generated triangulated categories and basic facts about them. Construction of homotopically projective resolutions using the Brown Representability Theorem. The derived category of small DG category. [Ya25], §4.3, §5.4,
[Ja21], §3.2.4, §4.2,
[Kr07], §4, §6
Dec 19Quasi-isomorphism of DG algebras (or, more generally, quasi-equivalences of small DG categories) induce triangle equivalences of their derived categories. Morita theorem for DG algebras (or small DG categories) and its specialization to the derived Morita theorem for ordinary algebras. Algebraic triangulated categories.[Ya25], §3.5, §4.5, §5.3,
[Kr07], §7

Literature

In addition to the sources listed below, there is an extensive list of literature available on the homepage of a similarly focused (albeit more elementary) Seminar on derived categories from the past.

[Ja21] G. Jasso, DG categories, unpublished notes, 2021.
[Ke94] B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. [link to the paper]
[Ke98] B. Keller, On the construction of triangle equivalences, in: Derived equivalences for group rings, 155–176. Lecture Notes in Math. 1685, Springer, 1998. [PDF file]
[Ke05]G. M. Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. No. 10 (2005). [link to the monograph]
[Kr07]H. Krause, Derived categories, resolutions, and Brown representability, in Interactions between homotopy theory and algebra, 101–139, Contemp. Math. 436, AMS, Providence, RI, 2007. [link to arXiv preprint] [link to exercises]
[Ma] J. P. May, The axioms for triangulated categories. [PDF file].
[Ri89] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. [link to the paper]
[We94] C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge, 1994.
[Ya25] D. Yang, Introduction to derived categories of differential graded algebras, unpublished notes, 2025.