Derived autoequivalences and a weighted Beilinson resolution

Given a smooth stacky Calabi-Yau hypersurface X in a weighted projective space, we consider the functor G which is the composition of the following two autoequivalences of Db(X): the first one is induced by the spherical object OX, while the second one is tensoring with OX(1). The main result of the paper is that the composition of G with itself w times, where w is the sum of the weights of the weighted projective space, is isomorphic to the autoequivalence “shift by 2”. The proof also involves the construction of a Beilinson type resolution of the diagonal for weighted projective spaces, viewed as smooth stacks.