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The explicit reciprocity law and the cohomology of Fontaine-Messing. (English) Zbl 0752.14015
After describing the Fontaine-Messing cohomology of schemes of mixed characteristic and defining the Milnor \(K\)-groups of a ring, this paper, completed in 1986, is concerned with certain regular local rings \(A\) of mixed characteristics \((0,p)\). One defines symbol maps from the Milnor \(K\)-groups of \(A\) to the étale cohomology groups of \(\text{Spec}{A\left[{1\over p}\right]}\) and to the Fontaine-Messing cohomology groups of \(\text{Spec}A\) realized with differential forms. Comparising these symbols, one obtains an explicit reciprocity law, which generalizes the classical one for Hilbert symbols as well as recent ones for local fields.

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
13H05 Regular local rings
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