Constructing Witt-Burnside rings |
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Authors: | Jesse Elliott |
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Institution: | California State University, Channel Islands, 1 University Drive, Camarillo, CA 93012, USA |
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Abstract: | The Witt-Burnside ring of a profinite group G over a commutative ring A generalizes both the Burnside ring of virtual G-sets and the rings of universal and p-typical Witt vectors over A. The Witt-Burnside ring of G over the monoid ring ZM], where M is a commutative monoid, is proved isomorphic to the Grothendieck ring of a category whose objects are almost finite G-sets equipped with a map to M that is constant on G-orbits. In particular, if A is a commutative ring and A× denotes the set A as a monoid under multiplication, then the Witt-Burnside ring of G over ZA×] is isomorphic to Graham's ring of “virtual G-strings with coefficients in A.” This result forms the basis for a new construction of Witt-Burnside rings and provides an important missing link between the constructions of Dress and Siebeneicher Adv. in Math. 70 (1988) 87-132] and Graham Adv. in Math. 99 (1993) 248-263]. With this approach the usual truncation, Frobenius, Verschiebung, and Teichmüller maps readily generalize to maps between Witt-Burnside rings. |
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Keywords: | 13K05 19A22 20E18 |
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