closes #106

Author: ecpeterson

quillen.tex

@@ -795,13 +795,13 @@ \section{Power operations for complex bordism}\label{QuillenPowerOpnsSection}
 \begin{enumerate}
     \item The following Tate objects vanish: \((MU \sm S(\C^m)_+)^{tC_p} \simeq *\).  As in \Cref{TateDestruction}, this is because the Tate construction vanishes on free \(C_p\)--cells.
     \item We can use \(\colim_{m \to \infty} S(\C^m)_+\) as a model for \(EC_p\), so that \[MU_{hC_p} = \left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right)_{hC_p}.\]
-    \item Coupling these two facts together, we get \[MU_{hC_p} = \left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right)_{hC_p} = \left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right)^{hC_p}.\]
+    \item Coupling these two facts together, we get \[MU_{hC_p} = \left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right)_{hC_p} = \colim_{m \to \infty} \left( \left(MU \sm S(\C^m)_+ \right)^{hC_p} \right).\]
     \item Commuting finite limits through sequential colimits in order to pull the fixed points functor out, this gives
     \begin{align*}
     MU^{tC_p} & = \cofib(MU_{hC_p} \to MU^{hC_p}) \\
-    & = \cofib\left(\left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right)^{hC_p} \to MU^{hC_p}\right) \\
-    & = \left(\cofib\left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right) \to MU \sm S^0\right)^{hC_p} \\
-    & = \left( \colim_{m \to \infty} MU \sm S^{\C^m} \right)^{hC_p}.
+    & = \cofib\left(\colim_{m \to \infty} \left( \left(MU \sm S(\C^m)_+ \right)^{hC_p} \right) \to MU^{hC_p}\right) \\
+    % & = \left(\cofib\left( \colim_{m \to \infty} MU \sm S(\C^m)_+ \right) \to MU \sm S^0\right)^{hC_p} \\
+    & = \colim_{m \to \infty} \left( \left( MU \sm S^{\C^m} \right)^{hC_p} \right).
     \end{align*}
 \end{enumerate}
 This last formula puts us in a position to calculate.  The Thom isomorphism for \(MU\) gives an identification \(MU \sm S^{\C} \simeq \Susp^2 MU\) as \(C_p\)--spectra, and the map \[(MU \sm S^0)^{hC_p} \to (MU \sm S^{\C})^{hC_p}\] can be identified with multiplication by the Thom class: \[MU^{\Susp^\infty BC_p} \xrightarrow{t \cdot -} (\Susp^2 MU)^{\Susp^\infty BC_p}.\]  In all, this gives