Auslander–Reiten sequences, locally free sheaves and Chebysheff polynomials

Abstract

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.