|
|
|
||
|
Non-repeated universal elective course.
In 2025/26: Purity and the spectrum of a module category (Isaac Bird)
Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (29.07.2025)
|
|
||
|
Oral exam. Last update: Jeřábek Emil, Mgr. et Mgr., Dr., Ph.D. (27.12.2023)
|
|
||
|
(a) Isaac Bird, The Ziegler spectrum of a module category:
(b) Jordan Williamson: Duality in tensor-triangular geometry
(1) P. Balmer, I. Dell’Ambrogio, and B. Sanders. Grothendieck-Neeman duality and the Wirthm¨uller isomorphism. Compos. Math., 152(8):1740–1776, 2016 (2) H. Fausk, P. Hu, and J. P. May. Isomorphisms between left and right adjoints. Theory Appl. Categ., 11:No. 4, 107–131, 2003 (3) T. Peirce, J. Williamson. Duality in tensor-triangular geometry via proxy-smallness. arXiv:2510.24415, 2025 Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)
|
|
||
|
The exam will be oral. Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (30.09.2024)
|
|
||
|
(a) Isaac Bird, The Ziegler spectrum of a module category:
How well can we understand the category of modules over a ring? In general it is pretty impossible, but for many problems there is a structure that gives a substantial insight and offers methods of classification. This is the pure structure on modules, which enables the encoding of substantial amounts of information about the module categoryinto a single topological space, the Ziegler spectrum, a partial generalisation of the Zariski spectrum in algebraic geometry. This course provides an introduction into the methods of pure homological algebra and the Ziegler spectrum. We will look at the pure structure, its properties, and its relationship to approximation theory. We will then turn our attention to defining the Ziegler spectrum, giving examples, applications and demonstrating the relationip between it and the Zariski spectrum. All these concepts will require the use of abelian category theory, which will be also be studied in the course. Summary: (1) Pure exact sequences, pure injective modules and duality pairs. (2) From modules to functors, finitely presented and fp-injective objects. (3) Localisation theory for Abelian categories and definable subcategories. (4) The Ziegler spectrum: properties and examples, and the fundamental correspondence. (5) Matlis’s structure theorem and the Zariski spectrum. (6) Definable functors and maps between Ziegler spectra. (7) Characters and endofinite modules.
(b) Jordan Williamson: Duality in tensor-triangular geometry
Tensor-triangular geometry provides a framework for studying classification problems across a range of areas. Originating in chromatic homotopy theory, its influence has since been seen in commutative algebra, geometry, and representation theory. In the course, we will see how one can prove powerful duality statements from a tensor-triangular perspective. We will focus on examples in commutative algebra, but the theory developed is rather broad, so other examples may be covered depending on the audience’s interests. Summary: (1) Brief recollections on triangulated categories (key examples, Brown representability, adjoint functor theorems) (2) Tensor-triangulated categories (definition, examples, compact objects, rigid objects, and proxy-small objects) (3) Abstract Grothendieck duality (geometric functors and their properties, the trichotomy theorem) (4) Abstract Matlis duality (Picard groups, Matlis dualising objects, connections between Matlis and Grothendieck duality) Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)
|
|
||
|
(a) Isaac Bird, The Ziegler spectrum of a module category: Beyond a familiarity with rings and modules, the course will be self contained. Concepts seen in the course Categories of Modules and Homological Algebra NMAG434 will be used throughout. Pure homological algebra can also be thought of as the model theory of modules, so there will be some connection with the algebraic parts of Model theory NMAG407.
(b) Jordan Williamson: Duality in tensor-triangular geometry: The course will assume some familiarity with the basics of triangulated categories although we will recall some key aspects. Although not a prerequisite, in some sense the course could be viewed as an unofficial sequel to Michal Hrbek’s course on Brown representability. Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)
|