Affine toric equivalence relations are effective
Author:
Claudiu Raicu
Journal:
Proc. Amer. Math. Soc. 138 (2010), 3835-3847
MSC (2010):
Primary 14A15, 14L30
DOI:
https://doi.org/10.1090/S0002-9939-2010-10416-1
Published electronically:
May 24, 2010
MathSciNet review:
2679607
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
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Additional Information
Claudiu Raicu
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720-3840 – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy, RO-014700 Bucharest, Romania
MR Author ID:
909516
Email:
claudiu@math.berkeley.edu
Keywords:
Equivalence relations,
toric varieties,
Amitsur complex,
monoid rings,
cohomology
Received by editor(s):
September 24, 2009
Received by editor(s) in revised form:
January 30, 2010
Published electronically:
May 24, 2010
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.