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Auslander–Reiten sequences, locally free sheaves and Chebysheff polynomials

Published online by Cambridge University Press:  13 March 2006

Roberto Martínez-Villa
Affiliation:
Instituto de Matematicas de la UNAM, Unidad Morelia, CP 61-3, 58089, Morelia Michoacan, Mexicomvilla@matmor.unam.mx
Dan Zacharia
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, NY 13244, USAzacharia@syr.edu
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Abstract

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Let R be the exterior algebra in n + 1 variables over a field K. We study the Auslander–Reiten quiver of the category of linear R-modules, and of certain subcategories of the category of coherent sheaves over Pn. If n > 1, we prove that up to shift, all but one of the connected components of these Auslander–Reiten quivers are translation subquivers of a ${\bf Z} A_{\infty}$-type quiver. We also study locally free sheaves over the projective n-space Pn for n > 1 and we show that each connected component contains at most one indecomposable locally free sheaf of rank less than n. Finally, using results from the theory of finite-dimensional algebras, we construct a family of indecomposable locally free sheaves of arbitrary large ranks, where the ranks can be computed using the Chebysheff polynomials of the second kind.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006