Generalized Burnside–Grothendieck ring functor and aperiodic ring functor associated with profinite groups

For every profinite group G, we construct two covariant functors ΔG and which are equivalent to the functor introduced in [A. Dress, C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, Adv. Math. 70 (1988) 87–132]. We call ΔG the generalized Burnside–Grothendieck ring functor and the aperiodic ring functor (associated with G). In case G is abelian, we also construct another functor ApG from the category of commutative rings with identity to itself as a generalization of the functor Ap introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, Adv. Math. 81 (1990) 1–29]. Finally, it is shown that there exist q-analogues of these functors (i.e., , and ApG) in case G is the profinite completion of the multiplicative infinite cyclic group .