  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.
