Tsen's theorem
In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes,{{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | page=181 | zbl=1130.12001 }} and more generally that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve.
The theorem was published by Chiungtze C. Tsen in 1933.
See also
References
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- {{citation | last1=Ding | first1=Shisun | last2=Kang | first2=Ming-Chang | last3=Tan | first3=Eng-Tjioe | title=Chiungtze C. Tsen (1898–1940) and Tsen's theorems | doi=10.1216/rmjm/1181070405 | year=1999 | journal=Rocky Mountain Journal of Mathematics | issn=0035-7596 | volume=29 | issue=4 | pages=1237–1269 | zbl=0955.01031 | mr=1743370 | doi-access=free }}
- {{citation | last=Lang | first=Serge | title=On quasi algebraic closure | journal=Annals of Mathematics |series=Second Series | volume=55 | pages=373–390 | year=1952 | issue=2 | issn=0003-486X | authorlink=Serge Lang |jstor=1969785 | zbl=0046.26202 | doi=10.2307/1969785}}
- {{citation | first=J. P. | last=Serre | authorlink=Jean-Pierre Serre | year=2002 |title=Galois Cohomology | publisher=Springer-Verlag | zbl=1004.12003 | others=Translated from the French by Patrick Ion | series=Springer Monographs in Mathematics | location=Berlin | isbn=3-540-42192-0 }}
- {{citation | last=Tsen | first=Chiungtze C. | authorlink=Chiungtze C. Tsen | title=Divisionsalgebren über Funktionenkörpern | language=German | journal=Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. | pages=335–339 | year=1933 | jfm=59.0160.01 | zbl=0007.29401 | url=https://eudml.org/doc/59436 }}
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Category:Theorems in algebraic geometry
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