The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

Keywords:

group cohomology, norm map, cyclic group, homotopy group cohomology, norm map, cyclic group, homotopy

MSC Classifications:

20J06, 20K01, 16W22, 18G35 show english descriptions Cohomology of groups
Finite abelian groups [For sumsets, see 11B13 and 11P70]
Actions of groups and semigroups; invariant theory
Chain complexes [See also 18E30, 55U15] 20J06 - Cohomology of groups
20K01 - Finite abelian groups [For sumsets, see 11B13 and 11P70]
16W22 - Actions of groups and semigroups; invariant theory
18G35 - Chain complexes [See also 18E30, 55U15]

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