# Affine Toric Equivalence Relations are Effective - Mathematics > Algebraic Geometry

Abstract: Any map of schemes $X\to Y$ defines an equivalence relation $R=X\times Y X\toX\times X$, the relation of -being in the same fiber-. We have shown elsewherethat not every equivalence relation has this form, even if it is assumed to befinite. By contrast, we prove here that every toric equivalence relation on anaffine toric variety does come from a morphism and that quotients by finitetoric equivalence relations always exist in the affine case. In special cases,this result is a consequence of the vanishing of the first cohomology group inthe Amitsur complex associated to a toric map of toric algebras. We prove moregenerally the exactness of the Amitsur complex for maps of commutative monoidrings.

Author: Claudiu Raicu

Source: https://arxiv.org/