The \(K\)-theory of fields in characteristic \(p\)
For a smooth quasi-projective variety \(X\) over a field one defines motivic cohomology \(H^p(X,{\mathbb Z}(q))\) as Bloch's higher Chow group \(\text{CH}^q(X,2q-p)\). In particular, in the underlying paper one is interested in the study of the \(p\)-part of the motivic cohomology groups \(H^i(k,{\mathbb Z}(n))\) and the higher algebraic \(K\)-groups \(K_n(k)\) of a field \(k\) of characteristic \(p\). The main result for such a field is the vanishing of the motivic cohomology groups \(H^i(k,{\mathbb Z}/p^r(n))\) for all \(i\neq n\), in other words \(H^i(k,{\mathbb Z}(n))\) is uniquely \(p\)-divisible for \(i\neq n\). There are several interesting consequences of the main result: \((i)\) Let \(K_n^M(k)\) denote the Milnor \(K\)-groups of \(k\). Then the natural map \(K_n^M(k)\rightarrow K_n(k)\) is an isomorphism up to uniquely \(p\)-divisible groups; \((ii)\) (due to \textit{B. Kahn}) For a semi-local ring \(R\), essentially smooth over a discrete valuation ring, Gersten's conjecture holds, i.e. there is an exact sequence \[ 0\rightarrow K_n(R,{\mathbb Z}/m)\rightarrow\bigoplus_{x\in R^{(0)}}K_n(k(x), {\mathbb Z}/m)\rightarrow\bigoplus_{x\in R^{(1)}}K_{n-1}(k(x),{\mathbb Z}/m) \rightarrow\cdots \] \((iii)\) Let \(k\) be a perfect field (of characteristic \(p\)), and let \(X\) be a smooth variety of dimension \(d\) over \(k\). Then \(K_n(X,{\mathbb Z}/p^r)=0\) for \(n>\dim(X)\). Furthermore, the main result also implies that Bloch's cycle complexes satisfy most of the Beilinson-Lichtenbaum axioms for motivic complexes, as extended by \textit{J. S. Milne} [Compos. Math. 68, No. 1, 59-102 (1988; Zbl 0681.14007)]. The paper ends with a variant of Hilbert's theorem \(90\): For a smooth variety \(X\) over a perfect field \(k\) of characteristic \(p\) one has \(R^{n+1}\varepsilon_*{\mathbb Z}_{(p)}(n)_{\text{ét}}=0\), where \({\mathbb Z}_{(p)}(n)={\mathbb Z}(n)\otimes{\mathbb Z}_{(p)}\), and \(\varepsilon:X_{\text{ét}}\rightarrow X_{\text{Zar}}\) is the change of topology map. The proof of the main result is done in a number of steps. An important ingredient are the logarithmic de Rham-Witt groups.
scientific article; zbMATH DE number 1443583