Affine Toric Equivalence Relations are Effective

TL;DR

This paper proves that all affine toric equivalence relations are induced by morphisms and that quotients by finite toric relations exist, extending understanding of toric algebraic structures and their quotients.

Contribution

It establishes that every affine toric equivalence relation arises from a morphism and that quotients by finite toric relations always exist in the affine case.

Findings

All affine toric equivalence relations are effective.

Quotients by finite toric equivalence relations exist in the affine case.

The Amitsur complex is exact for maps of monoid rings.

Abstract

Any map of schemes $X\to Y$ defines an equivalence relation $R=X\times_Y X\to X\times X$, the relation of "being in the same fiber". We have shown elsewhere that not every equivalence relation has this form, even if it is assumed to be finite. By contrast, we prove here that every toric equivalence relation on an affine toric variety does come from a morphism and that quotients by finite toric equivalence relations always exist in the affine case. In special cases, this result is a consequence of the vanishing of the first cohomology group in the Amitsur complex associated to a toric map of toric algebras. We prove more generally the exactness of the Amitsur complex for maps of commutative monoid rings.