q-Analog of the Mobius function and the cyclotomic identity associated to a profinite group

초록

Let G be a profinite group and q an indeterminate. In this paper, we introduce and study a q-analog of the Mobius function and the cyclotomic identity arising from the lattice of open subgroups of G. When q is any integer, we show that they have close connections with the functors W-G(q), Nr(G)(q), (Nr) over cap (q)(G) introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 257 (2007) 151-191]. In particular, we interpret the multiplicative property of the inverse of the table of marks and the Mobius function of G as a composition property of certain functors. Classification of W-G(q), Nr(G)(q), (Nr) over cap (q)(G) up to strict natural isomorphism as q varies over the set of integers and its application will be dealt with, too. (C) 2008 Elsevier Inc. All rights reserved.

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