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Singularities in derived categories

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Abstract

Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category Db(A) of finitely generated A-modules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an exact triangle N→M⊕Z→Z→N[1]. In this paper we define and study the type of singularity at every degeneration in the bounded derived categrory.

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References

  1. Bekkert, V., Drozd, Y.: Tame-wild dichotomy for derived categories. math.RT/ 0310352v1

  2. Bobiński, G., Zwara, G.: Normality of orbit closures for Dynkin quivers of type . Manuscripta Mathematica 105 (1), 103–109 (2001)

    Google Scholar 

  3. Bobiński, G., Zwara, G.: Schubert varieties and representations of Dynkin quivers. Colloquium Mathematicum 94 (2), 285–309 (2002)

    Google Scholar 

  4. Bongartz, K.: A geometric version of the Morita equivalence. J. Algebra 139, 159–171 (1991)

    Article  Google Scholar 

  5. Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Commentarii Mathematici Helvetici 69 (4), 575–611 (1994)

    Google Scholar 

  6. Crawley-Boevey, W., Schröer, J.: Irreducible components of varieties of modules. Journal für die Reine und Angewandte Mathematik 553, 201–220 (2002)

    Google Scholar 

  7. Hesselink, W.: Singularities in the nilpotent scheme of a classical group. Transactions of the American Mathematical Society 222, 1–32 (1976)

    Google Scholar 

  8. Huisgen-Zimmermann, B., Saorin, M.: Geometry of chain complexes and outer automorphisms under derived equivalence. Transactions of the American Mathematical Society 353 (12), 4757–4777 (2001)

    Article  Google Scholar 

  9. Jensen, B.T, Su, X., Zimmermann, A.: Degenerations for derived categories. Journal of Pure and Applied Algebra 198 (1–3), 281–295 (2005)

    Google Scholar 

  10. Le Potier, J.: Lectures on vector bundles. Translated by A. Maciocia. Cambridge Studies in Advanced Mathematics, 54. Cambridge University Press, Cambridge, 1997. viii+251 pp. ISBN: 0-521-48182-1

  11. Richmond, N.J.: A stratification for varieties of modules. The Bulletin of the London Mathematical Society 33 (5), 565–577 (2001)

    Google Scholar 

  12. Shafarevich, I.R.: Basic algebraic geometry. 2. Schemes and complex manifolds. Second edition, Translated from the 1988 Russian edition by Miles Reid. Springer-Verlag, Berlin, 1994. xiv+269 pp. ISBN: 3-540-57554-5

  13. Skowroński, A., Zwara, G.: Derived equivalences of selfinjective algebras preserve singularities. Manuscripta Mathematica 112 (2), 221–230 (2003)

    Article  Google Scholar 

  14. Zwara, G.: Smooth morphisms of module schemes. Proceedings of the London Mathematical Society 84, 539–558 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

    Bernt Tore Jensen

  2. LAMFA UMR 6140 du CNRS, Faculté de Mathématiques, Université de Picardie, 33 rue St Leu, F-80039, Amiens Cedex 1, France

    Xiuping Su

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  1. Bernt Tore Jensen
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  2. Xiuping Su
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Correspondence to Bernt Tore Jensen.

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Jensen, B., Su, X. Singularities in derived categories. manuscripta math. 117, 475–490 (2005). https://doi.org/10.1007/s00229-005-0572-3

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