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

 

Products of Brauer-Severi surfaces


Author: Amit Hogadi
Journal: Proc. Amer. Math. Soc. 137 (2009), 45-50
MSC (2000): Primary 14E05, 14M99; Secondary 14J25
Published electronically: July 25, 2008
MathSciNet review: 2439423
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Abstract: Let $ \{P_i\}_{1 \leq i \leq r}$ and $ \{Q_i\}_{1 \leq i \leq r}$ be two collections of Brauer-Severi surfaces (resp. conics) over a field $ k$. We show that the subgroup generated by the $ P_i$'s in $ Br(k)$ is the same as the subgroup generated by the $ Q_i$'s if and only if $ \prod P_i $ is birational to $ \prod Q_i$. Moreover in this case $ \prod P_i$ and $ \prod Q_i$ represent the same class in $ M(k)$, the Grothendieck ring of $ k$-varieties. The converse holds if $ \mathrm{char}(k)=0$. Some of the above implications also hold over a general noetherian base scheme.


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Additional Information

Amit Hogadi
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Address at time of publication: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India
Email: amit@math.princeton.edu, amit@math.tifr.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09450-1
PII: S 0002-9939(08)09450-1
Keywords: Brauer-Severi surfaces, Grothendieck ring, birational maps
Received by editor(s): December 29, 2006
Received by editor(s) in revised form: June 23, 2007, and November 30, 2007
Published electronically: July 25, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.