Tate diagonal



While the symmetric monoidal (∞,1)-category of spectra is not a cartesian monoidal (∞,1)-category, hence while the smash product of spectra does not admit diagonal morphisms, it turns out that for each prime number pp there is a analogue of the diagonal into the pp-fold smash product. This is the Tate diagonal (def. ). One way that this really does behave like a diagonal map is the construction of the Frobenius morphism on E-∞ rings (see this def. and this example)


For nn \in \mathbb{N} a natural number, write C p/nC_p \coloneqq \mathbb{Z}/n\mathbb{Z} for the cyclic group of order nn.


For XSpectraX \in Spectra a spectrum and nn a natural number, consider the Tate spectrum

(XX) tC nSpectra (X \wedge \cdots \wedge X)^{t C_n} \in Spectra

where the nn-fold smash product of spectra is regarded as equipped with the ∞-action by the cyclic group given by cyclic permutation of smash factors. This construction canonically extends to an (∞,1)-functor

T n:SpectraSpectra T_n \;\colon\; Spectra \longrightarrow Spectra

on the (∞,1)-category of spectra.


For pp a prime number, the ∞-groupoid of natural transformations

id spectraT p id_{spectra} \longrightarrow T_p

from the identity functor on the (∞,1)-category of spectra to the (∞,1)-functor T pT_p from def. is equivalent to the underlying space of T pT_p applied to the sphere spectrum:

Hom(id Specta,T p)Ω T p(𝕊)Hom Specta(𝕊,T p(𝕊)). Hom( id_{Specta}, T_p ) \;\simeq\; \Omega^\infty T_p(\mathbb{S}) \simeq Hom_{Specta}(\mathbb{S}, T_p(\mathbb{S})) \,.

(Nikolaus-Scholze 17, corollary III.1.3)


For pp a prime number, the Tate diagonal is the natural transformation

id Specta Δ p T p X (XX) tC p \array{ id_{Specta} &\overset{\Delta_p}{\longrightarrow}& T_p \\ X &\mapsto& (X \wedge \cdots X)^{t C_p} }

is the one which under the equivalence of prop. corresponds to the composite morphism

𝕊𝕊 C p𝕊 tC p \mathbb{S} \overset{}{\longrightarrow} \mathbb{S}^{C_p} \overset{}{\longrightarrow} \mathbb{S}^{t C_p}

where the first is the unit map into the homotopy fixed point spectrum, and the second is the defining one of the Tate spectrum.

(Nikolaus-Scholze 17, def. III.1.4)