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Abstract Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.
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The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors
For N∞ operads O and O' such that there is an inclusion of the associated indexing systems, there is a forgetful functor from incomplete Tambara functors over O' to incomplete Tambara functors over O. Roughly speaking, this functor forgets the norms in O' that are not present in O. The forgetful functor has both a left and a right adjoint; the left adjoint is an operadic tensor product, but the right adjoint is more mysterious. We explicitly compute the right adjoint for finite cyclic groups of prime order.
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- PAR ID:
- 10156171
- Date Published:
- Journal Name:
- Contemporary mathematics
- Volume:
- 729
- ISSN:
- 0271-4132
- Page Range / eLocation ID:
- 75-92
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1090/conm/729/14691
