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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. 

We introduce a computationally tractable way to describe the $$\mathbb Z$$-homotopy fixed points of a $$C_{n}$$-spectrum $$E$$, producing a genuine $$C_{n}$$ spectrum $$E^{hn\mathbb Z}$$ whose fixed and homotopy fixed points agree and are the $$\mathbb Z$$-homotopy fixed points of $$E$$. These form the bottom piece of a contravariant functor from the divisor poset of $$n$$ to genuine $$C_{n}$$-spectra, and when $$E$$ is an $$N_{\infty}$$-ring spectrum, this functor lifts to a functor of $$N_{\infty}$$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $$E^{hn\mathbb Z}$$, giving the homotopy groups of the $$\mathbb Z$$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $$\mathbb Z$$-homotopy fixed point case, giving us a family of new tools to simplify slice computations.