Quotients of triangulated categories - Dannetun,Eric (Master's Theses in Mathematical Sciences, 2024)
This thesis presents an exposition of some parts of the paper: Quotients of triangulated categories and equivalences of Buchweitz, Orlov and Amiot– Guo–Keller, by Iyama and Yang. Our contribution is to present the results and applications in a way that does not require too much knowledge or familiarity with the area and to include more detail than what is usually done. The Verdier quotient is a convenient way of obtaining a triangulated category by “quoten- ing” out a subcategory, however the downside is that the morphisms of the res- ulting category are hard to handle. The results of the paper are specifying con- ditions on a triangulated category T with a thick subcategory S which allows for finding an equivalence (as additive categories) between the Verdier quotient T/S and an ideal quotient Z/[P] (where Z and P are subcategories). With a further assumption an explicit triangulated structure of Z /[P] compatible with that of T /S is also given. We show that some earlier results, such as equi- valences by Keller–Vossieck and Amiot–Guo–Keller, can be viewed as special cases of these results.
Keywords: Homological algebra; Triangulated categories; Verdier localization; t-structures