Root stacks and periodic decompositions

Abstract

AbstractFor an effective Cartier divisor D on a scheme X we may form an $${n}^{\text {th}}$$ n th  root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is $$2n$$ 2 n -periodic. For $$n=2$$ n = 2 this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For $$n > 2$$ n > 2 we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (N-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.