nLab Tate diagonal

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Idea

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 $p$ there is a analogue of the diagonal into the $p$ -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 $n \in \mathbb{N}$ a natural number, write $C_p \coloneqq \mathbb{Z}/n\mathbb{Z}$ for the cyclic group of order $n$ .

For $X \in Spectra$ a spectrum and $n$ a natural number, consider the Tate spectrum

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

where the $n$ -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 \;\colon\; Spectra \longrightarrow Spectra

id_{spectra} \longrightarrow T_p

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

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

For $p$ a prime number, the Tate diagonal is the natural transformation

\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

\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.

Created on July 24, 2017 at 15:58:58. See the history of this page for a list of all contributions to it.