View source for Parshin's conjecture

In [[Mathematics|mathematics]], more specifically in [[Algebraic geometry|algebraic geometry]], '''Parshin's conjecture''' (also referred to as the '''Beilinson–Parshin conjecture''') states that for any smooth [[Projective variety|projective variety]] ''X'' defined over a [[Finite field|finite field]], the higher [[Algebraic K-theory|algebraic K-groups]] vanish up to torsion:<ref>Conjecture 51 in {{cite book|title=Handbook of K-Theory I|year=2005|publisher=Springer|pages=351–428|author=Kahn, Bruno|editor1=Friedlander, Eric |editor2=Grayson, Daniel |chapter=Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry}}</ref>
:<math>K_i(X) \otimes \mathbf Q = 0, \ \, i > 0.</math>

It is named after Aleksei Nikolaevich Parshin and [[Biography:Alexander Beilinson|Alexander Beilinson]].

==Finite fields==
The conjecture holds if <math>dim\ X = 0</math> by Quillen's computation of the K-groups of finite fields,<ref>{{cite journal|last=Quillen|first=Daniel|title=On the cohomology and K-theory of the general linear groups over a finite field|journal=Ann. of Math.|volume=96|year=1972|pages=552–586}}</ref> showing in particular that they are finite groups.

==Curves==
The conjecture holds if <math>dim\ X = 1</math> by the proof of Corollary 3.2.3 of Harder.<ref>{{cite journal|last=Harder|first=Günter|title=Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern|journal=Invent. Math.|year=1977|volume=42|pages=135–175|doi=10.1007/bf01389786}}</ref>
Additionally, by [[Biography:Daniel Quillen|Quillen]]'s finite generation result<ref>{{cite book|last=Grayson|first=Dan|title=Algebraic K-theory, Part I (Oberwolfach, 1980)|chapter=Finite generation of K-groups of a curve over a finite field (after Daniel Quillen)|url=http://www.math.uiuc.edu/~dan/Papers/FiniteGeneration.pdf|year=1982|volume=966|series=Lecture Notes in Math.|publisher=Springer|location=Berlin, New York}}</ref> (proving the [[Bass conjecture]] for the ''K''-groups in this case) it follows that the ''K''-groups are finite if <math>dim\ X = 1</math>.

==References==
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