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

Abstract

For every profinite group G, we construct two covariant functors Δ G and AP G which are equivalent to the functor W G 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 Δ G the generalized Burnside–Grothendieck ring functor and AP G the aperiodic ring functor (associated with G). In case G is abelian, we also construct another functor Ap G 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., W G , Δ G , AP G , and Ap G ) in case G is the profinite completion of the multiplicative infinite cyclic group C ˆ .