Abstract
Let $q$ be a non-negative integer. We prove that a perfect field $K$ has cohomological dimension at most $q+1$ if, and only if, for any finite extension $L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected algebraic group over $L$, the $q$-th Milnor $K$-theory group of $L$ is spanned by the images of the norms coming from finite extensions of $L$ over which $Z$ has a rational point. We also prove a variant of this result for imperfect fields.
Highlights
In 1986, in the article [11], Kato and Kuzumaki stated a set of conjectures which aimed at giving a diophantine characterization of cohomological dimension of fields
If Z is a scheme of finite type over L, one can introduce the subgroup Nq(Z/L) of KqM (L) generated by the images of the norm morphisms NL′/L when L′ runs through the finite extensions of L such that Z(L′) = ∅
For each non negative integer q, we introduce variants of the C1q property and we prove that, contrary to the C1q property, they characterize the cohomological dimension of fields
Summary
In 1986, in the article [11], Kato and Kuzumaki stated a set of conjectures which aimed at giving a diophantine characterization of cohomological dimension of fields. The theorems of Steinberg and Springer (see Section III.2.4 of [18]), which state that, if K is a perfect field with cohomological dimension at most one, every homogeneous space under a linear connected K-group has a zero-cycle of degree 1 (and even a rational point). (1) We first prove that, if P is a finite Galois module over a characteristic 0 field L of cohomological dimension ≤ q + 1 and α is an element in H2(L, P ), KqM (L) is spanned by the images of the norms coming from finite extensions of L trivializing α This requires to use the Bloch-Kato conjecture and some properties of norm varieties that have been established by Rost and Suslin.
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