Disciplina Discipline MAT5701
Feixes na Reta projetiva com pesos e tópicos relacionados

Sheaves on the Weighted Projetive Line and Related Tópics

Área de Concentração: 45131

Concentration area: 45131

Criação: 25/07/2019

Creation: 25/07/2019

Ativação: 31/07/2019

Activation: 31/07/2019

Nr. de Créditos: 1

Credits: 1

Carga Horária:

Workload:

Teórica

(por semana)

Theory

(weekly)

Prática

(por semana)

Practice

(weekly)

Estudos

(por semana)

Study

(weekly)

Duração Duration Total Total
10 0 5 1 semanas 1 weeks 15 horas 15 hours

Docentes Responsáveis:

Professors:

Eduardo do Nascimento Marcos

Hagen Meltzer

Objetivos:

O Objetivo é introduzir o estudo de retas projetivas com peso, e a teoria de feixes sobre a mesma, os feixes inclinantes e os complexos inclinantes na categoria derivada, bem como a ação do grupo de tranças sobre esse objetos.

Objectives:

The objective is to introduce the study of the weighted projective lines and the category of sheaves over it, as well the study of the derived category.

Justificativa:

O estudo desses objetos tem uma importância muito grande na teoria de representações. Essa disciplina representa uma disciplina avançada para os alunos de Álgebra.

Rationale:

These topics represent an important aspect of the representation theory. This is an advanced course which will be important for the students in Algebra.

Conteúdo:

Weighted projective lines and canonical algebras. 2. Exceptional modules for modules over canonical algebras and the braid group action. 3. Tubular mutations. 4. Tilting sheaves on weighted projective lines and concealed-canonical allgebras. 5. Tilting complexes for weighted projective lines

Content:

Weighted projective lines and canonical algebras. 2. Exceptional modules for modules over canonical algebras and the braid group action. 3. Tubular mutations. 4. Tilting sheaves on weighted projective lines and concealed-canonical allgebras. 5. Tilting complexes for weighted projective lines

Forma de Avaliação:

Exame oral

Type of Assessment:

Oral exam.

Bibliografia:

[1] I. Assem, D. Simson, A. Skowroński, Elements of Representation Theory of Associative Algebras, Volume 1. Techniques of Representation Theory, Lond. Math. Soc. Stud. Texts, vol. 65, Cambridge Univ. Press, Cambridge–New York, 2006.   [2] A. I. Bondal, Represntations of associative algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 53, No.1 25-44 (1989),  English translation: Math USSR, Izv. 35, No.3 (1990), 519-536.   [3] P. Dowbor, T. Hübner, A computer algebra approach to sheaves over weighted projective lines, In: Computational Methods for Representations of Groups and Algebras, in: Prog. Math., vol. 173, Birhäuser, (1999) 187–200, (1999).   [4] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory, In: Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 265-297 (1987).   [5] W. Geigle, H. Lenzing, Perpendicular categories with applications to representations and sheaves. J. Algebra 144, No.2, 273-343, (1991)   [6] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Lond. Math. Soc. Lecture Notes Ser., vol. 119, 1988.   [7] O. Kerner, Stable components of wild tilted algebras, J. Algebra 142, No. 1, 37-57, (1991).   [8] S.A. Kuleshov, D. O. Orlov, Exceptional sheaves on del Pezzo surfaces. Russ Acad, Sci. Izv. Math. 44, No.3, 479-513 (1995), Translation from Izv. Ross. Akad. Nauk, Ser. Mat. 58, No. 3, 53-87 (1994).  [9] H. Lenzing, H. Meltzer, Sheaves on a weighted projective line of genus one and representations of a tubular algeReferences     [10] H. Lenzing, H. Meltzer, Tilting sheaves and concealed-canonical algebras, In: Representation theory of algebras. Seventh international conference, August 22-26, 1994, Cocoyoc, Mexico, Bautista, Raymundo (ed.) et al., Providence, RI: American Mathematical Society. CMS Conf. Proc. 18, 455-473 (1996).   [11] H. Lenzing, H. Meltzer, The automorphism group of the derived category for a weighted projective line, Commun. Algebra 28,No.4, 1685-1700 (2000).   [12] H. Meltzer, Tubular mutations, Colloq. Math. 74, No.2, 267-274, (1997).   [13] H. Meltzer, Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines,References [14] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes Math., vol. 1099, Springer, (1984).   [15] A. N. Rudakov (ed.), Helices and vector bundles: Seminaaire Rudakov, London Mathematical Society Lecture Notes Series 148 Cambridge etc.: Cambridge University Press, (1990). bra, In: Representations of Algebras, Sixth International Conference, Ottawa, 1992, in: CMS Conf. Proc., vol. 14, 1992, pp. 317–337.

Bibliography:

[1] I. Assem, D. Simson, A. Skowroński, Elements of Representation Theory of Associative Algebras, Volume 1. Techniques of Representation Theory, Lond. Math. Soc. Stud. Texts, vol. 65, Cambridge Univ. Press, Cambridge–New York, 2006.   [2] A. I. Bondal, Represntations of associative algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 53, No.1 25-44 (1989),  English translation: Math USSR, Izv. 35, No.3 (1990), 519-536.   [3] P. Dowbor, T. Hübner, A computer algebra approach to sheaves over weighted projective lines, In: Computational Methods for Representations of Groups and Algebras, in: Prog. Math., vol. 173, Birhäuser, (1999) 187–200, (1999).   [4] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory, In: Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 265-297 (1987).   [5] W. Geigle, H. Lenzing, Perpendicular categories with applications to representations and sheaves. J. Algebra 144, No.2, 273-343, (1991)   [6] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Lond. Math. Soc. Lecture Notes Ser., vol. 119, 1988.   [7] O. Kerner, Stable components of wild tilted algebras, J. Algebra 142, No. 1, 37-57, (1991).   [8] S.A. Kuleshov, D. O. Orlov, Exceptional sheaves on del Pezzo surfaces. Russ Acad, Sci. Izv. Math. 44, No.3, 479-513 (1995), Translation from Izv. Ross. Akad. Nauk, Ser. Mat. 58, No. 3, 53-87 (1994).  [9] H. Lenzing, H. Meltzer, Sheaves on a weighted projective line of genus one and representations of a tubular algeReferences     [10] H. Lenzing, H. Meltzer, Tilting sheaves and concealed-canonical algebras, In: Representation theory of algebras. Seventh international conference, August 22-26, 1994, Cocoyoc, Mexico, Bautista, Raymundo (ed.) et al., Providence, RI: American Mathematical Society. CMS Conf. Proc. 18, 455-473 (1996).   [11] H. Lenzing, H. Meltzer, The automorphism group of the derived category for a weighted projective line, Commun. Algebra 28,No.4, 1685-1700 (2000).   [12] H. Meltzer, Tubular mutations, Colloq. Math. 74, No.2, 267-274, (1997).   [13] H. Meltzer, Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines,References [14] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes Math., vol. 1099, Springer, (1984).   [15] A. N. Rudakov (ed.), Helices and vector bundles: Seminaaire Rudakov, London Mathematical Society Lecture Notes Series 148 Cambridge etc.: Cambridge University Press, (1990). bra, In: Representations of Algebras, Sixth International Conference, Ottawa, 1992, in: CMS Conf. Proc., vol. 14, 1992, pp. 317–337.

Tipo de oferecimento da disciplina:

Presencial

Class type:

Presencial