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Summary: A mod- vanishing theorem of Beilinson-SoulŽe 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 CHi(X, n; Z/ ) = 0 when
n > 2i and = 0; this was known previously when i dim X and L is itself
algebraically closed. This "mod- " version of the Beilinson-SoulŽe 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 CHi(X, n) (for n > 2i) in certain cases.
1 Introduction
Let k be a field and X a smooth scheme over k. The Beilinson-SoulŽe conjecture
asserts that the motivic cohomology groups Hp
(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 CHi
(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 difficult, an analogous conjecture with finite
coefficients appears somewhat more tractable; indeed, the work of Suslin [Su2], Geisser
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