Summary: A mod­# vanishing theorem of Beilinson­Soul’e type
Reza Akhtar
Abstract
Let L be a field containing an algebraically closed field and X an equidimen­
sional quasiprojective scheme over L. We prove that CH i (X, n; Z/#) = 0 when
n > 2i and # #= 0; this was known previously when i # dimX and L is itself
algebraically closed. This ``mod­#'' version of the Beilinson­Soul’e conjecture
implies the equivalence of the rational and integral versions of the conjecture
for varieties over fields of this type and can be used to prove the vanishing of
the (integral) groups CH i (X, n) (for n > 2i) in certain cases.
1 Introduction
Let k be a field and X a smooth scheme over k. The Beilinson­Soul’e conjecture
asserts that the motivic cohomology groups H p (X, Z(q)) vanish when p < 0; by the
work of Suslin and Voevodsky (see [V2]), this is equivalent to requiring that the
higher Chow groups CH i (X, n) vanish when n > 2i. It follows from the definition
and from the calculations in [B1], Section 7 that the conjecture holds for i # 1.
While the conjecture itself seems very di#cult, an analogous conjecture with finite
coe#cients appears somewhat more tractable; indeed, the work of Suslin [Su2], Geisser
[G], and Geisser­Levine [GL] implies immediately that if X is an equidimensional
quasiprojective scheme over an algebraically closed field, then CH i (X, n; Z/#) = 0

Source: Akhtar, Reza - Department of Mathematics and Statistics, Miami University (Ohio)

Collections: Mathematics