Cohen-Macaulay modules over simple singularities

• Název práce v češtině: Cohen-Macaulayovy moduly nad jednoduchými singularitami
• Název v anglickém jazyce: Cohen-Macaulay modules over simple singularities
• Klíčová slova: Cohen-Macaulayův modul|jednoduchá singularita
• Klíčová slova anglicky: Cohen-Macaulay module|simple singularity
• Akademický rok vypsání: 2020/2021
• Typ práce: diplomová práce
• Jazyk práce: angličtina
• Ústav: Katedra algebry (32-KA)
• Vedoucí / školitel: doc. RNDr. Jan Šťovíček, Ph.D.
• Řešitel: Mgr. Yifan Zhang - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 16.03.2021
• Datum zadání: 21.04.2021
• Datum potvrzení stud. oddělením: 18.05.2021
• Datum a čas obhajoby: 03.02.2022 09:30
• Datum odevzdání elektronické podoby: 05.01.2022
• Datum odevzdání tištěné podoby: 10.01.2022
• Datum proběhlé obhajoby: 03.02.2022
• Oponenti: Isaac Bird, Ph.D.

Zásady pro vypracování

The student will aim at understanding the classification of indecomposable maximal Cohen-Macaulay modules over simple singularities. This result, originating from [3,4], is the main topic of the monograph [2]. It is put into the context of modern representation theory and homological algebra in [5], and this point of view also plays a major role in the survey [1]. References for the background in commutative algebra include [6,7,8]. The goal of the thesis will be a presentation of this classification in a suitable form. This may involve filling in details in proofs, describing the relation to modern representation theoretic and homological techniques, or outlining other possible classification results of this type.

Seznam odborné literatury

[1] O. Iyama, Tilting Cohen-Macaulay representations, Proc. of the ICM in Rio de Janeiro 2018, Vol. II. Invited lectures, 125-162, World Sci. Publ., Hackensack, NJ, 2018.

[2] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, LMS LNS 146, Cambridge University Press, Cambridge, 1990.

[3] H. Knörrer, Cohen-Macaulay modules on hypersurface singularities I, Invent. Math. 88(1987), no. 1, 153-164.

[4] R. O. Buchweitz, G. M. Greuel, F. O. Schreyer, Cohen-Macaulay modules on hypersurface singularities II, Invent. Math. 88 (1987), no. 1, 165-182.

[5] H. Kajiura, K. Saito, A. Takahashi, Matrix factorization and representations of quivers II, type ADE case, Adv. Math. 211 (2007), no. 1, 327-362.

[6] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.

[7] H. Matsumura, Commutative ring theory, 2nd edition, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989.

[8] D. Eisenbud, Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.