Abstract:
In general, for a finitely generated, abelian group G, not all units in the group ring Z[G] are trivial. For example, Z[C_5] has units which are not of the form \pm C_5. However, if one instead considers group rings over the sphere spectrum, then all strict units, i.e. ring maps from S[Z] \to S[G], are trivial. This was proved recently by Carmeli-Nikolaus-Yuan. Moreover, by work of Carmeli, the group of strict units in S is the trivial group. Thus, all strict units in S[G] are in the image of G. I will give an overview of these results, explain the role of lifts of the Frobenius and discuss an extension of Carmeli's theorem to S[1/p].