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Comparing Coxeter functors made from exceptional and spherical objects

A Coxeter element is a special isometry defined for some free abelian groups with a (not necessarily symmetric) bilinear pairing. For example, for a lattice with a basis of roots (i.e. vectors of square −2), the Coxeter element is the product of reflections along the basis roots. It depends on the ordering of the basis, but a permutation of the basis will lead to a conjugated Coxeter element. In various contexts, triangulated categories arise whose (numerical) K-groups, equipped with the Euler pairing, possess such a Coxeter element. By a Coxeter functor, we mean an autoequivalence of the category which lifts that Coxeter element. In the study of derived categories, the special role of exceptional collections was noticed very early. As shown in [BK], the existence of a full exceptional collection means that the category is built from blocks of the most simple type — the category of graded vector spaces. It is also well-established that such collections can be mutated, giving rise to a Coxeter functor on the category. More recently, collections of spherical objects have been found to be of importance. Spherical objects, in contrast to exceptional objects, are already interesting on their own, as they give rise to autoequivalences, the spherical twists. As originally noted by [ST], there are many cases where exceptional and spherical objects appear together, although in different categories. In our setup, we begin with two categories, one of which is generated by an exceptional collection whereas the other one is generated by spherical objects, with the two categories connected by a functor with reasonable properties. This situation occurs rather often, both in algebraic and geometric situations. The generators mentioned above give rise to two Coxeter-type functors which we compare in a nice way. Our applications include a purely algebraic statement in the language of Keller’s derived preprojective algebras made from a dg category generated by a full exceptional collection [Ke]. Geometric applications are obtained using (equivariant) cones (also see [Bri1]) and include the weighted projective lines of Geigle and Lenzing [GL]. As specific