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On the Braid Group Action on Exceptional Sequences for Weighted Projective Lines

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Abstract

We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line \(\mathbb {X}\) does not depend on the parameters of \(\mathbb {X}\). Finally we prove that the determinant of the matrix obtained by taking the values of n \(\mathbb {Z}\)-linear functions defined on the Grothendieck group \(\textrm{K}_0(\mathbb {X}) \simeq \mathbb {Z}^n \) of the elements of a full exceptional sequence is an invariant, up to sign.

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Acknowledgements

The first author was partially supported by Brazilian-French Network in Mathematics. The second author was partially supported by CNPq-302003/2018-5, by Grant Fapesp 2018/23690-6 and also by Brazilian-French Network in Mathematics. The third author was partially supported by Fapesp, 2018/08104-3, 2014/09310-5.

Funding

The first named author was partially supported by Brazilian-French Network in Mathematics. The second named author was partially supported by CNPq-302003/2018-5, by Grant Fapesp 2018/23690-6 and also by Brazilian-French Network in Mathematics. The third named author was partially supported by Fapesp, 2018/08104-3, 2014/09310-5.

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Authors and Affiliations

  1. Departamento de Matemática Universidade Federal do Parana, Curitiba, Brazil

    Edson Ribeiro Alvares

  2. Departamento de Matemática , IME, Universidade de Sao Paulo Brazil, Sao Paulo, Brazil

    Eduardo Nascimento Marcos

  3. Instytut Matematyki, Uniwersytet Szczeciński, 70451, Szczecin, Poland

    Hagen Meltzer

Authors
  1. Edson Ribeiro Alvares
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  2. Eduardo Nascimento Marcos
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  3. Hagen Meltzer
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Contributions

All authors reviewed the manuscript. The authors are Edson Ribeiro Alvares, Eduardo Nascimento Marcos and Hagen Meltzer. The work was done in joint discussions. It is impossible to say exactly who did what.

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Correspondence to Edson Ribeiro Alvares.

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Presented by: Christof Geiß

We dedicate this work to the memory of Andrzej Skowroński.

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Alvares, E.R., Marcos, E.N. & Meltzer, H. On the Braid Group Action on Exceptional Sequences for Weighted Projective Lines. Algebr Represent Theor 27, 897–909 (2024). https://doi.org/10.1007/s10468-023-10243-9

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