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Non-repeated universal elective course.
In the academic year 2024/25:
Auslander-Reiten theory and applications in representation theory and commutative algebra
Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (30.09.2024)
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The exam will be oral. Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (30.09.2024)
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[ASS] I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras, Vol. 1,London Math. Soc. Stud. Texts, vol. 65, Cambridge University Press, Cambridge, 2006. [Au1] M. Auslander, Isolated singularities and existence of almost split sequences, in: Representation Theory, II, Ottawa, Ont., 1984, in: Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986. [Au2] M. Auslander, Representation theory of Artin algebras II, Comm. Algebra 1 (1974) 269--310. [ARS] M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1997. [En1] H. Enomoto, Classifications of exact structures and Cohen-Macaulay-finite algebras, Adv. Math. 335 (2018), pp. 838-877. [En2] H. Enomoto, Relations for Grothendieck groups and representation-finiteness, J. Alg. 539 (2019), pp. 152-176. [Kr] H. Krause, A short proof for Auslander's defect formula, Special issue on linear algebra methods in representation theory, Linear Algebra Appl. 365 (2003), pp. 267-270. [LW] G. J. Leuschke, R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys and Monographs, vol. 181, American Mathematical Society. [Yo] Y. Yoshino, Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Mathematical Society, Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (30.09.2024)
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The course is completed with an oral exam. The requirements for the exam correspond to what is presented in lectures. Last update: Šťovíček Jan, doc. RNDr., Ph.D. (11.10.2017)
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In 2024/25: See https://www.karlin.mff.cuni.cz/~stovicek/index.php/en/2425zs-nmag498 for more details.
If one has a finite dimensional algebra or a commutative noetherian algebra over a field, one might ask how the category of finitely generated modules looks like. Although this is a very difficult problem in general, in many interesting cases one can describe the category in terms of generating homomorphisms and relations between them, using techniques developed by Maurice Auslander and Idun Reiten. These generating morphisms form a directed graph, which is called the Auslander-Reiten quiver, so one even can depict these categories graphically.
The aim of the course is to explain this piece of theory and illustrate it on examples of finite dimensional algebras and coordinate rings of isolated singularities. The plan is as follows:
1. Motivation. 2. Auslander-Reiten theory: the Jacobson radical of an additive category, irreducible morphisms, almost split morphisms and Auslander-Reiten sequences. 3. Examples from representation theory of finite dimensional algebras. 4. Background in commutative algebra: isolated singularities, Gorenstein rings, Cohen--Macaulay modules. 5. Examples from commutative algebra. Last update: Šťovíček Jan, doc. RNDr., Ph.D. (30.09.2024)
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