Funkce na abelovských a triangulovaných kategoriích

• Thesis title in Czech: Funkce na abelovských a triangulovaných kategoriích
• Thesis title in English: Functions on abelian and triangulated categories
• Academic year of topic announcement: 2024/2025
• Thesis type: dissertation
• Thesis language: čeština
• Department: Department of Algebra (32-KA)
• Supervisor: Alexandra Zvonareva
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 26.09.2024
• Date of assignment: 26.09.2024
• Confirmed by Study dept. on: 26.09.2024

Guidelines

Various notions of functions on abelian and triangulated categories are used to generalize the notion of dimension of a vector space. Among such functions are Sylvester rank functions [1, 2], Crawley-Boevey characters [3] and rank functions on triangulated categories recently introduced by Chuang and Lazarev [4]. It turns out that these functions can be related to various other notions appearing in the study of abelian and triangulated categories, such as Serre subcategories, thick subcategories, endofinite objects and localizations [5,6]. Many questions remain open in the study of these functions. One of such questions is contained in the paper by Chuang and Lazarev and concerns the connection between rank functions and the classification of functors from a given triangulated category to simple triangulated categories [4]. The project in question will focus on obtaining abstract results on the structure of the space of functions associated to an abelian or a triangulated category and to determining these spaces for interesting classes of examples arising in representation theory.

References

[1] Paul Cohn. Skew fields. Theory of general division rings., volume 57. Cambridge: Cambridge University Press, 2008
[2] Aidan Schofield. Representation of rings over skew fields, volume 92. Cambridge University Press, Cambridge. London Mathematical Society, London, 1985.
[3] William Crawley-Boevey. Additive functions on locally finitely presented Grothendieck categories. Communications in Algebra, 22(5):1629–1639, 1994.
[4] Joseph Chuang and Andrey Lazarev. Rank functions on triangulated categories. Journal für die reine und angewandte Mathematik (Crelles Journal), 2021(781):127–164, 2021.
[5] Henning Krause. Cohomological quotients and smashing localizations. American Journal of Mathematics, 127(6):1191–1246, 2005.
[6] Teresa Conde, Mikhail Gorsky, Frederik Marks, and Alexandra Zvonareva. A functorial approach to rank functions on triangulated categories. Journal für die reine und angewandte Mathematik (Crelles Journal) 0, 2024.