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\section*{Snaith theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{ForComplexTopologicalKTheory}{For complex topological K-theory}\dotfill \pageref*{ForComplexTopologicalKTheory} \linebreak
\noindent\hyperlink{preliminaries_2}{Preliminaries}\dotfill \pageref*{preliminaries_2} \linebreak
\noindent\hyperlink{the_ring_spectrum_of_the_circle_2group}{The ring spectrum of the circle 2-group}\dotfill \pageref*{the_ring_spectrum_of_the_circle_2group} \linebreak
\noindent\hyperlink{MapToKTheorySpacetum}{The map to the K-theory spectrum}\dotfill \pageref*{MapToKTheorySpacetum} \linebreak
\noindent\hyperlink{isomorphy}{Isomorphy}\dotfill \pageref*{isomorphy} \linebreak
\noindent\hyperlink{for_periodic_complex_cobordism}{For periodic complex cobordism}\dotfill \pageref*{for_periodic_complex_cobordism} \linebreak
\noindent\hyperlink{for_algebraic_ktheory}{For algebraic K-theory}\dotfill \pageref*{for_algebraic_ktheory} \linebreak
\noindent\hyperlink{for_smooth_spectra_and_differential_ktheory}{For smooth spectra and differential K-theory}\dotfill \pageref*{for_smooth_spectra_and_differential_ktheory} \linebreak
\noindent\hyperlink{ForMoravaETheory}{Snaith-like theorem for Morava $E$-theories}\dotfill \pageref*{ForMoravaETheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The original theorem by Snaith (\hyperlink{Snaith79}{Snaith 79}) identifies the [[complex K-theory spectrum]] [[KU]] simply as the localization of the [[∞-group ∞-ring]] $\mathbb{S}[B U(1)]$ of the [[circle 2-group]] away from the [[Bott element]] $\beta$:

\begin{displaymath}
KU \simeq \mathbb{S}[B U(1)][\beta^{-1}]
  \,.
\end{displaymath}
Later more instances of such characterizations of familiar [[E-∞ rings]] have been given:

\begin{itemize}%
\item in (\hyperlink{GepnerSnaith08}{Gepner-Snaith 08}) the analogous statement for [[algebraic K-theory]] is formulated in terms of [[motivic spectra]];


\item in (\hyperlink{Westerland12}{Westerland 12}) the tower of [[Morava E-theories]] (or some [[homotopy fixed point]] [[spectra]] of these) is shown to be localizations of the [[∞-group ∞-rings]] of $B^{n+1} \mathbb{Z}_p$, for all $n \in \mathbb{N}$.



\end{itemize}
\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\hypertarget{preliminaries}{}\subsubsection*{{Preliminaries}}\label{preliminaries}

For $A$ an [[abelian ∞-group]] write $E \coloneqq \mathbb{S}[A] = \Sigma^\infty_+ A$ for its [[∞-group E-∞ ring]].

\begin{defn}
\label{}\hypertarget{}{}
For $\beta \in \pi_n(E)$ an element of the $n$th stable [[homotopy group]], then \emph{multiplication by $\beta$} is a homomorphism

\begin{displaymath}
\beta_\ast
  \;\colon\;
  E
  \to
  \Sigma^{-n} E
  \,.
\end{displaymath}
The \emph{localization} of $E$ at $\beta$ is the [[homotopy colimit]] over the iterated multiplication with $\beta$

\begin{displaymath}
E[\beta^{-1}]
  \coloneqq
  \underset{\to}{\lim}
  \left[
    E \stackrel{\beta_\ast}{\to}
    \Sigma^{-n}E
    \stackrel{\Sigma^{-n} \beta_\ast}{\to}
   \Sigma^{-2n} E
   \to
   \cdots
  \right]
\end{displaymath}
which has the [[universal property]] that $\mu_\beta$ becomes an [[equivalence]] on $E[\beta^{-1}]$.

\end{defn}
\hypertarget{ForComplexTopologicalKTheory}{}\subsubsection*{{For complex topological K-theory}}\label{ForComplexTopologicalKTheory}

The original formulation of Snaith's theorem (\hyperlink{Snaith70}{Snaith 79, theorem 2.12}, spring) for complex [[topological K-theory]].

\hypertarget{preliminaries_2}{}\paragraph*{{Preliminaries}}\label{preliminaries_2}

Write $(Ho(Top), \times, \ast)$ for the [[classical homotopy category]], regarded as a [[symmetric monoidal category]] under forming [[product spaces]], with [[tensor unit]] the [[point space]].

Write $(Ho(Top^{\ast/}), \wedge , S^0)$ for the homotopy category of [[pointed topological spaces]] with [[tensor product]] the [[smash product]] of pointed spaces and [[tensor unit]] the [[0-sphere]]

Write $(Ho(Spectra), \wedge, \mathbb{S})$ be the [[stable homotopy category]] with its [[symmetric monoidal smash product of spectra]] $\wedge$ whose [[tensor unit]] is the [[sphere spectrum]] $\mathbb{S}$.

For $A,B \in Ho(Spectra)$ two [[spectra]], we write

\begin{displaymath}
[A,B] \coloneqq Hom_{Ho(Spectra)}(A,B) \in Ab
\end{displaymath}
for [[hom-set]] in the [[stable homotopy category]] and write

\begin{displaymath}
[A,B]_\bullet \coloneqq [\Sigma^\bullet A, B]
\end{displaymath}
for the corresponding $\mathbb{Z}$-graded group (\href{Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory}{this def.}).

There are two pairs of ([[derived functor|derived]]) [[adjoint functors]]

\begin{displaymath}
Ho(Spectra)
    \underoverset
      {\underset{\Omega^\infty}{\longrightarrow}}
      {\overset{\Sigma^\infty}{\longleftarrow}}
      {}
  Ho(Top^{\ast/})
    \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{(-)_+}{\longleftarrow}}
      {\bot}
  Ho(Top)
\end{displaymath}
which are [[strong monoidal functors]] (by \href{pointed+object#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}{this example} and \href{Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsIndeedStabilizationOfClassicalHomotopyCategory}{this prop}). The left adjoint composite

\begin{displaymath}
\mathbb{S}[-]
    \;\colon
  \Sigma^\infty((-)_+)
\end{displaymath}
hence takes [[H-spaces]] and in particular [[H-groups]] $G$ to [[ring spectra]].

More in detail, an [[H-space]] structure is a $G \in Ho(Top)$ equipped with morphisms

$\mu \;\colon\; G \times G \longrightarrow G$

$e \;\colon\; \ast \longrightarrow G$

satisfying [[associativity]] and [[unitality]] in $Ho(Spectra)$, and the corresponding ring spectrum has product

\begin{displaymath}
\left(\Sigma^\infty(G_+)\right)
    \wedge
  \left( \Sigma^\infty G_+\right)
   \;\simeq\;
  \Sigma^\infty( (G \times G)_+ )
    \overset{\Sigma^\infty (e_+)}{\longrightarrow}
  \Sigma^\infty (G_+)
\end{displaymath}
and unit

\begin{displaymath}
\mathbb{S}
    \simeq
  \Sigma^\infty( \ast_+)
    \overset{ \Sigma^\infty(e_+) }{\longrightarrow}
  \Sigma^\infty(G_+)
\end{displaymath}
We call

\begin{displaymath}
\mathbb{S}[G] \coloneqq \Sigma^\infty(G_+)
\end{displaymath}
equipped with this monoid structure the \emph{[[H-group ring spectrum]]} of $G$. See \href{∞-group+∞-ring#HGroupRingSpectra}{there} for more.

\hypertarget{the_ring_spectrum_of_the_circle_2group}{}\paragraph*{{The ring spectrum of the circle 2-group}}\label{the_ring_spectrum_of_the_circle_2group}

One such [[H-group]] is the \emph{[[circle 2-group]]}, hence (the [[homotopy type]] of) the [[classifying space]] $B U(1)$ for [[complex line bundles]], equivalently the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$, canonically presented by the [[complex projective space]] $\mathbb{C}P^\infty$

\begin{displaymath}
K(\mathbb{Z},2) \simeq B U(1) \simeq \mathbb{C}P^\infty
  \;\in\;
  Ho(Spaces)
  \,.
\end{displaymath}
This being the [[classifying space]] for [[complex line bundles]], it becomes an [[H-group]] via the map

\begin{displaymath}
B U(1) \times B U(1) \longrightarrow B U(1)
\end{displaymath}
which classifies the [[tensor product of vector bundles|tensor product of line bundles]], with inverses given by the map

\begin{displaymath}
B U(1) \longrightarrow B U(1)
\end{displaymath}
which form [[dual vector bundle|dual line bundles]].

Hence its [[H-group ring spectrum]] is

\begin{displaymath}
\mathbb{S}[B U(1)]
   \;=\;
  \Sigma^\infty( B U(1)_+ )
  \,.
\end{displaymath}
Therefore for $X \in Ho(Top^{\ast/})$ a [[pointed topological space]], then

\begin{displaymath}
[\Sigma^\infty X,\Sigma^\infty(G_+)]_\bullet
\end{displaymath}
is a graded ring, with the product of elements

\begin{displaymath}
\left(
    \Sigma^{n_i} \Sigma^\infty X
     \overset{\alpha_i}{\longrightarrow}
   \Sigma^\infty(G_+)
  \right)
  \;\in\;
  [\Sigma^\infty X, \Sigma^\infty(G_+)]_{n_i}
\end{displaymath}
for $i \in \{1,2\}$ given by

\begin{displaymath}
\Sigma^{n_1 + n_2}
  \Sigma^\infty X
    \overset{\Sigma^{n_1+ n_2} \Sigma^\infty \Delta_X}{\longrightarrow}
  \Sigma^{n_1 + n_2}
  \Sigma^\infty (X \wedge X)
   \simeq
  \left(
    \Sigma^{n_1} \Sigma^\infty X
  \right)
    \wedge
  \left(
     \Sigma^{n_2} \Sigma^\infty X
  \right)
    \overset{ \alpha_1 \wedge \alpha_2 }{\longrightarrow}
  \left(
    \Sigma^\infty (G_+)
  \right)
    \wedge
  \left(
    \Sigma^\infty (G_+)
  \right)
    \overset{\Sigma^\infty (\mu_+)}{\longrightarrow}
  \Sigma^\infty (G_+)
  \,.
\end{displaymath}
Here the isomorphism on the left is the combination of the strong monoidalness of $\Sigma^\infty$ with the respect of suspension $\Sigma$ for the smash product of spectra (the [[tensor triangulated category]] structure on $Ho(Spectra)$, \href{Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory}{this prop.}).

Observe that we have a splitting

\begin{displaymath}
\Sigma^\infty(B U(1)_+)
   \;\simeq\;
  \left(
    \Sigma^\infty (B U(1))
  \right)
    \;\oplus\;
  \mathbb{S}
\end{displaymath}
(by \href{∞-group+∞-ring#HGroupRingSpectrumSplitsAsDirectSumWithSphereSpectrum}{this remark}) and hence a canonical morphism

\begin{displaymath}
\Sigma^\infty (B U(1))
    \longrightarrow
  \Sigma^\infty (B U(1)_+)
  \,.
\end{displaymath}
Via this splitting, the morphism in $Ho(Top^{\ast/})$

\begin{displaymath}
h \colon S^2 \longrightarrow B U(1)
\end{displaymath}
in $[S^2, B U(1)] \simeq \pi_2(B U(1)) \simeq \mathbb{Z}$ which classifies the [[basic complex line bundle on the 2-sphere]] and represents $1 \in \mathbb{Z}$, induces a morphism

\begin{displaymath}
\beta
    \;\colon\;
  \Sigma^2 \mathbb{S}
    \simeq
  \Sigma^\infty S^2
    \overset{ \Sigma^\infty( h ) }{\longrightarrow}
  \Sigma^\infty B U(1)
    \overset{}{\longrightarrow}
  \Sigma^\infty (B U(1)_+)
\end{displaymath}
hence an element in $[\mathbb{S}, \Sigma^\infty(B U(1)_+)]_2$.

\hypertarget{MapToKTheorySpacetum}{}\paragraph*{{The map to the K-theory spectrum}}\label{MapToKTheorySpacetum}

The [[spectrum]] [[KU]] which [[Brown representability theorem|represents]] complex [[topological K-theory]] has in degree 0 the the [[product space]]

\begin{displaymath}
\Omega^\infty KU \simeq B U \times \mathbb{Z} \in Ho(Top^{\ast/})
\end{displaymath}
of the stable [[classifying space]] $B U$ for [[complex vector bundles]] and the [[integers]]. The base point is $(\ast, 0)$.

The [[projection]]

\begin{displaymath}
B U \times \mathbb{Z} \longrightarrow \mathbb{Z}
\end{displaymath}
classifies the virtual [[rank of a vector bundle|rank]] of [[virtual vector bundle]].

Hence the inclusion of classifying spaces

\begin{displaymath}
B U \simeq B U \times \{0\} \hookrightarrow  B U \times \mathbb{Z}
\end{displaymath}
classifies the inclusion $\tilde K_{\mathbb{C}}(-) \hookrightarrow K_{\mathbb{C}}(-)$ of [[reduced K-theory]].

There is a canonical morphism

\begin{displaymath}
\Sigma^\infty(B U(1)_+)
    \overset{\Phi}{\longrightarrow}
  K U
\end{displaymath}
in $Ho(Spectra)$, being the $(\Sigma^\infty \dashv \Omega^\infty)$-[[adjunct]] of

\begin{displaymath}
B U(1)_+ \longrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U
\end{displaymath}
in $Ho(Top^{\ast/})$, which in turn is the $((-)_+ \dashv U)$-[[adjunct]] of the canonical

\begin{displaymath}
\mathcal{O}(1) \;\colon\; B U(1) \longrightarrow B U \simeq B U \times \{1\} \hookrightarrow B U \times \mathbb{Z}
\end{displaymath}
in $Ho(Top)$.

Since the [[formal group law]] for K-theory says that $\mu^\ast ( \mathcal{O}(1) ) \simeq pr_1^\ast (\mathcal{O}(1)) \otimes pr_2^\ast(\mathcal{O}(2))$ $\Phi$ is a homomoprhism of ring spectra.

Under the above splitting, the morphism $\Phi$ decomposes as

\begin{displaymath}
\Phi
    \;\colon\;
  \Sigma^\infty (B U(1)_+)
    \simeq
  \Sigma^\infty B U(1)  \oplus \mathbb{S}
    \overset{ ( \tilde i , e_{KU})  }{\longrightarrow}
  KU
\end{displaymath}
Since, by the above, morphisms $\Sigma^\infty B U(1) \longrightarrow K U$ in $Ho(Spectra)$, hence equivalently morphisms $B U(1) \longrightarrow B U \times \mathbb{Z}$ in $Ho(Top^{\ast/})$, hence equivalently morphisms $B U(1) \to B U$ in $Ho(Top)$ correspond to the reduced K-theory of $B U(1)$, and since morphisms $\mathbb{S} \to KU$ in $Ho(Spectra)$, hence equivalently morphism $\ast \to B U \times \mathbb{Z}$ in $Ho(Top)$ correspond to the K-theory of the point, and since over (colimits of) [[compact topological spaces]] K-theory splits as $K(X) \simeq \tilde K(X) \oplus K(\ast)$ via $[E]- n \mapsto ([E] - rk(E)) + ( rk(E) - n )$ (\href{topological+K-theory#KGrupDirectSummandReducedKGroup}{this prop}) it follows that

\begin{enumerate}%
\item $\tilde i$ takes the canonical line bundle $\mathcal{O}(1)$ on $B U(1)$ to its image $\mathcal{O}(1)-1$ in reduced K-theory

\begin{displaymath}
B U(1) \overset{\mathcal{O}(1) \mapsto (\mathcal{O}(1)-1)}{\longrightarrow} B U \simeq B U \times \{0\} \hookrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U
\end{displaymath}

\item $e_{K U}$ is adjunct to $\ast \simeq (\ast,1) \hookrightarrow B U \times \mathbb{Z}$ ( is the ring spectrum unit of $K U$).



\end{enumerate}
Now observe that $\Phi$ takes multiplication by $\beta$ to multiplication with the [[Bott element]] $(h-1)$:

This is because multiplication by $\beta$ is the outer right boundary of the following diagram

\begin{displaymath}
\itexarray{
     \Sigma^\infty S^2 \wedge \Sigma^n \Sigma^\infty X
     \\
     {}^{\mathllap{ (h \oplus 0) \wedge id }}\downarrow
     \\
     \left(\Sigma^\infty( B U(1) ) \oplus \mathbb{S}\right) \wedge \Sigma^n \Sigma^\infty X
     \\
     {}^{\mathllap{id \wedge \alpha}}\downarrow
     \\
     \Sigma^\infty(B U(1)_+) \wedge \Sigma^n \Sigma^\infty( B U(1)_+ )
       &\overset{\Sigma^n\Sigma^\infty (\mu_+)}{\longrightarrow}&
     \Sigma^n \Sigma^\infty (B U(1)_+)
     \\
     \downarrow && \downarrow
     \\
     K U \wedge \Sigma^n K U &\longrightarrow & \Sigma^n K U
  }
\end{displaymath}
and since the bottom square commutes (since tensor product of line bundles corresponds to their product in K-theory) this is equivalent to the left and bottom boundary, which, by the above discussion, is multiplication with the [[Bott element]] $(h-1)$.

Since the Bott element is invertible in $K U$, this means for all $X \in Ho(Top^\ast/)$ that the morphism

\begin{displaymath}
[\Sigma^\infty X , \Sigma^\infty ( B (1)_+ ) ]_\bullet
    \overset{\Phi_X \coloneqq \Phi \circ (-)}{\longrightarrow}
  [\Sigma^\infty X, K U]_\bullet
    \simeq
  \tilde K_{\mathbb{C}}^\bullet(X)
\end{displaymath}
extends to the quotient ring

\begin{displaymath}
[\Sigma^\infty X, \Sigma^\infty ( B U(1)_+ )]_{\bullet}[\beta^{-1}]
\end{displaymath}
in which two elements are identified if they differ by multiplication by $\beta$, as above:

\begin{displaymath}
\itexarray{
    [\Sigma^\infty X, \Sigma^\infty( B U(1)_+ )]  &\overset{\Phi}{\longrightarrow}& \tilde K_{\mathbb{C}}^\bullet(X)
    \\
    \downarrow & \nearrow_{\mathrlap{\exists \Phi}}
    \\
    [\Sigma^\infty X, \Sigma^\infty( B U(1)_+ )]_\bullet [\beta^{-1}]
  }
  \,.
\end{displaymath}
\hypertarget{isomorphy}{}\paragraph*{{Isomorphy}}\label{isomorphy}

\begin{theorem}
\label{}\hypertarget{}{}
If $X \in Ho(Top^{\ast/})$ has the homotopy type of a [[finite CW-complex]], then the [[natural transformation]]

\begin{displaymath}
\Phi_X
    \;\colon\;
  [X, \Sigma^\infty ( B U(1)_+ )]_\bullet [\beta^{-1}]
    \longrightarrow
  \tilde K^\bullet(X)
\end{displaymath}
is a [[natural isomorphism]].

\end{theorem}
(\hyperlink{Snaith81}{Snaith 81, theorem 2.12}, \hyperlink{HopkinsMathew}{Hopkins-Mathew})

That (before localization) the map is an epimorphism is due to (\hyperlink{Segal73}{Segal 73, prop. 1}), see \href{complex+projective+space#HGroupRingSpectrumSurjectsOntoTopologicalKTheory}{this prop.}. The analog of this statement for [[real projective space]] $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$ instead of [[complex projective space]] $\mathbb{C}P^\infty \simeq B U(1)$ is the [[Kahn-Priddy theorem]].

\hypertarget{for_periodic_complex_cobordism}{}\subsubsection*{{For periodic complex cobordism}}\label{for_periodic_complex_cobordism}

\begin{theorem}
\label{}\hypertarget{}{}
The [[periodic complex cobordism spectrum]] is the [[∞-group ∞-ring]] of the [[classifying space]] for stable [[complex vector bundles]] (the classifying space for [[topological K-theory]]) localized at the [[Bott element]] $\beta$:

\begin{displaymath}
PMU \simeq (\mathbb{S}[B U])[\beta^{-1}]
     \,.
\end{displaymath}
\end{theorem}
(\hyperlink{Snaith81}{Snaith 81, theorem 2.7})

\hypertarget{for_algebraic_ktheory}{}\subsubsection*{{For algebraic K-theory}}\label{for_algebraic_ktheory}

\begin{remark}
\label{}\hypertarget{}{}
The analog of this result for the periodic [[algebraic cobordism spectrum]] and [[algebraic K-theory]] as [[motivic spectra]] is discussed in (\hyperlink{GepnerSnaith08}{GepnerSnaith 08}).

\end{remark}
\hypertarget{for_smooth_spectra_and_differential_ktheory}{}\subsubsection*{{For smooth spectra and differential K-theory}}\label{for_smooth_spectra_and_differential_ktheory}

Refinement of the Snaith theorem for [[KU]] to [[smooth spectra]] and to [[differential K-theory]] is in (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13, section 6.3}). See at \emph{\href{differential%20cohomology%20diagram#ViaSmoothSnaithTheorem}{differential cohomology diagram -- Smooth Snaith K-theory}}.

\hypertarget{ForMoravaETheory}{}\subsubsection*{{Snaith-like theorem for Morava $E$-theories}}\label{ForMoravaETheory}

Write $\Gamma_n$ for the [[Honda formal group]]. The [[automorphism group]] $Aut(\Gamma_n)$ induces for each prime $p$ a canonical [[determinant]] morphism

\begin{displaymath}
det \;\colon\; \mathbb{G}_n \longrightarrow \mathbb{Z}_p^\times
\end{displaymath}
from the [[Morava stabilizer group]] $\mathbb{G}_n \coloneqq Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) \ltimes Aut(\Gamma_n)$.

Write

\begin{displaymath}
S \mathbb{G}_n \coloneqq ker(det)
\end{displaymath}
for the [[kernel]]. This naturally [[infinity-action|acts]] on the [[Morava E-theory]] spectrum $E_n$. Write $E^{S\mathbb{G}_n}$ for the corresponding [[homotopy fixed point]] [[spectrum]]. (\hyperlink{Westerland12}{Westerland 12, 1.1}).

Write $L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p]$ for the $K(n)$-localization of the [[∞-group ∞-ring]] of the [[n-group|(n+1)-group]] $B^n \mathbb{Z}_p$.

\begin{theorem}
\label{}\hypertarget{}{}
There is a [[generalized element]] $\rho_n$ of the [[E-∞ ring]] $L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p]$ such that localization at that element yields the [[Morava E-theory]] spectrum $S\mathbb{G}_n$[[homotopy fixed points]]:

\begin{displaymath}
L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p] [\rho_n^{-1}]

  \stackrel{\simeq}{\longrightarrow}
  E_n^{S\mathbb{G}_n}
  \,.
\end{displaymath}
\end{theorem}
(\hyperlink{Westerland12}{Westerland 12, theorem 1.2})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[splitting principle]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The theorem is due to

\begin{itemize}%
\item [[Victor Snaith]], \emph{Algebraic Cobordism and K-theory}, Mem. Amer. Math. Soc. no 221 (1979)

\end{itemize}
with a simpler proof given in

\begin{itemize}%
\item [[Victor Snaith]], \emph{Localized stable homotopy of some classifying spaces}, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 325-330. MR 600247 (82g:55006) (\href{https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100058205}{pdf})

\end{itemize}
using results from

\begin{itemize}%
\item [[Graeme Segal]], \emph{The stable homotopy of complex of projective space}, The quarterly journal of mathematics (1973) 24 (1): 1-5. ([[Segal72.pdf:file]], \href{https://doi.org/10.1093/qmath/24.1.1}{doi:10.1093/qmath/24.1.1})

\end{itemize}
Another proof due to [[Mike Hopkins]] is in

\begin{itemize}%
\item [[Akhil Mathew]] (following [[Mike Hopkins]]), \emph{Snaith's construction of complex K-theory} (\href{http://math.uchicago.edu/~amathew/snaith.pdf}{pdf})

\end{itemize}
Refinement to [[smooth spectra]] and [[differential K-theory]] is in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 6.3 of \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})

\end{itemize}
Discussion of the [[E-∞ ring]] structure involved is around theorem 3.1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology]]}

\end{itemize}
A version for [[motivic spectra]] [[algebraic K-theory]] is discussed in

\begin{itemize}%
\item [[David Gepner]], [[Victor Snaith]], \emph{On the motivic spectra representing algebraic cobordism and algebraic K-theory}, Documenta Math. 2008 (\href{http://arxiv.org/abs/0712.2817}{arXiv:0712.2817})

\end{itemize}
and for [[motivic cohomology]] in

\begin{itemize}%
\item [[Markus Spitzweck]], [[Paul Arne Østvær]], \emph{The Bott inverted infinite projective space is homotopy algebraic K-theory} (\href{http://folk.uio.no/paularne/bott.pdf}{pdf})

\end{itemize}
Higher [[chromatic homotopy theory|chromatic]] analogs for [[Morava E-theory]] are discussed in

\begin{itemize}%
\item [[Craig Westerland]], \emph{A higher chromatic analogue of the image of J} (\href{http://arxiv.org/abs/1210.2472}{arXiv:1210.2472})


\item [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], \emph{A higher categorical analogue of topological T-duality for sphere bundles} (\href{http://arxiv.org/abs/1601.06285}{arXiv:1601.06285})



\end{itemize}
A unifying [[general abstract]] perspective is discussed in

\begin{itemize}%
\item [[Peter Arndt]], section 3.2 of \emph{Abstract motivic homotopy theory}, thesis 2017 (\href{https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full}{web}, \href{https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf}{pdf}, [[ArndtAbstractMotivic.pdf:file]]) exposition: lecture at \emph{\href{www.andrew.cmu.edu/user/fwellen/abstracts.html}{Geometry in Modal HoTT}}, 2019 (\href{https://www.youtube.com/watch?v=f0wpcNs8hQo}{recording I}, \href{https://www.youtube.com/watch?v=sTl8637a2Zo}{recording II})



\end{itemize}
See also at \emph{[[spherical T-duality]]}.

[[!redirects Snaith's theorem]]



\end{document}