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Parshin's conjecture

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In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion:[1]

K i ( X ) ⊗ Q = 0 ,   i > 0. {\displaystyle K_{i}(X)\otimes \mathbf {Q} =0,\ \,i>0.} {\displaystyle K_{i}(X)\otimes \mathbf {Q} =0,\ \,i>0.}

It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.

Finite fields

The conjecture holds if d i m   X = 0 {\displaystyle dim\ X=0} {\displaystyle dim\ X=0} by Quillen's computation of the K-groups of finite fields,[2] showing in particular that they are finite groups.

Curves

The conjecture holds if d i m   X = 1 {\displaystyle dim\ X=1} {\displaystyle dim\ X=1} by the proof of Corollary 3.2.3 of Harder.[3] Additionally, by Quillen's finite generation result[4] (proving the Bass conjecture for the K-groups in this case) it follows that the K-groups are finite if d i m   X = 1 {\displaystyle dim\ X=1} {\displaystyle dim\ X=1}.

References

  1. Conjecture 51 in Kahn, Bruno (2005). "Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry". In Friedlander, Eric; Grayson, Daniel (eds.). Handbook of K-Theory I. Springer. pp. 351–428.
  2. Quillen, Daniel (1972). "On the cohomology and K-theory of the general linear groups over a finite field". Ann. of Math. 96: 552–586.
  3. Harder, Günter (1977). "Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern". Invent. Math. 42: 135–175. Bibcode:1977InMat..42..135H. doi:10.1007/bf01389786.
  4. Grayson, Dan (1982). "Finite generation of K-groups of a curve over a finite field (after Daniel Quillen)". Algebraic K-theory, Part I (Oberwolfach, 1980) (PDF). Lecture Notes in Math. Vol. 966. Berlin, New York: Springer.