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 In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

''X'' defined over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

, the higher algebraic K-groups vanish up to torsion: :

K_i(X) \otimes \mathbf Q = 0, \ \, i > 0.

It is named after Aleksei Nikolaevich Parshin and

Alexander Beilinson Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...

Finite fields

The conjecture holds if

dim\ X = 0

by Quillen's computation of the K-groups of finite fields, 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. Additionally, by Quillen's finite generation result (proving the Bass conjecture for the ''K''-groups in this case) it follows that the ''K''-groups are finite if

dim\ X = 1

References

{{DEFAULTSORT:Parshin's conjecture Algebraic geometry Algebraic K-theory Conjectures