Inertial Chow rings of toric stacks

Abstract

For any vector bundle V on a toric Deligne–Mumford stack $${\mathcal X}$$ the formalism of Edidin et al. (Ann K-theory 1(1):85–108, 2016) defines two inertial products $$\star _{V^{+}}$$ and $$\star _{V^{-}}$$ on the Chow group of the inertia stack. We give an explicit presentation for the integral $$\star _{V^+}$$ and $$\star _{V^-}$$ Chow rings, extending earlier work of Borisov et al. (J Am Math Soc 18(1):193–215, 2005) and Jiang and Tseng (Math Z 264(1):225–248, 2010) in the orbifold Chow ring case, which corresponds to $$V = 0$$ . We also describe an asymptotic product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle V go to infinity.