Parshin's conjecture explained

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 .

It is named after Aleksei Nikolaevich Parshin and Alexander Beilinson.

Finite fields

The conjecture holds if

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

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 dim X = 1 .

Notes and References

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