Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
The goal of this lecture course is to give a modern point of view on some important duality theorems in algebra,
from the point of view of triangulated categories. This perspective also enables one to view these dualities not just
in an algebraic setting, but to transport them into other realms, such as geometry and topology. The main focus
will be on Grothendieck’s local duality theorem, which relates the Matlis dual of local cohomology to the ordinary
functional dual.
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
Cílem kurzu je poskytnout moderní pohled na některé důležité věty o dualitách v algebře z pohledu triangulovaných
kategorií. Tato perspektiva rovněž umožňuje nahlížet na tyto duality nejen v jazyce algebray, ale přenést je i do
jiných oblastí, jako je například geometrie či topologie. Hlavní pozornost bude věnována Grothendieckově větě o
lokální dualitě, která dává do souvislosti Matlisův duál lokálních kohomologií s obvyklým duálem funkcí.
Course completion requirements -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
The final exam will be an oral exam. For zápočet, students will have to get 50% of marks on 3 homework assignments.
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
The final exam will be an oral exam. For zápočet, students will have to get 50% of marks on 3 homework assignments.
Literature -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
(1) H. Krause. Localization theory for triangulated categories. In Triangulated categories, volume 375 of London Math. Soc. Lecture Note Ser., pages 161-235. Cambridge Univ. Press, Cambridge, 2010
(2) H. Krause. Homological theory of representations, volume 195 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2022
(3) M. Hovey, J. H. Palmieri, and N. P. Strickland. Axiomatic stable homotopy theory. Mem.Amer. Math. Soc., 128(610):x+114, 1997
(4) T. Barthel, D. Heard, and G. Valenzuela. Local duality in algebra and topology. Adv. Math., 335:563-663, 2018
(5) W. G. Dwyer and J. P. C. Greenlees. Complete modules and torsion modules. Amer. J. Math., 124(1):199-220, 2002
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
(1) H. Krause. Localization theory for triangulated categories. In Triangulated categories, volume 375 of London Math. Soc. Lecture Note Ser., pages 161-235. Cambridge Univ. Press, Cambridge, 2010
(2) H. Krause. Homological theory of representations, volume 195 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2022
(3) M. Hovey, J. H. Palmieri, and N. P. Strickland. Axiomatic stable homotopy theory. Mem.Amer. Math. Soc., 128(610):x+114, 1997
(4) T. Barthel, D. Heard, and G. Valenzuela. Local duality in algebra and topology. Adv. Math., 335:563-663, 2018
(5) W. G. Dwyer and J. P. C. Greenlees. Complete modules and torsion modules. Amer. J. Math., 124(1):199-220, 2002
Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
The course will give an introduction to triangulated categories, before turning to introducing local cohomology, firstly in the classical algebraic setting, and then in the triangulated realm and explaining how the latter recovers and generalises the former. We will then turn to exploring local duality in thetriangulated setting, which naturally leads us to consider other duality theorems such as Greenlees-May duality, and Warwick duality. We will show how one can recover the classical statement of Grothendieck local duality from this more general triangulated version.
• Introduction to duality (1 lecture):
philosophy behind duality statements;
some familiar examples;
statement of Grothendieck local duality.
• Triangulated categories (6 lectures):
definition and some examples (derived categories and stable module categories);
basic properties of triangulated categories;
tensor-triangulated categories and rigid objects;
statement of Brown representability and its consequences.
• Local cohomology in algebra (2 lectures):
recollections on derived functors;
the classical algebraic definition of local cohomology;
calculating some examples and proving some key properties.
• Local cohomology in triangulated categories (4 lectures):
local cohomology as a colocalization;
generalization from the algebraic setting to more general triangulated categories.
• Duality theorems in triangulated categories (3 lectures):
triangulated duality theorems (Greenlees-May duality and Warwick duality);
deduction of Grothendieck’s local duality theorem from Greenlees-May duality.
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
The course will give an introduction to triangulated categories, before turning to introducing local cohomology, firstly in the classical algebraic setting, and then in the triangulated realm and explaining how the latter recovers and generalises the former. We will then turn to exploring local duality in thetriangulated setting, which naturally leads us to consider other duality theorems such as Greenlees-May duality, and Warwick duality. We will show how one can recover the classical statement of Grothendieck local duality from this more general triangulated version.
• Introduction to duality (1 lecture):
philosophy behind duality statements;
some familiar examples;
statement of Grothendieck local duality.
• Triangulated categories (6 lectures):
definition and some examples (derived categories and stable module categories);
basic properties of triangulated categories;
tensor-triangulated categories and rigid objects;
statement of Brown representability and its consequences.
• Local cohomology in algebra (2 lectures):
recollections on derived functors;
the classical algebraic definition of local cohomology;
calculating some examples and proving some key properties.
• Local cohomology in triangulated categories (4 lectures):
local cohomology as a colocalization;
generalization from the algebraic setting to more general triangulated categories.
• Duality theorems in triangulated categories (3 lectures):
triangulated duality theorems (Greenlees-May duality and Warwick duality);
deduction of Grothendieck’s local duality theorem from Greenlees-May duality.
Entry requirements -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.09.2023)
Some familiarity with the following would be preferable, although we will review key aspects throughout the course as they become relevant.