Quotients of triangulated categories
(2024) In Master's Theses in Mathematical Sciences FMAM05 20241Mathematics (Faculty of Engineering)
- Abstract
- 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... (More)
- 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. (Less)
- Popular Abstract
- The origins of homological algebra can be traced back to the work of Henri Poincaré in the late 1800s, who introduced the concept of homology classes and relations to study topological spaces. In topology one usually considers two spaces equivalent if one is able to squish and deform any surface into the other without “tearing the space”, i.e creating holes. A famous analogy is that topologically a donut and a coffee mug are equivalent, as they (usually) contain the same number of holes and one can be deformed into the other. Mathematically it is not so easy to know how many holes a surface or shape contains as they are ususally described by a set of equations. However it is with the notion of homology one can make a mathematically... (More)
- The origins of homological algebra can be traced back to the work of Henri Poincaré in the late 1800s, who introduced the concept of homology classes and relations to study topological spaces. In topology one usually considers two spaces equivalent if one is able to squish and deform any surface into the other without “tearing the space”, i.e creating holes. A famous analogy is that topologically a donut and a coffee mug are equivalent, as they (usually) contain the same number of holes and one can be deformed into the other. Mathematically it is not so easy to know how many holes a surface or shape contains as they are ususally described by a set of equations. However it is with the notion of homology one can make a mathematically rigorous definition and this has for example lead to a general classification of compact surfaces. This type of mathematics belongs to the area of algebraic topology, which seeks to understand topology by using algebraic methods, such as homology. Since then, mathematicians discovered that the same techniques used to determine the number of holes of a sur- face can be used in a much more general setting and homological algebra is broadly speaking the study of these techniques, i.e the study of homology, in a more general algebraic setting.
Homological algebra often involves the study of chain complexes, that is (infinitely) long chains of mathematical objects and functions between them of the form
... X^i -> X^{i+1} -> X^{i+2} ...
(where each X^i is an objects and every arrow is a function between the objects). More specifically one is often interested in when there exists a function between two such complexes which gives that the homology of these chains are equal. The derived category consists precisely of such objects, that is we consider two chain complexes equal if there exists a function between them which gives that their homology are equivalent. Now it turns out that the derived category has the structure of a triangulated category (which is that it satisfies some abstract conditions). Studying triangulated categories is therefore interesting as it tells us about properties of derived categories but also about other mathematical objects which happens to have a similar structure. For triangulated categories there exists construction of Verdier quotients, which more or less is that one tries to remove a subset of objects while still keeping the triangulated
structure. One problem with this construction is however that the functions of this quotient are very hard to use and calculate with. This thesis treats a paper which gives simple conditions which a Verdier quotient may or may not satisfy. When the condi- tions are satisfied, one can interpret the Verdier quotient in a much simpler way and it is shown that some already known results are actually just special cases of this fact. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9162679
- author
- Dannetun, Eric LU
- supervisor
-
- Gustav Jasso LU
- organization
- course
- FMAM05 20241
- year
- 2024
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Homological algebra, Triangulated categories, Verdier localization, t-structures
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFMA-3549-2024
- ISSN
- 1404-6342
- other publication id
- 2024:E54
- language
- English
- id
- 9162679
- date added to LUP
- 2024-09-04 13:45:23
- date last changed
- 2024-09-04 13:45:23
@misc{9162679,
abstract = {{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.}},
author = {{Dannetun, Eric}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Quotients of triangulated categories}},
year = {{2024}},
}