Group cohomology of a finite cyclic group For the finite cyclic group of order with generator , the element in the associated group ring is a divisor of zero because its product with , given bygivesThis property can be used to construct the resolution of the trivial -module via the complexgiving the group cohomology computation for any -module . Note the augmentation map gives the trivial module its -structure byThis resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groupsshowing that applying the functor to the complex above (with removed since this resolution is a quasi-isomorphism), gives the computationforFor example, if , the trivial module, then , , and , hence Explicit cocycles Explicit cocycles for the group cohomology of a cyclic group using the Bar resolution can be given explicitlyprop 2.3. We get a complete set of generators of -cocycles for odd as the mapsgiven byfor odd, , a primitive -th root of unity, a field containing -th roots of unity, andfor a rational number denoting the largest integer not greater than . Also, we are using the notationwhere is a generator for . Note that for non-zero even indices the cohomology groups are trivial. Cohomology of free groups Using a resolution Given a set the associated free group has an explicit resolution of the trivial module which can be easily computed. Notice the augmentation maphas kernel given by the free submodule generated by the set , so.Because this object is free, this gives a resolutionhence the group cohomology of with coefficients in can be computed by applying the functor to the complex , givingthis is because the dual mapsends any -module morphismto the induced morphism on by composing the inclusion. The only maps which are sent to are -multiples of the augmentation map, giving the first cohomology group. The second can be found by noticing the only other mapscan be generated by the -basis of maps sending for a fixed , and sending for any . Using topology The group cohomology of free groups generated by letters can be readily computed by comparing group cohomology with its interpretation in topology. Recall that for every group there is a topological space , called the classifying space of the group, which has the propertyIn addition, it has the property that its topological cohomology is isomorphic to group cohomologygiving a way to compute some group