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Some observations on motivic cohomology of arithmetic schemes. (English) Zbl 0694.14005
Following S. Lichtenbaum [Invent. Math. 88, 183-215 (1987; Zbl 0615.14004)], one defines for arithmetic schemes X the motivic, i.e. universal, cohomology groups denoted by \(H^{i+2}(K,{\mathbb{Z}}(i))\), \(i\geq -1\), where K is the function field of X. For instance, \(H^ 3(K,{\mathbb{Z}}(1))\) is defined as \(H^ 2(K,{\mathbb{G}}_ m)\), the Brauer group of K. A wide range of known as well as new facts are obtained in a unified way with a general reciprocity homomorphism, which, if it is an isomorphism, establishes a duality between motivic cohomology group and an idele class group of X.
For \(i=0\), this duality amounts to higher dimensional class field theory; for \(i=-1\), it is a reciprocity uniqueness theorem. If X is regular of dimension 2, for \(i=2\) one has a certain Hasse principle for the existence of O-cycles on X of degree 1 and information about the kernel and cokernel of the reciprocity homomorphism for \(i=1\) is related to the Brauer-Grothendieck group of X.
Reviewer: J.H.de Boer

14F99 (Co)homology theory in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14G25 Global ground fields in algebraic geometry
Full Text: DOI EuDML
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