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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Affine toric equivalence relations are effective


Author: Claudiu Raicu
Journal: Proc. Amer. Math. Soc. 138 (2010), 3835-3847
MSC (2010): Primary 14A15, 14L30
Published electronically: May 24, 2010
MathSciNet review: 2679607
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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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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
Email: claudiu@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10416-1
PII: S 0002-9939(2010)10416-1
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.