Motivic Tambara functors

AbstractLet k be a field and denote by $$\mathcal {SH}(k)$$ SH ( k ) the motivic stable homotopy category. Recall its full subcategory $$\mathcal {SH}(k)^{{\text {eff}}\heartsuit }$$ SH ( k ) eff ♡ (Bachmann in J Topol 10(4):1124–1144. arXiv:1610.01346, 2017). Write $$\mathrm {NAlg}(\mathcal {SH}(k))$$ NAlg ( SH ( k ) ) for the category of $${\mathrm {S}\mathrm {m}}$$ S m -normed spectra (Bachmann-Hoyois in arXiv:1711.03061, 2017); recall that there is a forgetful functor $$U: \mathrm {NAlg}(\mathcal {SH}(k)) \rightarrow \mathcal {SH}(k)$$ U : NAlg ( SH ( k ) ) → SH ( k ) . Let $$\mathrm {NAlg}(\mathcal {SH}(k)^{{\text {eff}}\heartsuit }) \subset \mathrm {NAlg}(\mathcal {SH}(k))$$ NAlg ( SH ( k ) eff ♡ ) ⊂ NAlg ( SH ( k ) ) denote the full subcategory on normed spectra E such that $$UE \in \mathcal {SH}(k)^{{\text {eff}}\heartsuit }$$ U E ∈ SH ( k ) eff ♡ . In this article we provide an explicit description of $$\mathrm {NAlg}(\mathcal {SH}(k)^{{\text {eff}}\heartsuit })$$ NAlg ( SH ( k ) eff ♡ ) as the category of effective homotopy modules with étale norms, at least if $$char(k) = 0$$ c h a r ( k ) = 0 . A weaker statement is available if k is perfect of characteristic $$> 2$$ > 2 .