Compositio Mathematica

Auslander–Reiten sequences, locally free sheaves and Chebysheff polynomials

Roberto Martínez-Villa a1 and Dan Zacharia a2
a1 Instituto de Matematicas de la UNAM, Unidad Morelia, CP 61-3, 58089, Morelia Michoacan, Mexico
a2 Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA

Article author query
martinez-villa r   [Google Scholar] 
zacharia d   [Google Scholar] 


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.

(Published Online March 13 2006)
(Received December 21 2004)
(Accepted April 18 2005)

Key Words: Koszul algebras; linear modules; exterior algebra; coherent sheaves; locally free sheaves.

Maths Classification

14F05; 16E10; 16E65; 16G20; 16G70.

Dedicated to Claus M. Ringel on the occasion of his 60th birthday