Roots of Nakayama and Auslander-Reiten translations

Abstract

N) Abstract. We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an appli- cation we derive some new results on the structure of selfinjective algebras of canonical type. Throughout this paper K will denote a fixed algebraically closed field. We work in the derived category D b (X) of the category cohX of coherent sheaves on a weighted projective line X over K. We investigate whether, for a positive integer d, one of the automorphisms �T 2 ; ├T 2 ; �; that is, the Nakayama translation, a twisted Nakayama translation or the Auslander-Reiten translation, respectively, has a dth root in the automor- phism group of D b (X). Here, % denotes a rigid automorphism, that is, an automorphism of cohX—identified with a member of Aut(D b (X))—which preserves all Auslander-Reiten components and also the slope of indecom- posable objects; further, T denotes the translation shift in the derived cat- egory D b (X). Let Pic0 X denote the torsion group of the Picard group of X, and let AutX denote the automorphism group of X, identified with the group of all isomorphism classes of selfequivalences of the category cohX fixing the structure sheaf. It then follows from (9) that the rigid automorphisms form a subgroup of Pic0 Xo AutX, and, moreover, this group is finite if X has at least three exceptional points. Throughout the paper, by an automorphism we mean the isomorphism class of a selfequivalence of K-categories. When applied to a finite-dimen- sional basic K-algebra A, this means to identify automorphisms that differ by an inner automorphism. In particular, we say that an automorphism of A is non-trivial if it is not inner.