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

2004/05/21 by Young-Tak Oh, Oh, Young-Tak
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.RA #msc:11F03 #msc:11F22 #msc:17B70

paper · pdf · doi:10.48550/arxiv.math/0405400

minor corrections, 35 pages

arxiv created 2004/07/15 · arxiv updated 2009/12/01

Abstract

For every profinite group G, we construct two covariant functors Δ_G and \bf \mathcal AP_G from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor W_G introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, \it Adv. in Math. \bf70 (1988), 87-132]. We call Δ_G the generalized Burnside-Grothendieck ring functor and \bf \mathcal AP_G the aperiodic ring functor (associated with G). In case G is abelian, we also construct another functor \bf Ap_G from the category of commutative rings with identity to itself as a generalization of the functor \bf Ap introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, \it Adv. in Math. \bf 81 (1990), 1-29]. Finally it is shown that there exist q-analogues of these functors (i.e, W_G, Δ_G, \bf \mathcal AP_G, and \bf Ap_G) in case G= C the profinite completion of the multiplicative infinite cyclic group.

Related

• Gysin functors and the Grothendieck-Witt category, Part I
• The extended Burnside ring and module categories
• On the structure of Witt-Burnside rings attached to pro-p groups
• The Burnside ring of simple C-sets
• Grothendieck ring of pretriangulated categories