A generalization of the Dress construction for a Tambara functor, and its relation to polynomial Tambara functors

Abstract

For a finite group G, (semi-)Mackey functors and (semi-)Tambara functors are regarded as G-bivariant analogs of (semi-)groups and (semi-)rings respectively. In fact if G is trivial, they agree with the ordinary (semi-)groups and (semi-)rings, and many naive algebraic properties concerning rings and groups have been extended to these G-bivariant analogous notions.In this article, we investigate a G-bivariant analog of the semi-group rings with coefficients. Just as a coefficient ring R and a monoid Q yield the semigroup ring R[Q], our constrcution enables us to make a Tambara functor T [M ] out of a semi-Mackey functor M , and a coefficient Tambara functor T . This construction is a composant of the Tambarization and the Dress construction.As expected, this construction is the one uniquely determined by the righteous adjoint property. Besides in analogy with the trivial group case, if M is a Mackey functor, then T [M ] is equipped with a natural Hopf structure.Moreover, as an application of the above construction, we also obtain some G-bivariant analogs of the polynomial rings.

Spectrum of the Burnside Tambara functor on a cyclic $p$-group

“…We have ideals [5], fractions [4], and polynomials [6] of Tambara functors. These are mutually related, as they should be, and moreover in connection with the celebrated Dress construction [10], [7]. The most typical Tambara functor is the Burnside Tambara functor Ω G (Example 1.4), which plays a role just like Z in the ordinary commutative ring theory.…”