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A Homological Interpretation of the Transverse Quiver Grassmannians

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Abstract

In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second–named author to introduce a variety called transverse quiver Grassmannian and the first–named and third–named authors to consider the smooth loci of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext1(N, M/N) = 0. As a corollary we prove that the transverse quiver Grassmannian coincides with the smooth locus of the irreducible components of minimal dimension in the quiver Grassmannian.

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Authors and Affiliations

  1. Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy

    Giovanni Cerulli Irelli

  2. Département de Mathématiques, Université de Sherbrooke, 2500, Boul. de l’Université, J1K 2R1, Sherbrooke, QC, Canada

    Grégoire Dupont

  3. Dipartimento di Matematica Pura ed Applicata, Università degli studi di Padova, Via Trieste 63, 35121, Padova, Italy

    Francesco Esposito

Authors
  1. Giovanni Cerulli Irelli
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  2. Grégoire Dupont
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  3. Francesco Esposito
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Correspondence to Grégoire Dupont.

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Cerulli Irelli, G., Dupont, G. & Esposito, F. A Homological Interpretation of the Transverse Quiver Grassmannians. Algebr Represent Theor 16, 437–444 (2013). https://doi.org/10.1007/s10468-011-9314-2

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  • DOI: https://doi.org/10.1007/s10468-011-9314-2

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