\section{Introduction}
Bredon motivic cohomology (introduced in \cite{HOV} and \cite{HOV1}) is a generalization of motivic cohomology to the setting of smooth varieties with finite group action.   
Part of a larger group of motivic 
$C _2$-invariants, Hermitian K-theory, and motivic real cobordism play an essential role in equivariant motivic homotopy theory. 
One distinguishing feature is that Bredon motivic cohomology appears as the zero slice of the equivariant motivic sphere \cite{zeroslice}.  
   
 
 
Bredon motivic cohomology is ready for concrete computations, which will be crucial for applying the theory to other motivic and topological invariants. In this paper, we compute the Bredon motivic cohomology groups with finite coefficients of the complex numbers and $\mathbf{E} C _2$ and determine the ring structures with $\mathds{Z}/2$ coefficients. Using the usual methods of \cite{Sus}, \cite{YO}, \cite{YP} we generalize these results to an algebraically closed field of characteristic zero. These results can be seen as a first step in understanding the largely unknown and difficult to compute 
 $H^{*,*}_{C_2}(k,\ul{\mathds{Z}/2})$ for an arbitrary field $k$ (for partial results in this direction see \cite{Voi}) as well as the 
 $C_2$-equivariant motivic Steenrod algebra of cohomology operations. 
  
 
 
 We describe the main computations.  
 Bredon motivic cohomology is graded by a $4$-tuple of integers, written as $(a+p\sigma, b+q\sigma)$; 
 this is viewed as a pair of $C_2$-representations (here $\sigma$ denotes the sign representation). 
 If $X$ is a complex variety, Betti realization induces a comparison homomorphism
 \[
 \Re:H^{a+p\sigma, b+q\sigma}_{C_2}(X, \mathds{Z}/n) \to H^{a+p\sigma}_{\mathrm{Br}}(X(\C), \ul{\mathds{Z}/n})
 \]
 between Bredon motivic cohomology and topological Bredon cohomology. When $X=\spec(\C)$, this induces an isomorphism of bigraded rings, see \aref{prop:ptiso},
 \[
 H^{\star, 0}_{C_2}(\C, \mathds{Z}/n) \xrightarrow{\cong}
 H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n}).
 \]
 In particular via this isomorphism, we view $H^{\star,\star}_{C_2}(X,\mathds{Z}/n)$ as a module or algebra over $H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n})$.
 
  \begin{theorem}
  	There is an isomorphism of $ H^\star_\mathrm{Br}(\mathrm{pt}, \ul{\mathds{Z}/n})$-algebras
  	\[
  	H^{\star,\star}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n)  {\cong} H^\star_\mathrm{Br}(\mathrm{pt}, \ul{\mathds{Z}/n})[u^{\pm 1}, \lambda],
  	\]
  	where $|u| = (2\sigma-2, \sigma -1)$ and 
  	$|\lambda| =  (0,\sigma)$.
  	Furthermore, when $b+q\geq 0$,
  	\[
  	\Re:  H^{a+p\sigma, b+q\sigma}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n) \to  H^{a+p\sigma}_{C_2}(E C_2, \ul{\mathds{Z}/n})
  	\]
  	is an isomorphism if $a\leq 2b$ and is zero otherwise. 
  \end{theorem}
  
 This theorem generalizes (and uses) Voevodsky's computation of the motivic cohomology ring of $\mathbf{B} C_2$ \cite{Voc} over the complex numbers, which agrees with Edidin-Graham's equivariant higher Chow groups \cite{EG}.  
 
 
In \aref{sec:BC}, we determine  all Bredon motivic cohomology groups of the complex numbers and their Betti realizations. 
The Betti realization is an isomorphism when the weight is an actual representation. 
However, it can differ when the weight is a virtual representation. 
 
  \begin{theorem}
 Let $n\geq 2$.
 	\begin{enumerate}
 		\item If $b\geq 0$ and $b+q\geq 0$ then Betti realization induces an isomorphism for any $a,p$
 		\[
 		\Re: H^{a+p\sigma,b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\xrightarrow{\cong}
 		H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/n}).
 		\] 
 		
 		\item If $b\geq 0$ and $b+q<0$ then 
 		$H^{\star, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong 
 		\widetilde{H}^{\star, b+q\sigma}(\widetilde{\EG}  C_2, \mathds{Z}/n)$. 
 		In particular
 		\[
 		H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong \begin{cases}
 			\mathds{Z}/2 & 2\leq a \leq 2b+1\textrm{ and $n$ is even}\\
 			0 & \textrm{else}.
 		\end{cases}
 		\]
 		Moreover, if $a \leq 2b+1$ and $a< -p$, then  
 		$\Re$ is an isomorphism, it is injective if $a=-p$, and is zero for all other values of $a,p$.
 		
 		\item If $b<0$ and $b+q\geq 0$ then 
 		$H^{\star, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong H^{\star, b+q\sigma}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n)$. 		  	
 		In particular,
 		\[ 
 		H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong 
 	
 	
 	
 	
 	
 	
 		\begin{cases}
 			H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/n}) & a\leq 2b \\
 			0 & a = 2b+1 \\
 			H^{a+2q+(p-2q)\sigma}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n}) & 2b+2\leq a.
 		\end{cases}
 		\]
 		
 		Moreover, $\Re$ is an isomorphism if $a\leq 2b$ or if $2\leq a$. It is multiplication by $2$ if $2b+2\leq a\leq 1$ where $a$ is even and $a+p=0$, and it is zero for all other values of $a,p$.  
 		
 		\item If $b<0$ and $b+q<0$, then $H^{\star, b+q\sigma}_{C_2}(\C, \mathds{Z}/n)=0$. 
 	\end{enumerate}
 \end{theorem}
 
The main technique in these computations is to utilize the cofiber sequence
$C_{2+}\to S^{0}\to S^{\sigma}$ and the motivic isotropy separation sequence
$\mathbf{E} C_{2+}\rightarrow S^0\rightarrow \widetilde{\EG}  C _2$. These allow us to leverage non-equivariant computations, namely the motivic cohomology of $\spec(\C)$ and of $\mathbf{B} C_2$. 

The ring structure is determined in \aref{thm:ptring}. 
\begin{theorem}
	There is an isomorphism of $H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$-algebras
	\[
 H_{C_2}^{\star,\star}(\C,\mathds{Z}/2) \cong	H^{\star}_{\mathrm{Br}}(\mathrm{pt},  \ul{\mathds{Z}/2})[u,\lambda, \mu]/(u\mu-\alpha^2)\oplus\left(\bigoplus_{\substack{m,n\geq 1 \\ 0\leq d\leq 2n-1}} \mathds{Z}/2\{\frac{\theta \mu^n}{\alpha^d\lambda^m}\}\otimes\mathds{Z}/2[x^{\pm 1}]\right).
	\]
	Here $|u| = (2\sigma-2, \sigma -1)$, $|\lambda|=(0,\sigma)$, $|\mu| = (0, 1-\sigma)$, $|\theta| = (2-2\sigma, 0)$, $|\alpha|= (\sigma -1,0)$, and $|x| = (\sigma,0)$.  The products in the righthand summand are trivial. See \aref{thm:ptring} for a description of the action of the lefthand summand on the righthand summand. 
\end{theorem}






A brief outline of the paper is as follows. Sections 1 and 2 are devoted to the introduction and preliminaries. The main computations of Bredon motivic cohomology are carried out in Sections 3 and 4.  In the last section,  we generalize the results to any algebraically closed field of characteristic zero via a rigidity result for Bredon motivic cohomology.
  
  {\bf{Notation}} 
\begin{itemize}[label = {$\cdot$}]
\item $H^{a+p\sigma,b+q\sigma} _{C _2}(X,A)$ is the Bredon motivic cohomology of a $C _2$-smooth scheme, with coefficients $A$. 
\item $H^{n,q}(X,A)$ is motivic cohomology of a smooth scheme $X$.
\item $H^{a+b\sigma}_{\mathrm{Br}}(X,\ul{A})$ the Bredon cohomology of a $C_2$-topological space $X$ with coefficients in the constant Mackey functor $\underline{A}$.
\item $H^a_{\mathrm{sing}}(X,\mathds{Z})$ the singular cohomology of a topological space $X$.
\item In general we use the notation $\star$ for a $C _2$-representation and $*$ for a natural number. So for example, $H^{\star,\star} _{C _2}(X,\mathds{Z})=\underset{a,b,p,q}{\oplus}  H^{a+p\sigma,b+q\sigma} _{C _2}(X,\mathds{Z})$ and $H^{*,*}(X,\mathds{Z})=\underset{a,b}{\oplus} H^{a,b}(X,\mathds{Z})$.
  \item  $S^\sigma$ the sphere associated to the sign representation $\sigma$.
\item  All $C _2$-varieties are over the field of complex numbers (unless stated otherwise) and we view $C_2$ as a group scheme by $C_2=\spec(\C)\sqcup \spec(\C)$. 
  \end{itemize}
   {\bf{Acknowledgement}} The authors wish to thank Institut Mittag-Leffler, Stockholm, where the research of this paper started in 2017 during the program on Algebro-Geometric and Homotopical Methods. Heller and {\O}stv{\ae}r thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program on K-theory, algebraic cycles and motivic homotopy theory in 2020.
   
   
   
   
   
   
   
     
   
 \section{Preliminaries}
 We record some background on Bredon motivic  cohomology. 

 
 \subsection{Equivariant motivic homotopy} 
 The stable equivariant motivic homotopy category $\mathrm{SH}^{C_2}(k)$ is the stabilization of Voevodsky's category of equivariant motivic spaces  \cite{Del}, with respect to Thom spaces of representations. We recall a few key facts we use in the case $G=C_2$. See \cite{Hoy}, \cite{HOV1}, or \cite{GH} for details. 
 
 
  
 Let $V=a+p\sigma$ be  a $C _2$-representation, where $a$ denotes the $a$-dimensional trivial representation and $p\sigma$ is the $p$-dimensional sign representation. 
 We write $\mathds{A}(V)$ and $\P(V)$ for the  $C_2$-schemes $\mathds{A}^{\dim(V)}$ and $\P^{\dim(V)}$ equipped with the corresponding action coming from $V$. The associated motivic representation sphere is 
 \[
 T^V:=\P(V\oplus 1)/\P(V).
 \] 
 The following spheres  play a role in our indexing
 $S^1$, $S^\sigma$, $S_t=(\mathds{A}^1\setminus\{0\},1)$, and $S^{\sigma}_t=(\mathds{A}^1\setminus\{0\},1)$ with the $C _2$-action $x\rightarrow x^{-1}$.  We write
 \[
 S^{a+p\sigma,b+q\sigma}:=S^{a-b}\wedge S^{(p-q)\sigma}\wedge S^b _t\wedge S^{q\sigma} _t.
 \]
 In this indexing, we have $T\simeq S^{2,1}$ and $T^\sigma\simeq S^{2\sigma, \sigma}$. The stable equivariant motivic homotopy category
 $\mathrm{SH}^{C_2}(k)$ is the stabilization of (based) $C_2$-motivic spaces with respect to the motivic sphere 
 $T^{\rho}$ corresponding to the regular representation $\rho= 1 +\sigma$. 
 
 We make use of two fundamental cofiber sequences in 
 $\mathrm{SH}^{C_2}(k)$. The first is
 \begin{equation}\label{eqn:cof1}
 C_{2+}\to S^{0}\to S^{\sigma}.
 \end{equation}
 The second is
 \begin{equation}\label{eqn:cof2}
 \mathbf{E} C_{2+}\rightarrow S^0\rightarrow \widetilde{\EG}  C _2.
 \end{equation}
 Here, $\mathbf{E} C_{2}$ is the universal free $C_2$-space, which can be expressed as a colimit $\mathbf{E} C _2\simeq \colim _n\mathds{A}(n\sigma)\setminus \{0\}$, see \cite[Section 3]{GH}. It follows that 
   $\widetilde{\EG}  C _2=\colim _n S^{2n\sigma,n\sigma}$.
 In particular, the maps $S^0\to T^\sigma$ and $S^0\to S^\sigma$ induce equivalences
 \[
 	\widetilde{\EG}  C_2 \xto{\simeq} T^{\sigma}\wedge\widetilde{\EG}  C_2\,\,\,\textrm{  and  }\,\,\,
 	\widetilde{\EG}  C_2 \xto{\simeq} S^{\sigma}\wedge\widetilde{\EG}  C_2,
 \]
  see \cite[Proposition 2.9]{HOV1}.
 
  
 Equipping a variety with trivial action $\mathrm{Sm}_{k}\to \mathrm{Sm}^{C_2}_k$ induces 
a functor $\mathrm{SH}(k)\to \mathrm{SH}^{C_2}(k)$.
 
 \subsection{Bredon motivic cohomology}


 Bredon motivic cohomology is represented in $\mathrm{SH}^{C_2}(k)$ by  
 the spectrum $M\ul{A}$ associated to an abelian group $A$, where 
 $M\underline{A} _n=A _{tr,C _2}(T^{n\rho})$ is the free presheaf with equivariant transfers, see \cite{HOV1} for details.


  \begin{definition}[\cite{HOV1}] The Bredon motivic cohomology of a motivic $C _2$-spectrum $E$ with coefficients in an abelian group $A$ is defined by 
  $$\widetilde{H}^{a+p\sigma,b+q\sigma} _{C _2}(E,A)=[E,S^{a+p\sigma,b+q\sigma}\wedge M\underline{A}] _{\mathrm{SH}^{C _2}(k)}.$$
  \end{definition}



If $X\in \mathrm{Sm}^{C_2}_k$ we typically write
\[
H^{a+p\sigma,b+q\sigma}_{C _2}(X, A) := \widetilde{H}^{a+p\sigma,b+q\sigma}_{C_2}(X_+, A).
\]
 When $A$ is a ring, then $H^{\star,\star} _{C _2}(X, A)$ is a graded commutative ring by \cite[Proposition 3.24]{HOV1}. Specifically this means that if $x\in H^{a+p\sigma, b+q\sigma} _{C _2}(X, A)$ and $y\in H^{c+s\sigma, d+t\sigma} _{C _2}(X, A)$, then
\[
x\cup y = (-1)^{ac+ps}y\cup x.
\]


If  $E$ is in the image of $\mathrm{SH}(k)\to \mathrm{SH}^{C_2}(k)$, i.e. it has ``trivial action", then
$\widetilde{H}^{a,b}_{C _2}(E, A) \cong \widetilde{H}^{a,b}(E, A)$ is ordinary motivic cohomology. 
If $E\in \mathrm{SH}^{C_2}(k)$, then $\widetilde{H}^{a+p\sigma, b+q\sigma}_{C_2}(C_{2+}\wedge E, A) \cong \widetilde{H}^{a+p, b+q}(E, A)$.


\subsection{Betti realization}



  The map of sites 
  $\mathrm{Sm} _\C^{C_2}\rightarrow {\rm Top}^{C_2}$, given by $X\rightarrow X(\C)$, where the set of complex points is equipped with the analytic topology,  extends to a functor
  $\Re:\mathrm{SH}^{C _2}(\C)\rightarrow \mathrm{SH}^{C _2}$ between the stable equivariant motivic homotopy category over $\C$ and the classical stable equivariant homotopy category. We refer to this functor as the Betti realization.
  
  
   The indexing of the spheres above was chosen to interact well with complex Betti realization; we have 
$\Re (S^{a+p\sigma,b+q\sigma})\simeq S^{a+p\sigma}$.  
  
  Betti realization of the Bredon motivic $C _2$-spectrum  represents  Bredon cohomology of a $C_2$-topological space, by \cite[Theorem A.29]{HOV1}. 
   In particular, for any smooth $C _2$-scheme over $\C$ there is a functor 
\[
\Re :H^{a+p\sigma,b+q\sigma} _{C _2}(X,A)\rightarrow H^{a+p\sigma}_{\mathrm{Br}}(X(\C),\underline{A}).
\] 
   
Using the Beilinson-Lichtenbaum theorem proved by Voevodsky and Rost \cite{Voev:miln},\cite{Voev:BK}, it is shown in \cite{HOV1} that Betti realization is an isomorphism in a suitable range, on Bredon cohomology of smooth schemes.  
 \begin{theorem}[\cite{HOV1}] \label{BLe} 
Let $X$ be a smooth complex $C _2$-variety and $A$ be a finite abelian group. The comparison map 
 \[
 \Re : H^{a+p\sigma,b+q\sigma}_{C _2}(X,A)\rightarrow H^{a+p\sigma}_{\mathrm{Br}}(X(\C),\ul{A})
 \]
 is an isomorphism if $a+p\leq b+q$ and $a\leq min\{b-q,b\}$ and a monomorphism if $a+p=b+q+1$ and $a\leq \min\{b-q,b\}+1$.
 \end{theorem}

In \aref{sec:BC}, we will see that a stronger result holds for $X=\spec(\C)$. For the moment, we note that in nonnegative integer weights, we always have an isomorphism for finite coefficients. 
In particular, Betti realization induces an isomorphism of rings $H^{\star,0}_{C _2}(\C,\mathds{Z}/n)\cong H^{\star} _{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n})$ and 
so $H^{\star,\star}_{C_2}(X,\mathds{Z}/n)$ is a module over $H^{\star}_{\mathrm{Br}}(\mathrm{pt},\mathds{Z}/n)$. 
In fact, by \cite{Voi},  Betti realization is an isomorphism in weight zero even with $\mathds{Z}$-coefficients.

\begin{proposition}\label{prop:ptiso}
Let $A$ be a finite abelian group and $b\geq 0$. Betti realization induces an isomorphism for any $a,p$
\[
H^{a+p\sigma, b}_{C_2}(\C,A)\xrightarrow{\cong} H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt},\ul{A}).
\]
\end{proposition}
\begin{proof}
If $b\geq 0$, then $H^{a,b}(\C, A)\to H^{a}_{\mathrm{sing}}(\C, A)$ is an isomorphism for all $a$. In particular the result holds for $p=0$.  Using the comparison long exact sequence \eqref{eqn:les1} and the five lemma, the result holds for all $p$ by induction. 
\end{proof}
   
   
  
  When $k=\C$, Betti realization takes the cofiber sequences \eqref{eqn:cof1} and \eqref{eqn:cof2}
  to the corresponding ones in $\mathrm{SH}^{C_2}$. 
  In particular, for any $\mathcal{X}$ in $\mathrm{SH}^{C_2}(\C)$ we have comparisons of long exact sequences
  \begin{equation}\label{eqn:les1}
  \begin{tikzcd}[column sep=small]
  \cdots \ar[r] & \widetilde{H}^{a+(p-1)\sigma, b +q\sigma}_{C_2}(\mathcal{X}, A)  \ar[r, "i^*"]\ar[d] & \widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\mathcal{X}, A) \ar[r, "p^*"]\ar[d] & \widetilde{H}^{a+p, b +q}_{}(\mathcal{X}, A) \ar[r, "\delta"]\ar[d] & \cdots \\
  \cdots \ar[r] & \widetilde{H}^{a+(p-1)\sigma}_{\mathrm{Br}}(\Re(\mathcal{X}),\ul A) \ar[r, "i^*"]& \widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\Re(\mathcal{X}),\ul A) \ar[r, "p^*"] &   \widetilde{H}^{a+p}_{\mathrm{sing}}(\Re(\mathcal{X}), A) \ar[r, "\delta"] & \cdots
  \end{tikzcd}
  \end{equation}
  and
  \begin{equation}\label{eqn:les2}
  \begin{tikzcd}[column sep=small]
  \cdots \ar[r] & \widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\widetilde{\EG}  C_2 \wedge \mathcal{X}, A) \ar[r]\ar[d] & \widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\mathcal{X}, A) \ar[r]\ar[d] & \widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\mathbf{E} C_{2+}\wedge\mathcal{X}, A) \ar[r]\ar[d] & \cdots 
  \\
  \cdots \ar[r] & \widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\wt\mathrm{E} C_2\wedge \Re(\mathcal{X}), \ul A) \ar[r]& \widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\Re(\mathcal{X}),\ul A) \ar[r] & \widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_{2+}\wedge \Re(\mathcal{X}),\ul A) \ar[r] & \cdots .
  \end{tikzcd}
  \end{equation} 





\subsection{Borel motivic cohomology}\label{sub:Borel}

We sometimes refer to the groups $\widetilde{H}^{a+p\sigma,b+q\sigma} _{C_2}(\mathcal{X}\wedge \mathbf{E} C_{2+},A)$
as the \emph{Borel motivic cohomology} of $\mathcal{X}$. 
The quotient  
$\mathbf{E} C _2/C _2\simeq \colim _n \left(\mathds{A}(n\sigma)\setminus \{0\}\right)/C _2$ is the geometric classifying space $\mathbf{B} C_2$ constructed by Morel-Voevodsky \cite{MVo} and Totaro \cite{T}. In particular, when $\mathcal{X}$ is a smooth $C_2$-scheme, the Borel motivic cohomology of $\mathcal{X}$ is  isomorphic in integer bidegrees to Edidin-Graham's equivariant higher Chow groups \cite{EG}, see \cite{HOV1}.  

 
From \cite[Proposition 3.16]{HOV1}, we have an isomorphism
   $H^{a,b} _{C _2}(\mathbf{E} C _2,A)\cong H^{a,b}(\mathbf{B} C _2,A)$. In case $k=\C$, this isomorphism fits into the commutative diagram.
   \[
   \begin{tikzcd}
   H^{a,b} _{C _2}(\mathbf{E} C _2,A)\ar[d,"\Re"']\ar[r,"\cong"] & H^{a,b}(\mathbf{B} C _2,A)\ar[d, "\Re"] \\
   H^{a} _{\mathrm{Br}}(\mathrm{E} C _2,\ul{A})\ar[r, "\cong"] & H^{a}_{\mathrm{sing}}({\rm B} C _2,A)
   \end{tikzcd}
   \]
    The motivic cohomology of $\mathbf{B} C_2$ was computed by Voevodsky.  
  \begin{theorem}[{\cite[Theorem 6.10]{Voc}}] \label{Voeth} For $k$ a perfect field with $char(k)\neq 2$ we have that $$H^{*,*}(\mathbf{B} C _2,\mathds{Z}/2)\cong H^{*,*}(k,\mathds{Z}/2)[f,d]/({f}^2=d\tau+\rho f)$$ with $f\in H^{1,1}(\mathbf{B} C _2,\mathds{Z}/2), d\in H^{2,1}(\mathbf{B} C _2,\mathds{Z}/2)$.
  \end{theorem}
 In the case $k=\C$, $\rho=0$ and so  this isomorphism becomes 
 \[
 H^{*,*}(\mathbf{B} C _2,\mathds{Z}/2)\cong \mathds{Z}/2[\tau][f,d]/({f}^2=d\tau).
 \]

\begin{lemma}\label{lema:leqb}
	Let $A$ be a finite abelian group. Betti realization
	\[
	\Re:H^{a,b}(\mathbf{B} C_2, A)\to H^{a}_{\mathrm{sing}}({\rm B} C_2, A)
	\]
	is an isomorphism if $a\leq 2b$. 
\end{lemma}
\begin{proof}
By \cite[Section 6]{Voc}, $\mathbf{B} C_{2+}$ sits in the cofiber sequence
\[
\mathbf{B} C_{2+}\to \P^{\infty}_+ \to {\rm Th}(\mcal{O}(-2)).
\]
The lemma follows  from the comparison of long exact sequences induced by this cofiber sequence, the five lemma, the Thom isomorphism, and that $\Re:H^{a,b}(\P^{\infty};A)\to H^{a}_{\mathrm{sing}}(\C P^{\infty}, A)$ is an isomorphism if $a\leq 2b$.
\end{proof}

 
 By \cite[Theorem 5.4]{HOV1}, the cohomology $H^{\star,\star}_{C_2}(\mathbf{E} C_2, A)$  is  $(2\sigma-2,\sigma-1)$-periodic. The periodicity is obtained by multiplication with an invertible element 
 $u\in H^{2\sigma-2,\sigma-1}(\mathbf{E} C_2,A)$. 
 
 
 
 For later use we record a computation of 
 $\widetilde{H}^{\star,\star} _{C _2}(\widetilde{\EG}  C _2,A)$. 
 
 \begin{proposition}\label{crt} 
 	The cohomology of $\widetilde{\EG}  C _2$ as an 
 	$H^{*, *}_{C_2}(\C,A)$-module is 
 	\[
 	\widetilde{H}^{\star,\star} _{C _2}(\widetilde{\EG}  C _2,A)\cong \widetilde{H}^{*,*}(\Sigma \mathbf{B} C _2,A)[x^{\pm 1},\lambda^{\pm 1}],
 	\]
 	 where $|x|= (\sigma,0)$ and $|\lambda|=(0,\sigma)$. 
 \end{proposition}
 \begin{proof} 
 	Since $\widetilde{\EG}  C_2\simeq S^{0,\sigma}\wedge \widetilde{\EG}  C_2$ and $\widetilde{\EG}  C_2\simeq S^{\sigma,0}\wedge \widetilde{\EG}  C_2$ we have isomorphisms
 	\begin{align*}
 	\widetilde{H}^{*+p\sigma,*+q\sigma}_{C _2}(\widetilde{\EG}  C _2,A)& \cong \widetilde{H}^{*+(p+1)\sigma,*+q\sigma} _{C _2}(\widetilde{\EG}  C _2,A) \textrm{ and} \\
 	\widetilde{H}^{*+p\sigma,*+q\sigma} _{C _2}(\widetilde{\EG}  C _2,A) & \cong \widetilde{H}^{*+p\sigma,*+(q+1)\sigma} _{C _2}(\widetilde{\EG}  C _2,A)
 	\end{align*}
 	given respectively by multiplication by elements 
 	$x\in \widetilde{H}^{\sigma, 0} _{C _2}(\widetilde{\EG}  C _2,A)$ and 
 	$\lambda\in \widetilde{H}^{0,\sigma} _{C _2}(\widetilde{\EG}  C _2,A)$.
 	It thus suffices to show that 
 	$\widetilde{H}^{*,*}_{C_2}(\widetilde{\EG}  C_2, A)\cong \widetilde{H}^{*,*}(\Sigma^1 \mathbf{B} C _2,A)$, 
 	The result then follows from the long exact sequence induced by \eqref{eqn:cof2}. Indeed, by 
 	\cite[Proposition 3.16]{HOV1} the map  
 	$H^{*,*}_{C_2}(k,A)\to H^{*,*}_{C_2}(\mathbf{E} C_2,A)$ is isomorphic to the split monomorphism
 	$p^*:H^{*,*}_{C_2}(k,A)\to H^{*,*}_{C_2}(\mathbf{B} C_2,A)$. Thus $\widetilde{H}^{*+1,*}_{C_2}(\widetilde{\EG}  C_2,A)$ is isomorphic to the cokernel of $p^*$, which is $\widetilde{H}^{*+1,*}(\Sigma \mathbf{B} C_2,A)$. 
 	

 \end{proof}


 
 
 
    



 \subsection{Topological Bredon cohomology}\label{sec:tbc}
 
The Bredon cohomology ring of a point with $\mathbb{Z}/2$ coefficients was originally computed by Stong in unpublished work. The computation appears in \cite{Car:op} and with $\mathds{Z}$ coefficients in \cite{D}, a recent discussion of these computations appears in \cite{Greenlees:4}. 
This ring with $\mathds{Z}/2$  coefficients is displayed below in \aref{point2}. 





\begin{theorem}\label{crpoint} 
	\,
	\begin{enumerate}
	\item There is an additive decomposition of $H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$ as the direct sum: 
	\[
	H^{\star}_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/2})\cong
	\mathds{Z}/2[\sigma, \alpha]\oplus \bigoplus_{n,m\geq 0}\mathds{Z}/2\left\{\frac{\theta}{\alpha^n\sigma^m}\right\}
	\]
	where $|\sigma| = \sigma$, $|\alpha| = \sigma -1$, and $|\theta| = 2-2\sigma$. 
	\item 
	 $H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$ is a strictly commutative ring whose product structure is determined as follows: $\mathds{Z}/2[\alpha,\sigma]$ is the polynomial ring, all products between the elements  $\frac{\theta}{\alpha^n\sigma^m}$ are zero, and the remaining products are determined by
	\begin{align*}
		\sigma^i\smash\cdot\frac{\theta}{\alpha^n\sigma^m}
		& = \begin{cases}
			\displaystyle{\frac{\theta}{\alpha^n\sigma^{m-i}}} & i\leq m \vspace{1mm} \\
			0 & \textrm{else}
		\end{cases} \\
		\alpha^i \smash\cdot \frac{\theta}{\alpha^n\sigma^m} & =
		\begin{cases}
			\displaystyle{\frac{\theta}{\alpha^{n-i}\sigma^{m}} } & i\leq n 
			\vspace{1mm}
			\\
			0 & \textrm{else}.
		\end{cases}
	\end{align*}
\end{enumerate}
\end{theorem}
 
 
 
 \begin{remark}\label{rem:peg}
 	\begin{enumerate}
 		\item This ring can be written  as
 		\[
 		H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2}) \cong \mathds{Z}/2[\sigma, \alpha] \oplus \Sigma^{2-2\sigma}\mathds{Z}/2[\sigma^{-1}, \alpha^{-1}],
 		\]
 		with the multiplication between summands as described above. 
 		\item The Bredon cohomology ring of $\mathrm{E} C_2$ is
 		\[
 		H^\star_{\mathrm{Br}}(\mathrm{E} C_2,\ul{\mathds{Z}/2}) \cong \mathds{Z}/2[\sigma, \alpha^{\pm 1}]
 		\]
 		where $|\alpha| = \sigma -1$ and $|\sigma| = \sigma$, see e.g., \cite[Lemma 27]{Car:op}. 
 		
 		\item The Bredon cohomology of $\wt{\mathrm{E}}C_2$ is
 		\[
 		H^\star_{\mathrm{Br}}(\wt{\mathrm{E} }C_2,\ul{\mathds{Z}/2}) \cong 
 		\oplus_{n\geq 0}\mathds{Z}/2\left\{\frac{\theta}{\alpha^n} \right\}\otimes \mathds{Z}/2[x^{\pm 1}]
 		\]
 		where  $|x|=\sigma$ see e.g., 
 		\cite[Lemma 28]{Car:op}. The $H^{\star}_{\Br}(\pt, \ul{\Z/2})$-module structure is given by 
 		\begin{align*}
 			\sigma \smash\cdot \frac{\theta}{\alpha^n}x^i & = \frac{\theta}{\alpha^n}x^{i+1} 
 			\vspace{1mm}
 			 \\
 			\alpha\smash\cdot \frac{\theta}{\alpha^n}x^i & = 
			\begin{cases}
 				\displaystyle{\frac{\theta}{\alpha^{n-1}}x^i } & n\geq  1 \vspace{1mm} \\
 					0 & n=0
 			\end{cases} \\
 		\frac{\theta}{\alpha^s\sigma^t}\smash\cdot\frac{\theta}{\alpha^n}x^i & = 0.
 		\end{align*}
 		\item  The ring map $H^{\star}_{\Br}(\pt, \ul{\Z/2})\to H^{\star}_{\mathrm{Br}}(\mathrm{E} C_2,\ul{\mathds{Z}/2})$ is identified with the map 
 		\[
 		\mathds{Z}/2[\sigma,\alpha]\oplus\bigoplus_{m,n\geq 0} \mathds{Z}/2\left\{\frac{\theta}{\alpha^n\sigma^m}\right\} \to 
 		\mathds{Z}/2[\sigma,\alpha^{\pm 1}]. 
 		\]
 		which sends $\alpha\mapsto \alpha$, $\sigma\mapsto \sigma$ and $\displaystyle{\frac{\theta}{\alpha^n\sigma^m} }\mapsto 0$. 
 	\end{enumerate}
 \end{remark}
 
 
The Bredon cohomology ring with $\mathds{Z}/2$-coefficients is depicted below. 



\begin{figure}[H]
	\begin{tikzpicture}[scale=0.8]
		\path[style=myfill] (0,0) -- (-4.0,4.0) -- (0,4.0) -- (0,0) ;
		\path[style=myfill] (2,-2) -- (2,-4.0) -- (4.0,-4.0) -- (2,-2) ;
		\draw[style=mygrid] (-4.0,-4.0) grid (4.0,4.0);
		\draw[gray] (4.5,0) node[below, black] {\small $a$};
		\draw[gray] (0,4.5) node[left, black] {\small $p$};
		\foreach \x in {-4,...,-1,1,2,...,4}
		\draw [font=\tiny, gray] (\x cm,2pt) -- (\x cm,-2pt) node[anchor=north] {$\x$};
		\foreach \y in {-4,...,-1,1,2,...,4}
		\draw [font=\tiny, gray] (2pt,\y cm) -- (-2pt,\y cm) node[anchor=east] {$\y$};
		
		\foreach \y in {0,...,4}
		\fill (0,\y) circle(2pt);
		\foreach \y in {1,...,4}
		\fill (-1,\y) circle(2pt);
		\foreach \y in {2,...,4}
		\fill (-2,\y) circle(2pt);
		\foreach \y in {3,...,4}
		\fill (-3,\y) circle(2pt);
		\foreach \y in {4,...,4}
		\fill (-4,\y) circle(2pt);
		
		\foreach \y in {0,...,2}
		\fill (2,-\y-2) circle(2pt);
		\foreach \y in {1,...,2}
		\fill (3,-\y-2) circle(2pt);
		\foreach \y in {2,...,2}
		\fill (4,-\y-2) circle(2pt);

		\draw (2,-4) node[left]{\small $\frac{\theta}{\sigma^2}$};
		\draw (4.1,-4) node[below, right]{\small $\frac{\theta}{\alpha^2}$};

		\draw(-2.05,1.95) node[below, left]{\small $\alpha^2$};
		\draw(-3.05,2.95) node[below, left]{\small $\alpha^3$};
		\draw(-4.05,3.95) node[below, left]{\small $\alpha^4$};
		\draw (0,2) node[right]{\small $\sigma^2$};
		\draw (0,3) node[right]{\small $\sigma^3$};
		\draw (0,4) node[right]{\small $\sigma^4$};
		\draw (-1,1) node[left] {\small $\alpha$};
		\draw (0,1) node[right] {\small $\sigma$};
		\draw (2.1,-2) node[right] {\small $\theta$};
		\draw (3.1,-3) node[right] {\small $\frac{\theta}{\alpha}$};
		\draw (2.1,-3) node[left] {\small $\frac{\theta}{\sigma}$};
		
		\draw[dashed, red] (2, -2) -- (4,-4);
		\draw[dashed, red] (0, 0) -- (-4,4);
		\draw[dashed, red] (0, 1) -- (-3,4);
		\draw[dashed, red] (0, 2) -- (-2,4);
		\draw[dashed, red] (0, 3) -- (-1,4);
		\draw[dashed, red] (2, -3) -- (3,-4);
		
		\draw[blue] (0,0)  to  (0,4);
		\draw[blue] (-1,1)  to  (-1,4);
		\draw[blue] (-2,2)  to  (-2,4); 
		\draw[blue] (-3,3)  to  (-3,4);
		\draw[blue] (2,-2)  to  (2,-4);
		\draw[blue] (3,-3)  to  (3,-4);
		\draw (-4, -5) node[anchor = west] {\smash{$\bullet$ denotes the group $\mathds{Z}/2$} } ;
	\end{tikzpicture} \hfill 
	\caption{$H^{a+p\sigma} _{\mathrm{Br}}(\mathrm{pt};\ul{\mathds{Z}/2})$,  dashed lines indicate multiplication by $\alpha$ and solid lines indicate multiplication by $\sigma$.}
	\label{point2}
\end{figure}


With $\mathds{Z}$-coefficients (see \cite{D})
we have
\[
H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}})\cong 
\begin{cases}
	\mathds{Z} & a=-p\textrm{, $a$ even} \\
	\mathds{Z}/2 & p<a\textrm{, $a$ even and $a\leq 0$} \\
	\mathds{Z}/2 & a\leq -p\textrm{, $a$ odd and $3\leq a$} \\
	0 & \textrm{else}.
\end{cases}
\]
The ring structure is succinctly described as 
\[
H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \mathds{Z}) \cong \mathds{Z}[\sigma]/(2\sigma)[y] \oplus y^{-1}\smash\cdot\left(\left(2\mathds{Z}\oplus \Sigma^{1-\sigma}\mathds{Z}/2[\sigma^{-1}]\right)[y^{-1}]\right),
\]
where $|y|=2\sigma -2$ and $|\sigma|= \sigma$. Multiplication is as suggested by the notation. In particular, if we write $\theta$ for the generator in degree $2-2\sigma$, then $y\cup \theta =2$.  We are interested mostly in the case of finite coefficients.





 \subsection{Vanishing of  Bredon motivic cohomology}

An important feature of motivic cohomology is its vanishing regions. If $X\in \mathrm{Sm}_k$  then 
$H^{a,b}(X,\mathds{Z}/n) = 0$ in any of the following cases
\begin{enumerate}
	\item $a>2b$, 
	\item $a> b+\dim(X)$, or
	\item $b<0$.
\end{enumerate}

The vanishing regions for $H^{\star,\star}_{C_2}$ are more complicated. 





\begin{proposition} \label{van} 
	Let $X\in \mathrm{Sm}^{C_2}_k$ and suppose that $b+q<0$ and $b<0$.  Then
	\[
	H^{\star,b+q\sigma}_{C_2}(X,\mathds{Z}/n)=0.
	\]
\end{proposition}
\begin{proof} 
	Since $b<0$, we have  
	$\widetilde{H}^{a+p\sigma,b+q\sigma}(\widetilde{\EG}  C _2\wedge X_+)\cong \widetilde{H}^{a,b}(\widetilde{\EG}  C _2\wedge X_+)=0$, see 
	\cite[Lemma 4.2]{HOV1}.  
	Using the cofiber sequence \eqref{eqn:cof2}, we 
	  obtain 
	  \[
	H^{a+p\sigma,b+q\sigma}(X)\xto{\cong} 
	H^{a+p\sigma, b+q\sigma}(\mathbf{E} C _2\times X).
	  	\]
	  	 Since $H^{a+p\sigma, b+q\sigma}(\mathbf{E} C _2\times X)\cong H^{a+2q+(p-2q)\sigma, b+q}(\mathbf{E} C _2\times X)$, it suffices to see that  
	  	 $H^{a+p\sigma, n}(\mathbf{E} C _2\times X)=0$ for $n<0$. This follows from the case $p=0$, by induction using \eqref{eqn:cof1}. 
	  	 

\end{proof}

\begin{proposition} \label{tra}
	If $a\geq 2b+2$ then for any $X\in \mathrm{Sm}^{C_2}_k$,
	\begin{enumerate}
		\item $H^{a+p\sigma,b+q\sigma}_{C _2}(X,\mathds{Z}/n)\cong H^{a+p\sigma,b+q\sigma} _{C _2}(X\times \mathbf{E} C _2,\mathds{Z}/n)$
		\item $\widetilde{H}^{a+p\sigma,b+q\sigma} _{C _2}(X_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)=0$.
	\end{enumerate}
\end{proposition}
\begin{proof} 
	The two statements are equivalent by \eqref{eqn:cof2}. Therefore, we will establish the second one. 
	
Since	
$\widetilde{H}^{a+p\sigma,b+q\sigma} _{C _2}(X_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n) \cong \widetilde{H}^{a,b} _{C _2}(X_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)$, we can assume that $p=q=0$.  We can assume that $X$ has trivial action, since $\widetilde{H}^{a,b} _{C _2}(X^{C_2}_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)\cong \widetilde{H}^{a,b} _{C _2}(X_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)$ by \cite[Proposition 4.10]{GH}.


Consider the exact sequence from \eqref{eqn:cof2}.
	\[
	\cdots \to H^{a-1,b} _{C _2}(X\times \mathbf{E} C _2,\mathds{Z}/n) \rightarrow \widetilde{H}^{a,b} _{C _2}(X_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)\rightarrow
	 H^{a,b} _{C _2}(X,\mathds{Z}/n)\rightarrow H^{a,b} _{C _2}(X\times \mathbf{E} C _2,\mathds{Z}/n)\to\cdots .
	 \]
	Now $H^{a,b} _{C _2}(X\times \mathbf{E} C_2,\mathds{Z}/n)\cong H^{a,b}(X\times_{C_2}\mathbf{E} C_2,\mathds{Z}/n)$, 
	by \cite[Proposition 3.16]{HOV1}. If $a>2b$ this last group is zero and the proposition follows. 
	
\end{proof}
\begin{proposition} \label{1van} 
	For any $X\in \mathrm{Sm}^{C_2}_k$, if 
	$a\geq 2b+2$ and $p\geq 2q$ then we have that  
	\[
	H^{a+p\sigma,b+q\sigma}_{C_2}(X,\mathds{Z}/n)=0.
	\] 	
\end{proposition}
\begin{proof} 
  By \aref{tra}, it suffices to show that 
  $H^{a+p\sigma,b+q\sigma}_{C_2}(X\times \mathbf{E} C_2,\mathds{Z}/n)=0$.
  By periodicity of Borel motivic cohomology we have  
  \[
  H^{a+p\sigma,b+q\sigma}_{C_2}(X\times \mathbf{E} C_2)
  =H^{a+2q+(p-2q)\sigma,b+q} _{C _2}(X\times \mathbf{E} C _2).
  \]
  If $p=2q$,  then $H^{a+2q+(p-2q)\sigma,b+q} _{C _2}(X\times \mathbf{E} C _2) \cong H^{a+2q,b+q} (X\times_{C_2} \mathbf{E} C _2)$. This group vanishes when $a>2b$. To conclude the proposition, we use 
  the long exact sequence obtained from \eqref{eqn:cof1} and induction on $p\geq 2q$.
  

	
	     
\end{proof}







The following example shows that the vanishing range in the previous proposition is the best possible in general.

\begin{example} 
	For any $p$, we have 
	\[
	H^{2+p\sigma, 1-2\sigma}_{C_2}(\P^1, \mathds{Z}/n)\neq 0.
	\]
	To see this, we first note that 
	$H^{\star,1-2\sigma}(X\times \mathbf{E} C _2,\mathds{Z}/n)=0$ (see the proof of \aref{neq}) and therefore from \eqref{eqn:cof2} we see that
	$\widetilde{H}^{2+p\sigma,1-2\sigma}(\P^1_+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)\cong H^{2+p\sigma,1-2\sigma}(\P^1,\mathds{Z}/n)$. Since $\P^1_+\simeq T\vee S^0$, we thus have
	\[
	\widetilde{H}^{2+p\sigma,1-2\sigma}(\P^1 _+\wedge \widetilde{\EG}  C _2,\mathds{Z}/n)\simeq  \widetilde{H}^{1,1}(\widetilde{\EG}  C _2,\mathds{Z}/n)\oplus \widetilde{H}^{2,1}(\widetilde{\EG}  C _2,\mathds{Z}/n).
	\]
	This group is nonzero by \aref{crt}.
	

\end{example}  













\section{Bredon motivic cohomology of \texorpdfstring{$\mathbf{E} C_2$}{EC2}}\label{5}
In this section, we compute the Bredon motivic cohomology ring of $\mathbf{E} C_2$. 



\begin{proposition}\label{Bocal}
Suppose $b+q\geq 0$. Then
\[
H^{a+p\sigma,b+q\sigma}_{C_2}(\mathbf{E} C _2, \mathds{Z}/n)
\cong 
\begin{cases}
	H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_2,\ul{\mathds{Z}/n}) & a\leq 2b \\
	0 & a = 2b+1 \\
	H^{a+2q+(p-2q)\sigma}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n}) & 2b+2\leq a.
\end{cases}
\]
Furthermore, Betti realization is an isomorphism if $a\leq 2b$. It is   multiplication by  $2$ if $2b+2\leq a$, $p=-a$, and $a$ is even. All other Betti realizations are zero. 



\end{proposition}
\begin{proof} 
		Since $H^{\star, \star}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n)$ is $(2-2\sigma, 1-\sigma)$ periodic and the statement of the proposition is compatible with this periodicity,
		it suffices to treat the case $q=0$. We now assume that $q=0$.
		
		When  $p=0$, then $H^{a,b}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n)\cong H^{a,b}(\mathbf{B} C_2, \mathds{Z}/n)=0$ if $a>2b$ and 
		by \aref{lema:leqb}, Betti realization is an isomorphism $ a\leq 2b$. 
	
	
	
	
We suppress the coefficient group for the remainder of the proof for typographical simplicity and proceed by induction on $p$. 
To begin with, we use the comparison of exact sequences \eqref{eqn:les1}
	\[
	\begin{tikzcd}[column sep=small]
	\cdots \ar[r] & H^{i+p, b}_{}(\C) \ar[r]\ar[d] & H^{i+p\sigma, b }_{C_2}(\mathbf{E} C_2) \ar[r]\ar[d] & H^{i+(p+1)\sigma, b }_{C_2}(\mathbf{E} C_2) \ar[r]\ar[d] & H^{i+1+p, b}_{}(\C) \ar[r]\ar[d]& \cdots 
	\\
	\cdots \ar[r] &H^{i+p}_{\mathrm{sing}}(\mathrm{pt}) \ar[r]& H^{i+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_2) \ar[r] & H^{i+(p+1)\sigma}_{\mathrm{Br}}(\mathrm{E} C_2) \ar[r] & H^{i+1+p}_{\mathrm{sing}}(\mathrm{pt}) \ar[r]& \cdots .
	\end{tikzcd}
	\]
	A straightforward induction shows that for $p\geq 0$, Betti realization 
	$H^{i+p\sigma, b}_{C_2}(\mathbf{E} C_2)\to H^{i+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)$ 
	is an isomorphism if $i\leq 2b$ and 
	$H^{i+p\sigma, b}_{C_2}(\mathbf{E} C_2)=0$ if $i>2b$. This  establishes the result in case $p\geq 0$. 
	
	Now we establish the result for $p\leq 0$.
	  Using the comparison of exact sequences \eqref{eqn:les1} and the five lemma, we find that if the map
	   $H^{i+n\sigma, b}_{C_2}(\mathbf{E} C_2)\to H^{i+n\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)$ is an isomorphism for all $i\leq 2b$ when $n=p+1$, then this map is also an isomorphism for all $i\leq 2b$ when $n=p$. By downward induction on $p$, starting with $p=0$, we deduce the computation for $a\leq 2b$. 
	
	Now assume that $H^{2b+1+n\sigma, b}(\mathbf{E} C_2)=0$ for $n=p+1$. If $p\geq -(2b+1)$, if follows from the exact sequence induced by \eqref{eqn:cof1} that this group vanishes for $n=p$ as well. Thus downward induction  implies the result for $p<-(2b+1)$ once we treat the case $p=-(2b+1)$. Consider the comparison of exact sequences
	\[
	\begin{tikzcd}[column sep=small]
	H^{2b-2b\sigma}_{C_2}(\mathbf{E} C_2) \ar[r,"\phi"]\ar[d, "\cong"'] & H^{0, b}_{}(\C) \ar[r]\ar[d, "\cong"] & H^{2b+1-(2b+1)\sigma, b }_{C_2}(\mathbf{E} C_2) \ar[r]\ar[d] & H^{2b+1 -2b\sigma, b }_{C_2}(\mathbf{E} C_2) = 0 \ar[d]   
	\\
{H^{2b-2b\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)} \ar[r, "\cong"] & {H^{0}_{\mathrm{sing}}(\mathrm{pt})} \ar[r]& {H^{2b+1-(2b+1)\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)} \ar[r] & {H^{2b+1 -2b\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)}. 
	\end{tikzcd}
	\]
	That the bottom left horizontal arrow is an isomorphism can be seen by noting that this map can be identified with the restriction to the fiber homomorphism $H^{2b}_{\mathrm{sing}}({\rm Th}(\gamma), \mathds{Z}/n)\to H^{2b}_{\mathrm{sing}}(S^{2b},\mathds{Z}/n)$, where $\gamma$ is the vector bundle on 
$\mathrm{B} C_2$ determined by the $b$-dimensional complex sign representation. This map is an isomorphism because  $\gamma$ is orientable.
	It follows that the map labeled $\phi$ is an isomorphism and so $H^{2b+1-(2b+1)\sigma, b}_{C_2}(\mathbf{E} C_2) =0$. 
	
	
	If $a\geq 2b+2$, then $H^{a+p\sigma, b}_{C_2}(\C)\cong H^{a+p\sigma, b}_{C_2}(\mathbf{E} C_2)$, since $H^{a+p\sigma, b}(\wt\mathbf{E} C_2)\cong 
	\widetilde{H}^{a, b}(\Sigma \mathbf{B} C_2)=0$ for $a\geq 2b+2$. The case $a\geq 2b+2$ thus follows from \aref{prop:ptiso}. 
	
	
For the last statement about Betti realization, we have already checked that it is an isomorphism if $a\leq 2b$. The remaining part of the statement follows from the commutative diagram, where $2b+2\leq a$ 
	\[
	\begin{tikzcd}
		H^{a+p\sigma, b}_{C_2}(\C) \ar[r, "\cong"]\ar[d, "\cong"'] & H^{a+p\sigma, b }_{C_2}(\mathbf{E} C_2) \ar[d] \\
		{H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt})} \ar[r, "\smash\cdot 2"]& 
		{H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)}.
	\end{tikzcd}
	\]
To see that the bottom arrow is multiplication by $2$, note that for $2\leq a$,
$H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}})\cong H_{-a-p\sigma}^{\mathrm{Br}}(\mathbf{E} C_2,\ul{\mathds{Z}})$, see e.g.,~\cite{Greenlees:4} for details, and under this identification the lower arrow is induced by the norm map $H\mathds{Z}_{hC_2}\to H\mathds{Z}^{hC_2}$.
 

	
 \end{proof}



\begin{proposition} \label{neq} If $b+q<0$ then 
	$H^{\star,
		b+q\sigma} _{C _2}(\mathbf{E} C _2, \mathds{Z}/n) = 0$.
\end{proposition}
\begin{proof} 
	From  $(2\sigma-2,\sigma-1)$ periodicity we have that $H^{a+p\sigma,b+q\sigma} _{C _2}(\mathbf{E} C _2,\mathds{Z}/n)\cong H^{a+2q+(p-2q)\sigma,b+q}_{C _2}(\mathbf{E} C _2, \mathds{Z}/n)$. 
	Using  the vanishing $H^{a+2q,b+q}_{C_2}(\mathbf{E} C _2, \mathds{Z}/n) \cong H^{a+2q,b+q}(\mathbf{B} C _2, \mathds{Z}/n) =0$ together with the exact sequence induced by \eqref{eqn:cof1}, the result follows by induction. 
\end{proof}
 

The groups $H^{a+p\sigma, b}_{C_2}(\mathbf{E} C_2,\mathds{Z}/2)$, for different values of $p$, $q$, are displayed in \aref{Borelne} below.






\begin{figure}[H]
	
\begin{tikzpicture}[scale=0.6]
		
		\draw[gray] (-5,0) -- (7.5,0) node[below, black] {\small $a$};
		\draw[gray] (0,-5.1) -- (0,4.4) node[left, black] {\small $b$};
	
		
		
		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
		\draw [font=\tiny, gray] (5 cm,2pt) -- (5 cm,-2pt) node[anchor=north east] {$-p$};
		\draw [font=\tiny,gray] (2 cm,2pt) -- (2 cm,-2pt) node[anchor=north] {$2$};
		\draw [font=\tiny, gray] (2pt,-4cm) -- (-2pt,-4cm) node[below left] {$-q$};
		\draw [font=\tiny,style = myline] (-5,-2.5)--(8,4);
		\draw [font=\tiny,gray](7,3.5) node[below = 1pt] {$a=2b$};
		\draw [font=\tiny,style = myline] (-5,-3.5)--(8,3); 
		\draw [font=\tiny,gray] (7,1.8) node {$a=2b+2$};
		\draw[style = myline] (5,-5.1)--(5,4.2);
		
		\draw[style = myline](-5,-4)--(8,-4);		

		\path[style=myfill] (5,2.5) -- (5,4) -- (8,4) -- (5,2.5) ;
		\path[style=myfill] (-5,-4) -- (5,-4) -- (5,1.5) -- (-5, -3.5) -- (-5,-4) ;
		\path[style=myhatch] (-5,-4) -- (5,-4) -- (5,1.5) -- (-5, -3.5) -- (-5,-4) ;
		\draw[style=mygrid] (-5.0,-5.0) grid (8.0,4.0);

		
		\foreach \y in {-4,...,1}
		\fill (5,\y) circle(2pt);
		\foreach \y in {3,...,4}
		\fill (5,\y) circle(2pt);
		\foreach \y in {4,...,4}
		\fill (7,\y) circle(2pt);
		\foreach \y in {3,...,4}
		\fill (6,\y) circle(2pt);
		\foreach \y in {4,...,4}
		\fill (8,\y) circle(2pt);
	
		
		
		
		
		\foreach \y in {-4,...,1}
		\fill (4,\y) circle(2pt);
		
		\foreach \y in {-4,...,0}
		\fill (3,\y) circle(2pt);
		\foreach \y in {-4,...,0}
		\fill (2,\y) circle(2pt);
		
		\foreach \y in {-4,...,-1}
		\fill (1,\y) circle(2pt);
		\foreach \y in {-4,...,-1}
		\fill (0,\y) circle(2pt);   
		\foreach \y in {-4,...,-2}
		\fill (-1,\y) circle(2pt);  
		\foreach \y in {-4,...,-2}
		\fill (-2,\y) circle(2pt);   
		\foreach \y in {-4,...,-3}
		\fill (-3,\y) circle(2pt);  
		\foreach \y in {-4,...,-3}
		\fill (-4,\y) circle(2pt);  
  
		
			\draw (6.4, -4.7) node[fill = white, font= \footnotesize, align=left] { $q\geq 0$\\$p-2q\leq -2$};
		
		
		
		
	\end{tikzpicture}
	\begin{tikzpicture}[scale=0.6]
		\path[style=myfill] (-2,-1) -- (-2,4) -- (6,4) -- (6,3) -- (-2,-1) ;
		\path[style=myfill] (-6,-4) -- (-2,-4) -- (-2,-2) -- (-6,-4) ;
		\path[style=myhatch] (-6,-4) -- (-2,-4) -- (-2,-2) -- (-6,-4) ;
		\draw[style=mygrid] (-6.0,-5.0) grid (6.0,4.0);
		\draw[gray] (-6,0) -- (5.5,0) node[below, black] {\small $a$};
		\draw[gray] (0,-5.1) -- (0,4.4) node[left, black] {\small $b$};
		
		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
		\draw [font=\tiny, gray] (-2 cm,2pt) -- (-2 cm,-2pt) node[below left] {$-p$};
		\draw [font=\tiny, gray] (2pt,-4cm) -- (-2pt,-4cm) node[below left] {$-q$};
		\draw [style = myline] (-6,-3)--(6,3);
 		\draw [font=\tiny, gray] (5.1,2.5) node[below] {$a=2b$};
		\draw [style = myline] (-6,-4)--(6,2);
		\draw [font=\tiny, gray] (5,1.2) node[below=2.4pt] {$a=2b+2$};
		\draw [font=\tiny,gray] (2 ,2pt)--(2,-2pt) node[anchor=north] {$2$};
		\draw[style = myline] (-2,-5)--(-2,4.1);
		\draw[style = myline](-6.1,-4)--(6,-4);

		
		\foreach \y in {0,...,4}
		\fill (-1,\y) circle(2pt);
		\foreach \y in {-4,...,4}
		\fill (-2,\y) circle(2pt);
		\foreach \y in {-3,...,-4}
		\fill (-3,\y) circle(2pt);
		\foreach \y in {-3,...,-4}
		\fill (-4,\y) circle(2pt);

		\foreach \y in {0,...,4}
		\fill (0,\y) circle(2pt);
		\foreach \y in {1,...,4}       
		\fill(1,\y) circle(2pt);      

		
		
		
		
		\foreach \y in {1,...,4}
		\fill (2,\y) circle(2pt);
		
		\foreach \y in {2,...,4}
		\fill (3,\y) circle(2pt);
		\foreach \y in {2,...,4}
		\fill (4,\y) circle(2pt);
		
		\foreach \y in {3,...,4}
		\fill (5,\y) circle(2pt);
		\foreach \y in {3,...,4}
		\fill (6,\y) circle(2pt);   

		\foreach \y in {-6,...,-5}
		\fill (\y,-4) circle(2pt);  
		

	
		
		\draw (4.7, -4.7) node[fill = white,font= \footnotesize, align= left] {$q\geq 0$\\$p-2q\geq 0$};
		
	\end{tikzpicture}\hfill
\newline

	\begin{tikzpicture}[scale=0.6]
		\path[style=myfill] (-2,2) -- (-2,5) -- (10,5)--(4,2) --(-2,2) ;
		\draw[gray] (-3.0,0) -- (9.5,0) node[below, black] {\small $a$};
		\draw[gray] (0,-2.1) -- (0,5.5) node[left, black] {\small $b$};
		\draw[style=mygrid] (-3.0,-2.0) grid (10.0,5.0);
		
		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
		\draw [font=\tiny, gray] (2 cm,2pt) -- (2 cm,-2pt) node[anchor=north] {$2$};
		\draw [font=\tiny, gray] (-2 cm,2pt) -- (-2 cm,-2pt) node[below left] {$-p$};
		\draw [font=\tiny, gray] (2pt,2cm) -- (-2pt,2cm) node[below left] {$-q$};

		
		\draw [style = myline] (-3,-1.5)--(10,5);
		\draw [font=\tiny,gray, dashed] (9.5, 4.7) node[below] {$a=2b$};
		\draw [style = myline] (-2.1,-2.1)--(10,4);
		\draw [font=\tiny, gray, dashed] (9.1, 3.2) node[below] {$a=2b+2$};
		
		\draw[style = myline] (-2,-2.1)--(-2,5.5);
		\draw[style = myline](-3,2)--(10,2);
		
		
		\foreach \y in {2,...,5}
		\fill (-1,\y) circle(2pt);
	
		\foreach \y in {2,...,5}
		\fill (-2,\y) circle(2pt);

		\foreach \y in {2,...,5}
		\fill (0,\y) circle(2pt);
		\foreach \y in {2,...,5}       
		\fill(1,\y) circle(2pt);      

		
		
		
		\foreach \y in {2,...,5}
		\fill (2,\y) circle(2pt);
		
		\foreach \y in {2,...,5}
		\fill (3,\y) circle(2pt);
		\foreach \y in {2,...,5}
		\fill (4,\y) circle(2pt);
		
		\foreach \y in {3,...,5}
		\fill (5,\y) circle(2pt);
		\foreach \y in {3,...,5}
		\fill (6,\y) circle(2pt);   
		\foreach \y in {4,...,5}
		\fill (7,\y) circle(2pt);  
		\foreach \y in {4,...,5}
		\fill (8,\y) circle(2pt);   
		\foreach \y in {4,...,5}
		\fill (8,\y) circle(2pt);  
		\foreach \y in {5,...,5}
		\fill (9,\y) circle(2pt);  
		\fill (10,5) circle(2pt);  
		
		\draw  (8, -1.5) node[fill = white, font= \footnotesize, align= left, align=left] {$q<0$\\ $p\geq 0$};

	\end{tikzpicture}
	\begin{tikzpicture}[scale=0.6]
		\path[style=myfill] (5,2.5) -- (5,5) -- (10,5) --(5,2.5) ;
		\draw[style=mygrid] (-2,-2.0) grid (10.0,5.0);
		\draw[gray] (-2,0) -- (9.5,0) node[below, black] {\small $a$};
		\draw[gray] (0,-2.1) -- (0,5.5) node[left, black] {\small $b$};
		
		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
\draw [font=\tiny, gray] (2 cm,2pt) -- (2 cm,-2pt) node[anchor=north] {$2$};
		\draw [font=\tiny, gray] (5 cm,2pt) -- (5 cm,-2pt) node[below left] {$-p$};
		\draw [font=\tiny, gray] (3pt,2cm) -- (-3pt,2cm) node[below left] {$-q$};
		\draw [style = myline] (-2,-1)--(10,5);
		\draw [font=\tiny,gray, dashed] (9.5, 4.7) node[below] {$a=2b$};
		\draw [style = myline] (-2,-2)--(10,4);
		\draw [font=\tiny, gray, dashed] (9.1, 3.2) node[below] {$a=2b+2$};
		
		\draw[style = myline] (5,-2.1)--(5,5.5);
		\draw[style = myline](-2,2)--(10,2);
		
		


	
		
		\foreach \y in {3,...,5}
		\fill (5,\y) circle(2pt);
		\foreach \y in {3,...,5}
		\fill (6,\y) circle(2pt);   
		\foreach \y in {4,...,5}
		\fill (7,\y) circle(2pt);  
		\foreach \y in {4,...,5}
		\fill (8,\y) circle(2pt);   
		\foreach \y in {4,...,5}
		\fill (8,\y) circle(2pt);  
		\foreach \y in {5,...,5}
		\fill (9,\y) circle(2pt);  
		\fill (10,5) circle(2pt);
	
		\draw (8, -1.5) node[fill = white, align=left, font= \footnotesize, align= left] {$q<0$\\$p\leq -1$};
		
		
	\end{tikzpicture}	


$\bullet$ denotes the group $\mathds{Z}/2$\hfill\,
\caption{$H^{*+p\sigma,*+q\sigma} _{C _2}(\mathbf{E} C _2, \mathds{Z}/2)$, Betti realization is an isomorphism on the upper cone (solidly shaded region) and it is zero on the lower cone (hatched shaded region). }
	\label{Borelne}
\end{figure}



 \begin{notation}\label{namedelts}
 We introduce certain  elements in the
 cohomology of  $\spec(\C)$ and $\mathbf{E} C_2$. Note that the named elements in $H^\star_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$ introduced in \aref{sec:tbc} give rise to elements in $H^{\star, 0}_{C_2}(\C, \mathds{Z}/2)$ via \aref{prop:ptiso}. 
  The stated isomorphisms between the cohomology of  $\spec(\C)$ and $\mathbf{E} C_2$ all follow from the exact sequences associated to \eqref{eqn:cof2} together with the vanishing of the Bredon motivic cohomology of $\widetilde{\EG}  C_2$ in the relevant degrees, see \aref{crt}.
 
 \begin{itemize} 
 	\item $\lambda\in H^{0,\sigma} _{C _2}(\C,\mathds{Z}/2)\cong
 	 H^{0,\sigma} _{C _2}(\mathbf{E} C _2,\mathds{Z}/2)\cong \mathds{Z}/2$ is the generator. 
 	
 	\item  $u\in H^{2\sigma-2,\sigma-1} _{C _2}(\C,\mathds{Z}/2)\cong H^{2\sigma-2,\sigma-1}_{C _2}(\mathbf{E} C _2,\mathds{Z}/2)\cong \mathds{Z}/2$ is the generator. Multiplication by $u$ on the 
 	Bredon motivic cohomology of $\mathbf{E} C _2$ induces the $(2\sigma-2,\sigma-1)$ periodicity.
 	
 	\item  $\alpha\in H^{\sigma-1,0} _{C _2}(\C,\mathds{Z}/2)\cong H^{\sigma-1,0}_{C_2}(\mathbf{E} C _2,\mathds{Z}/2)\cong \mathds{Z}/2$ is the generator. 
 	 
 	\item  $\sigma\in H^{\sigma,0} _{C _2}(\C,\mathds{Z}/2)\cong 
 	H^{\sigma,0} _{C _2}(\mathbf{E} C _2,\mathds{Z}/2) \cong\mathds{Z}/2$ is the generator.
 	
 	
 	
 	\item $\theta\in H^{2-2\sigma,0} _{C _2}(\C,\mathds{Z}/2)\cong 
 	H^{2-2\sigma ,0}_{C_2}(\mathbf{E} C _2,\mathds{Z}/2) \cong \mathds{Z}/2$ is the generator.
 
 	\item Define $\mu := \displaystyle{\frac{\alpha^2}{u}}\in H^{0,1-\sigma}_{C_2}(\C) \cong H^{0,1-\sigma}_{C_2}(\mathbf{E} C_2)$.
 \end{itemize}

\end{notation}

\begin{remark}
	In terms of these elements we have
	\[
	\tau = \frac{\alpha^2\lambda}{u}\in H^{0,1}_{C_2}(\C,\mathds{Z}/2)\cong H^{0,1}(\C,\mathds{Z}/2).
	\]
	(this can be seen, for example, by noting that $\Re(\frac{\alpha^2\lambda}{u})=1$).
\end{remark}


     \begin{remark}\label{rem:beff}
\,
    \begin{enumerate}
    	\item For both $\Re:H^{\star,\star}_{C_2}(\C,\mathds{Z}/2)\to H^\star_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/2})$ and $\Re:H^{\star,\star}_{C_2}(\mathbf{E} C_2,\mathds{Z}/2)\to H^\star_{\mathrm{Br}}(\mathrm{E} C_2,\ul{\mathds{Z}/2})$ we have 
    	\begin{itemize}
    		\item $\Re(\lambda) = 1$,
    		\item $\Re(\alpha)= \alpha$ and $\Re(\sigma) = \sigma$, 
    		\item $\Re(u) = \alpha^2$. 
    	\end{itemize}
      \item For  $\Re:H^{\star,\star}_{C_2}(\C,\mathds{Z}/2)\to H^\star_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/2})$  we have
      $\Re(\theta)=\theta$.
      \item For $\Re :H^{\star,\star}_{C_2}(\mathbf{E} C_2,\mathds{Z}/2)\to H^\star_{\mathrm{Br}}(\mathrm{E} C_2,\ul{\mathds{Z}/2})$ we have
      $\Re (\theta)= 0$. 

    \end{enumerate} 
     \end{remark}
    
 

 Recall that 
 $H^{\star,0}_{C_2}(\C,\mathds{Z}/2)\cong H^\star_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$, see \aref{prop:ptiso}. In particular, we view 
 $ H^{\star,\star}_{C_2}(\mathbf{E} C_2, \mathds{Z}/2)$ as an 
 $H^\star_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$-module via this isomorphism.
 
\begin{theorem}\label{thm:was4.8}
	Let $n\geq 2$. The canonical map is an isomorphism of rings
\[
H^\star_\mathrm{Br}(\mathrm{pt}, \ul{\mathds{Z}/n})[u^{\pm 1}, \lambda] \xrightarrow{\cong} H^{\star,\star}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n).
\]

\end{theorem} 
 \begin{proof}
 	Since $H^{\star,0}_{C_2}(\widetilde{\EG}  C_2,\mathds{Z}/n) = 0$, we have $H^{\star,0}_{C_2}(\C,\mathds{Z}/n)\cong 
 	H^{\star,0}_{C_2}(\mathbf{E} C_2,\mathds{Z}/n)$.
 	Thus, together with periodicity, we have an isomorphism
 	\[
 	H^{\star,0}_{C_2}(\C, \mathds{Z}/n)[u^{\pm 1}]\xrightarrow{\cong} \bigoplus_{a,p,b}H^{a+p\sigma, b-b\sigma}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n).
 	\] 	
 	The result now follows from \aref{lem:was4.5}.
 \end{proof}

 \begin{corollary}
 	There is an isomorphism of rings
 	\[
 	H^{\star,\star}_{C_2}(\mathbf{E} C_2, \mathds{Z}/2)\cong \mathds{Z}/2[\sigma, \alpha][\lambda, u^{\pm 1}]\oplus \bigoplus_{\substack{c,m,n\geq 0 \\ b\in \mathds{Z}}} \mathds{Z}/2\left\{\frac{\theta u^b\lambda^c}{\sigma^n\alpha^m}\right\}.
 	\]
 	  The left hand summand acts on the right hand summand in the obvious way i.e. 	  
	  \begin{align*}
 			u^k\frac{\theta u^b\lambda^c}{\sigma^n\alpha^m} =\frac{\theta u^{b+k}\lambda^c}{\sigma^n\alpha^m}, k\in\mathds{Z},
 			\vspace{1mm}
 			 \\
			 \lambda \frac{\theta u^b\lambda^c}{\sigma^n\alpha^m}=\frac{\theta u^b\lambda^{c+1}}{\sigma^n\alpha^m},
			 \vspace{1mm}
			 \\
 			\alpha\smash\cdot  \frac{\theta u^b\lambda^c}{\sigma^n\alpha^m} = 
			\begin{cases}
 				\displaystyle{\frac{\theta u^b\lambda^c}{\sigma^n\alpha^{m-1}}} & m\geq  1 \vspace{1mm} \\
 					0 & m=0,
 			\end{cases} \\
			\vspace{1mm}
			 \\
 			\sigma\smash\cdot  \frac{\theta u^b\lambda^c}{\sigma^n\alpha^m} = 
			\begin{cases}
 				\displaystyle{\frac{\theta u^b\lambda^c}{\sigma^{n-1}\alpha^{m}}} & n\geq  1 \vspace{1mm} \\
 					0 & n=0,
 			\end{cases} \\
 		 		\end{align*}

	  
	  
 	  and all multiplications in the right hand summand are trivial. 
 \end{corollary}


\begin{remark}
Recall that $H^{*,*}(\mathbf{E} C_2, \mathds{Z}/2) \cong H^{*,*}(\mathbf{B} C_2, \mathds{Z}/2)\cong \mathds{Z}/2[\tau][f,d]/\{f^{2}=d\tau\} $, see \aref{sub:Borel}. In terms of the generators that appear in \aref{thm:was4.8} we have
$f=\frac{\sigma\alpha\lambda}{u},d=\frac{\sigma^2\lambda}{u},\tau=\frac{\alpha^2\lambda}{u}$.

\end{remark}


 \begin{lemma}\label{lem:was4.5}
 	Let $b+q\geq 0$. Then
 	\[
 	\cdot \lambda: H^{a+p\sigma, b+q\sigma}_{C_2}(\mathbf{E} C_2,\mathds{Z}/n)\xrightarrow{\cong} H^{a+p\sigma, b+(q+1)\sigma}_{C_2}(\mathbf{E} C_2,\mathds{Z}/n)
 	\]
 	is an isomorphism for all $a,p\in\mathds{Z}$. 
 \end{lemma}
 \begin{proof}
 	By periodicity, it suffices to treat the case $q=0$. 
 	
 	
 	
 	If $a\leq 2b$ this follows from \aref{Bocal} since $\Re(\lambda)=1$. 
 	 	The result holds for $a=2b+1$ since these groups are zero in this case. 
 	 	If $a\geq 2b+2$, then $\widetilde{H}^{a+p\sigma, b+q\sigma}_{C_2}(\widetilde{\EG}  C_2,\mathds{Z}/n)=0$ by \aref{crt} and so 
 	$H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong H^{a+p\sigma, b+q\sigma}_{C_2}(\mathbf{E} C_2,\mathds{Z}/n)$.  
 	Multiplication by $\lambda$ fits into 
 	the commutative triangle
 	\[
 	\begin{tikzcd}
 	H^{a+p\sigma, b}_{C_2}(\C,\mathds{Z}/n)\ar[d, "\cong"']\ar[r, "\smash\cdot \lambda"] & H^{a+p\sigma, b+\sigma}_{C_2}(\C,\mathds{Z}/n)\ar[dl] \\
 	H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n}).
 	\end{tikzcd}
 	\]
 	It follows that 
 	$\smash\cdot \lambda: H^{a+p\sigma, b}_{C_2}(\mathbf{E} C_2,\mathds{Z}/n)\to H^{a+p\sigma, b+\sigma}_{C_2}(\mathbf{E} C_2,\mathds{Z}/n)$ is injective. 
 But by \aref{Bocal} and \aref{crpoint}, either both of these groups are $\mathds{Z}/n$ or both are $0$. Thus the map is an isomorphism.
 \end{proof}












We end this section with the following.

\begin{proposition}\label{prop:wteg}
	There is an isomorphism of 
	$H^{\star}_{\mathrm{Br}}(\mathrm{pt},  \ul{\mathds{Z}/2})$-modules
	\[	
	\bigoplus_{\subalign{1&\leq i \vspace{1pt} \\ 0&\leq j\leq 2i-1}\hspace{-2em}}\mathds{Z}/2 \{\frac{\theta \mu^i}{\alpha^j} \} \otimes \mathds{Z}/2[x^{\pm 1}, \lambda^{\pm 1}]\xrightarrow{\cong}
	\widetilde{H}^{\star,\star}_{C_2}(\widetilde{\EG}  C _2,\mathds{Z}/2)
	\]
	where $|x|=(\sigma,0)$ and the other elements are as above.\\
		
		
	The $H^{\star}_{\mathrm{Br}}(\mathrm{pt},  \ul{\mathds{Z}/2})$-module structure is determined by
	\begin{align*}
		\sigma \smash\cdot \frac{\theta\mu^m}{\alpha^i}x^j\lambda^k & = \frac{\theta\mu^m}{\alpha^i}x^{j+1}\lambda^k \\
		\alpha\smash\cdot \frac{\theta\mu^m}{\alpha^i}x^j\lambda^k & = 
		\begin{cases}
			\displaystyle{\frac{\theta\mu^m}{\alpha^{i-1}}x^j\lambda^k} & i\geq  1 \vspace{1mm} \\
			0 & i=0
		\end{cases} \\
		\frac{\theta}{\alpha^s\sigma^t}\smash\cdot\frac{\theta\mu^m}{\alpha^i}x^j\lambda^k & = 0.
	\end{align*}
	 
\end{proposition}
\begin{proof}

It follows from \aref{crt} and \aref{lema:leqb} that  Betti realization 
\[
\Re:\widetilde{H}^{a+p\sigma, b+q\sigma}_{C_2}(\widetilde{\EG}  C_2, \mathds{Z}/2)\to 
\widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\wt \mathrm{E} C_2, \ul{\mathds{Z}/2})
\]
is an isomorphism if $b\geq 0$ and $a\leq 2b+1$  
and $\widetilde{H}^{a+p\sigma, b+q\sigma}_{C_2}(\widetilde{\EG}  C_2, \mathds{Z}/2) = 0$ if $b<0$ or $a>2b+1$. In particular
\[
\widetilde{H}^{a+p\sigma, b+q\sigma}_{C_2}(\widetilde{\EG}  C_2, \mathds{Z}/2)\cong 
\begin{cases}
	\mathds{Z}/2 & 2\leq a \leq 2b+1 \\
	0 & \textrm{else}.
\end{cases}
\]


Consider the  diagram
\[
\begin{tikzcd}
	\left(\oplus\mathds{Z}/2\left\{\frac{\theta \mu^i}{\alpha^j}\right\} \otimes \mathds{Z}/2[\lambda^{\pm 1}, x^{\pm 1}]\right)^{(a+p\sigma, b+q\sigma)}
	\ar[r ]\ar[dr] & \widetilde{H}^{a+p\sigma,b+q\sigma}_{C_2}(\wt \mathbf{E} C_2, \mathds{Z}/n)\ar[d] \\
	& \widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\wt \mathrm{E} C_2,\ul{\mathds{Z}/n}),
\end{tikzcd}
\]
where the diagonal arrow is defined by $\alpha\mapsto \alpha$, $\theta\mapsto\theta$, $\mu\mapsto 1$, $\lambda\mapsto 1$, and $x\mapsto \sigma$. 
The horizontal arrow is an isomorphism of graded $\mathds{Z}/2$ vector spaces and
if $b\geq 0$ and $a\leq 2b+1$ the diagonal map is an isomorphism. The 
 $H^{\star}_{\Br}(\pt, \ul{\Z/2})$-module structure on $H^{a+p\sigma}_{\mathrm{Br}}(\wt \mathrm{E} C_2, \ul{\mathds{Z}/2})$ then follows from  
\aref{rem:peg}.
 
\end{proof}




\section{Bredon motivic cohomology of \texorpdfstring{$\C$}{C}}\label{sec:BC}
 
  This section identifies the Bredon motivic cohomology ring of the complex numbers with $\mathds{Z}/2$ coefficients. 
  
  
  We begin with the additive computation.
  
  \begin{theorem}\label{thm:additive}
  Let $n\geq 2$ be a natural number.
  	\begin{enumerate}
  		\item If $b\geq 0$ and $b+q\geq 0$ then Betti realization induces an isomorphism for any $a,p$
  		\[
  		\Re : H^{a+p\sigma,b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\xrightarrow{\cong}
  		H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/n}).
  		\] 
  		
  		\item If $b\geq 0$ and $b+q<0$ then 
  		$H^{\star, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong 
  		\widetilde{H}^{\star, b+q\sigma}_{C_2}(\widetilde{\EG}  C_2, \mathds{Z}/n)$. 
  		In particular
  			\[
  			H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong \begin{cases}
  			\mathds{Z}/2 & 2\leq a \leq 2b+1\textrm{ and $n$ is even}\\
  			0 & \textrm{else}.
  			\end{cases}
  			\]


  			Moreover, if $a \leq 2b+1$ and $a< -p$, then  
  		$\Re$ is an isomorphism, it is injective if $a=-p$, and is zero for all other values of $a,p$.
  		
  		\item If $b<0$ and $b+q\geq 0$ then 
  		$H^{\star, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong H^{\star, b+q\sigma}_{C_2}(\mathbf{E} C_2, \mathds{Z}/n)$. 		  	
  		 In particular,
  		 \[ 
  		 H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/n)\cong 
  	 \begin{cases}
  	 	H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt},\ul{\mathds{Z}/n}) & a\leq 2b \\
  	 	0 & a = 2b+1 \\
  	 	H^{a+2q+(p-2q)\sigma}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/n}) & 2b+2\leq a.
  	 \end{cases}
  		 \]
  		 
  		 Moreover, $\Re $ is an isomorphism if $a\leq 2b$ or if $2\leq a$. It is multiplication by $2$ if $2b+2\leq a\leq 1$ where $a$ is even and $a+p=0$, and it is zero for all other values of $a,p$.  
  		
  \item If $b<0$ and $b+q<0$, then $H^{\star, b+q\sigma}_{C_2}(\C, \mathds{Z}/n)=0$. 
\end{enumerate}
  \end{theorem}
 \begin{proof}
 	
 We make use of the comparison of long exact sequences, obtained from \eqref{eqn:cof1} and \eqref{eqn:cof2} 
 \begin{equation}\label{eqn:6cof1}
 \begin{tikzcd}[column sep=small]
 \cdots \ar[r] & H^{a+p, b +q}_{}(\C) \ar[r]\ar[d] & H^{a+p\sigma, b +q\sigma}_{C_2}(\C) \ar[r]\ar[d] & H^{a+(p+1)\sigma, b +q\sigma}_{C_2}(\C) \ar[r]\ar[d] & H^{a+1+p, b+q}(\C)\ar[r]\ar[d] & \cdots \\
 \cdots \ar[r] & H^{a+p}_{\mathrm{sing}}(\mathrm{pt}) \ar[r]& H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt}) \ar[r] & H^{a+(p+1)\sigma}_{\mathrm{Br}}(\mathrm{pt}) \ar[r] & H^{a+1+p}_{\mathrm{sing}}(\mathrm{pt})\ar[r] & \cdots
 \end{tikzcd} 
 \end{equation}
 and
 \begin{equation}\label{eqn:6cof2}
 \begin{tikzcd}[column sep=small]
 	\cdots \ar[r] & \widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\widetilde{\EG}  C_2) \ar[r]\ar[d] & H^{a+p\sigma, b +q\sigma}_{C_2}(\C) \ar[r]\ar[d] & H^{a+p\sigma, b +q\sigma}_{C_2}(\mathbf{E} C_{2}) \ar[r]\ar[d] & \cdots 
 	\\
 	\cdots \ar[r] & \widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\wt\mathrm{E} C_2) \ar[r]& H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt}) \ar[r] & H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_{2}) \ar[r] & \cdots .
 \end{tikzcd}
 \end{equation}
  
 First we note that (4) follows since $ H^{a+p\sigma, b +q\sigma}_{C_2}(\mathbf{E} C_{2})=0$ if $b+q <0$ (see \aref{neq}) and
 $\widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\widetilde{\EG}  C_2) = 0$ if $b<0$ (see \aref{crt}).
  
  
 To establish (1), we first observe that it 
 suffices to show that $H^{a,b+q\sigma}_{C_2}(\C)\to H^{a}_{\mathrm{Br}}(\mathrm{pt})$ is an isomorphism for all $a$. Indeed the general case follows from the $p=0$ case by induction (upwards and downwards) on $p$, using \eqref{eqn:6cof1}, since $H^{*,n}(\C)\to H^*_{\mathrm{sing}}(\mathrm{pt})$ is an isomorphism when $n\geq 0$. 
 Next, we note that using \eqref{eqn:6cof2} together with \aref{Bocal} and
 \aref{crt}, we have that $H^{a, b +q\sigma}_{C_2}(\C)\to H^{a}_{\mathrm{Br}}(\mathrm{pt})$ is an isomorphism for $a\leq 2b$. 
  Since $b\geq 0$, for this map to be an isomorphism for $a> 2b$, we need to see that $H^{a, b +q\sigma}_{C_2}(\C) =0$ for $a>2b$. This also follows from loc. cit., which finishes (1).
  
  
 For part (2), consider  the commutative diagram
 \[
 \begin{tikzcd}[column sep=small]
 	\widetilde{H}^{a+p\sigma, b +q\sigma}_{C_2}(\widetilde{\EG}  C_2) \ar[r, "\cong"]\ar[d] & H^{a+p\sigma, b +q\sigma}_{C_2}(\C) \ar[d] 
 	\\
 	\widetilde{H}^{a+p\sigma}_{\mathrm{Br}}(\wt\mathrm{E} C_2) \ar[r]& H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{pt}).
 \end{tikzcd}
 \]
 The upper horizontal arrow is an isomorphism  
 since 
  $H^{\star, b+q\sigma}(\mathbf{E} C_2, \mathds{Z}/n)=0$ if $b+q<0$.  For $a\leq 2b+1$,  the left hand vertical arrow is an isomorphism 
by \aref{lema:leqb} and \aref{crt}.  
 If $a+p<0$ then $H^{a+p\sigma}_{\mathrm{Br}}(\mathrm{E} C_2)=0$, so the lower horizontal map is an isomorphism and hence the right vertical map is as well. 
 If $a+p=0$, 
 then the lower horizontal map is injective, implying it is an isomorphism (as the domain and codomain have the same finite number of elements). Hence the right vertical map is an isomorphism as well. 
 Finally, by inspection, we see that if $a+p>0$, the right vertical map is always zero (either the domain or codomain is always zero in this case).  
     
  All of the statements of (3) follow from \aref{Bocal} except that $\Re$ is an isomorphism for $a\geq 2$. 
  This follows by induction (upwards and downwards) on $p$ using \eqref{eqn:6cof1}, starting with $p=0$. 
   
 \end{proof}
  


 

 
 The groups $H^{a+p\sigma, b+q\sigma}_{C_2}(\C, \mathds{Z}/2)$ are displayed, for various values of $p,q$, in \aref{Bredon4}.
 
 \begin{figure}[H]
 	\begin{tikzpicture}[scale=0.6]
 		\path[style=myfill] (6,2.5) -- (6,4) -- (9,4) --(6,2.5) ;
 		\path[style=myfill] (2,0.5) -- (2,7) -- (6,7) --(6,2.5)--(2,0.5) ;
 		\draw[style=mygrid] (-2.0,-2.0) grid (10.0,7.0);
 		
 		\draw[gray] (-2,0) -- (10.2,0) node[below, black] {\small $a$};
 		\draw[gray] (0,-2) -- (0,7.2) node[left, black] {\small $b$};
 		
 		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
 		\draw [font=\tiny, gray] (2 cm,2pt) -- (2 cm,-2pt) node[below left] {$2$};
 		\draw [font=\tiny, gray] (6 cm,2pt) -- (6 cm,-2pt) node[below left] {$-p$};
 		\draw [font=\tiny, gray] (2pt,5cm) -- (-2pt,5cm) node[below left] {$-q$};
 		\draw [font=\tiny, gray, style=myline] (-2,-1)--(10,5);
 		\draw [font=\tiny, gray] (9.3,4.6) node[anchor=north] {$a=2b$};
 		\draw [font=\tiny, gray, style=myline] (-2,-2)--(10,4);
 		\draw [font=\tiny, gray] (9.1, 3.4) node[anchor=north] {$a=2b+2$};
 		\draw [font=\tiny,gray, style=myline] (2,-2)--(2,7) node[anchor=north] {$a=2$};
 		
 		
 		\draw[style=myline] (6,-2)--(6,7);
 		\draw[style=myline](-2,5)--(10,5);
 		
 		
 		
 		
 		\foreach \y in {2,...,7}
 		\fill (5,\y) circle(2pt);
 		\foreach \y in {1,...,7}
 		\fill (2,\y) circle(2pt);
 		\foreach \y in {2,...,7}
 		\fill (3,\y) circle(2pt);
 		\foreach \y in {2,...,7}
 		\fill (4,\y) circle(2pt);
 		\foreach \y in {5,...,7}
 		\fill (5,\y) circle(2pt);
 		\foreach \y in {3,...,7}
 		\fill (6,\y) circle(2pt);
 		\foreach \y in {3,...,4}
 		\fill (7,\y) circle(2pt);  
 		\fill(8,4) circle(2pt);
 		
 		
 		
 		\fill (5,2) circle(2pt);
 		
 		\fill (7,3) circle(2pt);
 		
 		\fill (9,4) circle(2pt);
 		
 		
 		\fill (3,1) circle (2pt);
 		
 		
 		\draw (9.1, -1.5) node[fill = white, align=left, font= \footnotesize, align= left] {$q<0$\\$p\leq -1$};
 		
 		
 		
 		
 		
 		
 		
 	\end{tikzpicture}
 \hfill
 	\begin{tikzpicture}[scale=0.6]
 		\path[style=myfill] (0,4) -- (-4,4) --(-4,7)--(0,7)--(0,4) ;
 		\path[style=myfill] (2,0.5) -- (2,3) -- (7,3) --(2,0.5) ;

 		\draw[style=mygrid] (-5.0,-2.0) grid (7.0,7.0);
 		
 		\draw[gray] (-5,0) -- (7.2,0) node[below, black] {\small $a$};
 		\draw[gray] (0,-2) -- (0,7.3) node[left, black] {\small $b$};
 		
 		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
 		\draw [font=\tiny, gray] (2 cm,2pt) -- (2 cm,-2pt) node[anchor=north] {$2$};
 		\draw [font=\tiny, gray] (-4 cm,2pt) -- (-4 cm,-2pt) node[below left] {$-p$};
 		\draw [font=\tiny, gray] (2pt,4cm) -- (-2pt,4cm) node[below left] {$-q$};
 		\draw [font=\tiny,gray,  style = myline] (-4,-2)--(7,3.5);
 		\draw [font=\tiny,gray](6.3,3.8) node[anchor=north] {$a=2b$};
 		\draw [font=\tiny,gray,  style = myline] (-2,-2)--(7,2.5);
 		\draw [font=\tiny,gray] (6.1,1.9) node[anchor=north] {$a=2b+2$};
 		\draw [font=\tiny,gray, style = myline] (2,-2)--(2,7);
 		
 		
 		\draw[ style = myline] (-4,-2)--(-4,7);
 		
 		\draw[ style = myline](-5,4)--(7,4);
 		
 		
 		
 		\foreach \y in {4,...,7}
 		\fill (-2,\y) circle(2pt);
 		\foreach \y in {4,...,7}
 		\fill (-1,\y) circle(2pt);
 		\foreach \y in {1,...,3}
 		\fill (2, \y) circle(2pt);  
 		\foreach \y in {1,...,3}
 		\fill (3, \y) circle(2pt);   
 		\foreach \y in {2,...,3}
 		\fill (4, \y) circle(2pt);  
 		\foreach \y in {2,...,3}
 		\fill (5, \y) circle(2pt);  
 		\foreach \y in {3,...,3}
 		\fill (6, \y) circle(2pt);  
 		\fill (7,3) circle(2pt);
 		
 		\foreach \y in {4,...,7}
 		\fill (0,\y) circle(2pt);
 		\foreach \y in {4,...,7}
 		\fill (-3,\y) circle(2pt);
 		\foreach \y in {4,...,7}
 		\fill (-4,\y) circle(2pt);
 		
 		
 		
 		\draw (6, -1.5) node[fill = white, align=left, font= \footnotesize, align= left] {$q<0$\\$p\geq 0$};
 		
 		
 		
 		
 	\end{tikzpicture}
 	\newline
 	\begin{tikzpicture}[scale=0.6]
 		\path[style=myfill] (2,-4) -- (2,4)--(6,4)--(6,-3)--(2,-3) ;
 		\path[style=myfill] (-4,-3) -- (2,-3) -- (2,0) --(-4,-3) ;
 		\path[style = myhatch] (1.5, -3)--(1.5, -0.25) -- (-4, -3) -- (-2.5, -3);
 		\draw[style=mygrid] (-5.0,-6.0) grid (7.0,4.0);
 		
 		\draw[gray] (-5,0) -- (7.2,0) node[below, black] {\small $a$};
 		\draw[gray] (0,-6) -- (0,4.3) node[left, black] {\small $b$};
 		
 		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
 		\draw [font=\tiny, gray] (2 cm,2pt) -- (2 cm,-2pt) node[below left] {$2$};
 		\draw [font=\tiny, gray] (6 cm,2pt) -- (6 cm,-2pt) node[below left] {$-p$};
 		\draw [font=\tiny, gray] (3pt,-3 cm) -- (-3pt,-3 cm) node[below left] {$-q$};
 		\draw [font=\tiny,gray,  style = myline] (-5,-2.5)--(7,3.5);
 		\draw [font=\tiny,gray] (6.5,3.0) node[anchor=north] {$a=2b$};
 		\draw [font=\tiny,gray,  style = myline] (-5,-3.5)--(7,2.5);
 		\draw [font=\tiny,gray] (6.1,2.0) node[anchor=north] {$a=2b+2$};
 		\draw [style=myline] (2,-6)--(2,4);
 		
 		
 		\draw[ style = myline] (6,-6)--(6,4);
 		
 		\draw[ style = myline](-5,-3)--(7,-3);
 		
 		\foreach \y in {-3,...,4}
 		\fill (5,\y) circle(2pt);
 		\foreach \y in {-3,...,4}
 		\fill (2,\y) circle(2pt);
 		\foreach \y in {-3,...,4}
 		\fill (3,\y) circle(2pt);
 		\foreach \y in {-3,...,4}
 		\fill (4,\y) circle(2pt);
 		\foreach \y in {-3,...,4}
 		\fill (5,\y) circle(2pt);
 		\foreach \y in {-3,...,4}
 		\fill (6,\y) circle(2pt);
 		\foreach \y in {-3,...,-1}
 		\fill (1,\y) circle(2pt);
 		\foreach \y in {-3,...,-1}
 		\fill (0,\y) circle(2pt);
 		\foreach \y in {-3,...,-2}
 		\fill (-2,\y) circle(2pt);
 		\foreach \y in {-3,...,-3}
 		\fill (-3,\y) circle(2pt);
 		\foreach \y in {-3,...,-3}
 		\fill (-4,\y) circle(2pt);
 	
 	
 	
 		\foreach \y in {-3,...,-2}
 		\fill (-1,\y) circle(2pt);
 		
 		
 		
 	
 		
 		
 		\fill (5,2) circle(2pt);
 		\draw (6.1, -5.5) node[fill = white, align=left, font= \footnotesize, align= left] {$q\geq 0$\\$p\leq -1$};
 		
 	\end{tikzpicture}
 \hfill
 
 
 
 	%
 	%
 
 
 	\begin{tikzpicture}[scale=0.6]
 		\path[style=myfill] (-3,-1.5) -- (0,0)--(0,4)--(-3,4)--(-3,-1.5) ;
 		\path[style=myfill] (-3,-2.5) -- (-3,-5) -- (-8,-5) --(-3,-2.5) ;
 		\path[style=myhatch] (-3,-2.5) -- (-3,-5) -- (-8,-5) --(-3,-2.5) ;
 		\draw[style=mygrid] (-9.0,-6.0) grid (3.0,4.0);
 		
 		\draw[gray] (-9,0) -- (3.2,0) node[below, black] {\small $a$};
 		\draw[gray] (0,-6) -- (0,4.3) node[left, black] {\small $b$};
 		
 		\draw [font=\tiny, gray] (1 cm,2pt) -- (1 cm,-2pt) node[anchor=north] {$1$};
 		\draw [font=\tiny, gray] (2 cm,2pt) -- (2 cm,-2pt) node[anchor=north] {$2$};
 		\draw [font=\tiny, gray] (-3 cm,2pt) -- (-3 cm,-2pt) node[below left] {$-p$};
 		\draw [font=\tiny, gray] (3pt,-5 cm) -- (-3pt,-5 cm) node[below left] {$-q$};
 		\draw [font=\tiny,gray, style=myline] (-9,-4.5)--(3,1.5); 	
 		\draw [font=\tiny,gray] (2,1) node[anchor=north] {$a=2b$};
 		\draw [font=\tiny,gray,  style = myline] (-9,-5.5)--(3,0.5); \draw [font=\tiny,gray] (2,-0.5) node[anchor=north] {$a=2b+2$};
 	
 		
 		
 		\draw[ style = myline] (-3,-6)--(-3,4);
 		
 		\draw[ style = myline](-8,-5)--(3,-5);
 		
 		\foreach \y in {-1,...,4}
 		\fill (-2,\y) circle(2pt);
 		\foreach \y in {-4,...,-3}
 		\fill (-3,\y) circle(2pt);
 		\foreach \y in {-1,...,4}
 		\fill (-3,\y) circle(2pt);
 		\foreach \y in {-5,...,-3}
 		\fill (-4,\y) circle(2pt);
 		\foreach \y in {-5,...,-4}
 		\fill (-5,\y) circle(2pt);
 		
 		
 		\foreach \y in {0,...,4}
 		\fill (0,\y) circle(2pt);
 		\foreach \y in {-4,...,-3}
 		\fill (-3,\y) circle(2pt);
 		\foreach \y in {0,...,4}
 		\fill (-1,\y) circle(2pt);
 		\foreach \y in {-4,...,-4}
 		\fill (-5,\y) circle(2pt);
 		\fill(-6,-4) circle(2pt); 
 		
 		
 		
 	
 		
 		
 		\fill (-3,-5) circle(2pt);
 		\fill (-4,-5) circle(2pt);
 		\fill (-5,-5) circle(2pt);
 		\fill (-6,-5) circle(2pt);
 		\fill (-7,-5) circle(2pt);
 		\fill (-8,-5) circle(2pt);
 		
 		
 		\draw (2, -5.5) node[fill = white, align=left, font= \footnotesize, align= left] {$q\geq 0$\\$p\geq 0$};
 		
 	\end{tikzpicture}
 
 $\bullet$ denotes the group $\mathds{Z}/2$\hfill\,
 	\caption{$H^{*+p\sigma,*+q\sigma} _{C _2}(\C, \mathds{Z}/2)$, Betti realization is an isomorphism on the solidly shaded regions and it is zero on the hatched shaded regions. }
 	\label{Bredon4}
 \end{figure}
 



The elements in the theorem below were introduced after \aref{neq}.
 
 \begin{theorem}\label{thm:ptring}
 	\,
 	\begin{enumerate}
 		\item There is an $H^{\star}_{\mathrm{Br}}(\mathrm{pt}, \ul{\mathds{Z}/2})$-algebra isomorphism 
 		\[
 		\phi:H^{\star}_{\mathrm{Br}}(\mathrm{pt},  \ul{\mathds{Z}/2})[u,\lambda, \mu]/(u\mu-\alpha^2)\xto{\cong} \bigoplus_{b+q\geq 0}H_{C_2}^{\star,b+q\sigma}(\C,\mathds{Z}/2)
 		\]
 		defined by $u\mapsto u$, $\lambda\mapsto \lambda$, $\mu\mapsto \alpha^2/u$.
 		 
 		\item 
 		There is an 
 		$H^{\star}_{\mathrm{Br}}(\mathrm{pt},  \ul{\mathds{Z}/2})$-algebra isomorphism 
 		\[
 		H^{\star}_{\mathrm{Br}}(\mathrm{pt},  \ul{\mathds{Z}/2})[u,\lambda, \mu]/(u\mu-\alpha^2)\oplus\left(\bigoplus_{\substack{m,n\geq 1 \\ 0\leq d\leq 2n-1}} \mathds{Z}/2\{\frac{\theta \mu^n}{\lambda^m\alpha^d}\}\otimes\mathds{Z}/2[x^{\pm 1}]\right)\xto{\cong} H_{C_2}^{\star,\star}(\C,\mathds{Z}/2)
 		\]
 		where $|x| = (\sigma, 0)$. The ring $H_{C_2}^{\star,\star}(\C,\mathds{Z}/2)$ is the commutative ring whose product structure is determined as follows, the left hand summand is the displayed quotient of a polynomial ring, all products in the summand 
 		$\oplus \mathds{Z}/2\{\frac{\theta \mu^n}{\lambda^m\alpha^d}\}\otimes\mathds{Z}/2[x^{\pm 1}]$ are trivial, and the remaining products are determined by
 		\begin{align*}
 		\lambda \smash\cdot\frac{\theta\mu^nx^l}{\lambda^m\alpha^d}& =
 		\begin{cases}
 		\displaystyle{\frac{\theta\mu^nx^l}{\lambda^{m-1}\alpha^d}} & m\geq 2 \vspace{1mm} \\
 		0 & m=1, l>0 \vspace{1mm}\\
 		\displaystyle{\frac{\theta\mu^nx^l}{\alpha^d} }& m=1, l\leq 0
 		\end{cases}
 		\\
 		\alpha \smash\cdot\frac{\theta\mu^nx^l}{\lambda^m\alpha^d}& =
 		 \begin{cases}
 			\displaystyle{\frac{\theta\mu^nx^l}{\lambda^{m}\alpha^{d-1}}} &  1\leq d\leq 2n-1 \vspace{1mm} \\ 0 & d=0 
 			\end{cases}
 			\\
 		 \mu \smash\cdot\frac{\theta\mu^nx^l}{\lambda^m\alpha^d} & = \frac{\theta\mu^{n+1}x^{l}}{\lambda^m\alpha^d} 
 		\\
 		\sigma\smash\cdot \frac{\theta\mu^nx^l}{\lambda^m\alpha^d} &= \frac{\theta\mu^{n}x^{l+1}}{\lambda^m\alpha^d}
 		 		\\
 		u\smash\cdot \frac{\theta\mu^nx^l}{\lambda^m\alpha^d} & =
 		\begin{cases}
 			\displaystyle{\frac{\theta\mu^{n-1}\sigma^l}{\lambda^m\alpha^{d-2}}} & d\geq 2, n\geq 2 \vspace{1mm} \\
 			0 & \textrm{else}
 		\end{cases}
 		\end{align*}

 	\end{enumerate}
 \end{theorem}
 \begin{proof}
 For the first part, examining	 the weights of elements, we see that
 \[ 
 \left( H^{\star}_{\Br}(\pt, \ul{\Z/2})[u,\lambda, \mu]/(u\mu-\alpha^2)\right)^{(\star, b+q\sigma)} \cong \begin{cases}
 	H^{\star}_{\Br}(\pt, \ul{\Z/2})\cdot\{\mu^b\lambda^{q+b} \} & b\geq 0 \vspace{2mm}\\
 	H^{\star}_{\Br}(\pt, \ul{\Z/2})\cdot \{u^b\lambda^{q-b}\} & b\leq 0.
 \end{cases}
 \]
  
 If $b\geq 0$,  consider the commutative diagram
 \[
 \begin{tikzcd}
 	\left( H^{\star}_{\Br}(\pt, \ul{\Z/2})[u,\lambda, \mu]/(u\mu-\alpha^2)\right)^{(\star, b+q\sigma)}\ar[r, "\phi"]\ar[dr, "\cong"'] & H^{\star,\star}_{C_2}(\C, \mathds{Z}/2)\ar[d, "\cong"] \\
 	& H^{\star}_{\Br}(\pt, \ul{\Z/2}).
 \end{tikzcd}
 \]
 The right vertical arrow is an isomorphism by \aref{thm:additive} and thus so is $\phi$. If $b\leq 0$, consider the commutative diagram
 \[
 \begin{tikzcd}
 	\left( H^{\star}_{\Br}(\pt, \ul{\Z/2})[u,\lambda, \mu]/(u\mu-\alpha^2)\right)^{(\star, b+q\sigma)}\ar[r, "\phi"]\ar[dr, "\cong"'] & H^{\star,\star}_{C_2}(\C, \mathds{Z}/2)\ar[d, "\cong"] \\
 	& H^{\star, \star}_{C_2}(\mathbf{E} C_2,\mathds{Z}/2). 
 \end{tikzcd}
 \]
 The vertical arrows are isomorphisms and therefore, so is $\phi$. 
 
 For the second part, we have  
  $\oplus_{b+q<0}\widetilde{H}^{\star,b+q\sigma}_{C_2}(\wt\mathbf{E} C_{2}, \mathds{Z}/2)\xto{\cong} \oplus_{b+q<0}H^{\star, b+q\sigma}_{C_2}(\C, \mathds{Z}/2)$. The stated module isomorphism then follows from \aref{prop:wteg}. 
  
  
  The ring $H_{C_2}^{\star,\star}(\C,\mathds{Z}/2)$ is graded commutative and since in each group $H_{C_2}^{a+p\sigma, b+q\sigma}(\C,\mathds{Z}/2)$  is either $\mathds{Z}/2$ or $0$, the ring is commutative. 
  Products in $\widetilde{H}^{\star,\star}_{C_2}(\wt\mathbf{E} C_2, \mathds{Z}/2)$ are trivial, since they are determined by products in $\widetilde{H}^{*,*}_{C_2}(\wt\mathbf{E} C_2, \mathds{Z}/2)\cong \widetilde{H}^{*,*}(\Sigma \mathbf{B} C_2, \mathds{Z}/2)$, which are trivial. 
  The products involving $\sigma, \alpha$ are determined by \aref{prop:wteg}. The products involving $\lambda$ and $\mu$ are determined by \aref{rhon} below. The products involved are determined by those involving $\alpha$ and $\mu$, since $\mu=\alpha^2/u$.
  

  


 \end{proof}
 
\begin{remark}\label{rem:similar}
We restrict here to the case of $\mathds{Z}/2$ coefficients for simplicity because this is the most interesting case. However, similar techniques yield the general case of $\mathds{Z}/2^i$ coefficients with appropriate changes to \aref{prop:wteg} and \aref{crpoint}.
\end{remark}

 \begin{lemma} \label{rhon} 
 	Let $b+q<0$ with $b\geq 0$.
 	\begin{enumerate}
 		\item Multiplication by $\lambda$, 
 		$H^{a+p\sigma,b+q\sigma}_{C_2}(\C,\mathds{Z}/2)\xto{\smash\cdot \lambda} H^{a+p\sigma,b+(q+1)\sigma} _{C _2}(\C,\mathds{Z}/2)$
 		 is 
 		 \begin{enumerate}
 		 	\item   an isomorphism  if  $b+q<-1$ and $a\leq 2b+1$,
 		 	\item  an isomorphism  if  $b+q = -1$, $a\leq 2b+1$ , and $a\leq -p$,
 		 	\item zero otherwise. 
 		 \end{enumerate} 
 		  
 		 
 		
 		 \item  Multiplication by $\mu$,
 		 $ H^{a+p\sigma,b+(q-1)\sigma}_{C _2}(\C,\mathds{Z}/2)\xto{\smash\cdot \mu} H^{a+p\sigma,b+1+(q-2)\sigma}_{C _2}(\C,\mathds{Z}/2)$
 		 is an isomorphism if $a\leq 2b+1$.
  	\end{enumerate}

 \end{lemma}
 \begin{proof} 
 	
  If $b+q<-1$, $a+p\leq 0$ and $a\leq 2b+1$,  Betti realization is an isomorphism by \aref{thm:additive}. We have the commutative diagram (since $\Re(\lambda)=1$)
 	\[
 	\begin{tikzcd}
 		H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/2)\ar[r, "\smash\cdot \lambda"]\ar[dr, "\cong"'] & H^{a+p\sigma,b+(q+1)\sigma}_{C_2}(\C, \mathds{Z}/2)\ar[d, "\cong"] \\
 		& (H^{\star}_{\Br}(\pt, \ul{\Z/2}))^{a+p\sigma},
 	\end{tikzcd}
 	\]
 	and so multiplication by $\lambda$ is an isomorphism in this case.  If $a+p>0$,  consider the diagram where $a+p+k\leq 0$,
 	\[
 	\begin{tikzcd}
 			H^{a+p\sigma, b+q\sigma} _{C _2}(\C,\mathds{Z}/2) \ar[r, "\smash\cdot \lambda"] 
 			\ar[d, "\smash\cdot \sigma^k"'] & 
 		H^{a+p\sigma, b+(q+1)\sigma}_{C_2}(\C,\mathds{Z}/2)\ar[d, "\smash\cdot \sigma^k"]
 		\\
 		H^{a+(p+k)\sigma,b+q\sigma} _{C _2}(\C,\mathds{Z}/2) \ar[r, "\smash\cdot \lambda"] & 
 		H^{a+(p+k)\sigma,b+(q+1)\sigma} _{C _2}(\C,\mathds{Z}/2).
 	\end{tikzcd}
 	\]
 	The vertical arrows are isomorphisms by \aref{prop:wteg} and \aref{thm:additive}.
 	The bottom horizontal arrow is an isomorphism, and therefore, so is the top. 
 	
 	
 	Now, if $b+q=-1$, we again consider the  commutative diagram
 	\[
 	\begin{tikzcd}
 		H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/2)\ar[r, "\smash\cdot \lambda"]\ar[dr] & H^{a+p\sigma,b+(q+1)\sigma}_{C_2}(\C, \mathds{Z}/2)\ar[d, "\cong"] \\
 		& (H^{\star}_{\Br}(\pt, \ul{\Z/2}))^{a+p\sigma},
 	\end{tikzcd}
 	\]
 	where the righthand vertical arrow is an isomorphism because $b+q+1=0$, by \aref{thm:additive}. If $a+p\leq 0$ and $a\leq 2b+1$ then diagonal map is an isomorphism and so multiplication by $\lambda$ is also an isomorphism. If $a+p>0$, then the diagonal map is zero and so multiplication by $\lambda$ is zero in this case. Lastly, since $H^{a+p\sigma, b+q\sigma}_{C_2}(\C,\mathds{Z}/2)$ is zero if $a>2b+1$, multiplication by $\lambda$ is zero if $a>2b+1$.
 	
 	 
 	
 
 	A similar argument shows that multiplication by $\mu$ is an isomorphism when $b+q<0$ and $a\leq 2b+1$ 
 \end{proof}
 





\section{Bredon motivic cohomology of algebraically closed fields} 
   In this section we consider an algebraically closed field $k$ and a natural number $n>1$ coprime to $char(k)$. 
   Let $V$ be a $C _2$-equivariant smooth scheme over $k$.
   First we note a rigidity theorem for rational points:
   
   \begin{theorem} \label{sec} For a connected smooth scheme $X$ over $k$ and $k$-rational points $x_{0}$, $x_{1}$ of $X$, 
   $$(x_ 0) _*=(x_ 1) _*: H^{\star,\star} _{C _2}(V\times X,\mathds{Z}/n)\rightarrow H^{\star,\star} _{C _2}(V,\mathds{Z}/n).$$ 
   \end{theorem}  
 
 According to \cite{YO}, Theorem \ref{sec} follows if the functor $F(-)=H^{\star,\star} _{C _2}(V\times -,\mathds{Z}/n)$ is a homotopy invariant presheaf on $Sm/k$ with weak transfers in the sense of \cite{YP}. The four conditions that need to be fulfilled according to \cite{YO}  are:
 
 1) Additivity: For $X=X_0\sqcup X _1$ with corresponding embeddings $i _m:X _m\hookrightarrow X$ for $m=0,1$ and $f: X\rightarrow Y$ a map in $Sm/k$, 
     we have $f _*=(fi _0) _*i _0^*+(fi _1) _*i _1^*$
 
 2) Base change: For every finite flat map $f$, closed embedding $g$, and cartesian diagram: 
 
  \[
\xymatrixrowsep{1cm}
\xymatrixcolsep {1cm}
\xymatrix@-0.9pc{ 
X' \ar[r]^{g _1}\ar[d]^{f _1}& 
Y'\ar[d]^{f} 
\\
X\ar[r]^{g} &
Y
}
\]

 we have $g^*f _*=f_{1*}{g _1}^*$
 
3) Normalization: If $f$ is the identity map on $k$ then $f _*=id_{H^{\star,\star}_{C_2}(V,\mathds{Z}/n)}$

4) Homotopy invariance: The rational points 0 and 1 of the affine line $\mathbb{A}^1 _k$ with trivial $C _2$-action yield equal pullback maps
$$0_*=1_*:H^{\star,\star} _{C _2}(V\times _k\mathbb{A}^1 _k)\rightarrow H^{\star,\star} _{C _2}(V).$$
 
 The functor $F$ fulfills all four conditions above as it is a homotopy invariant presheaf with equivariant transfers (\cite{HOV1}). 
 Moreover, because it is a homotopy invariant presheaf with equivariant transfers, according to \cite{Sus}, \cite{YO}, \cite{YP}, 
 we have the following theorem:
 
 \begin{theorem} \label{ext} Suppose $k\subset K$ is an extension of algebraically closed fields and $X$ is a smooth $C_2$-equivariant scheme. 
 If $n$ is coprime to $char(k)$, then $\pi: Spec(K)\rightarrow Spec(k)$ induces an isomorphism: 
 $$\pi^*: H^{\star,\star} _{C _2}(X,\mathds{Z}/n)\simeq H^{\star,\star} _{C _2}(X _K,\mathds{Z}/n)$$
 \end{theorem}
 \begin{proof} We can write $Spec(K)=lim _U (U)$, where $U$ is an affine smooth variety over $Spec(k)$. There is an induced map $$\pi^*:H^{\star,\star} _{C _2}(X,\mathds{Z}/n)\rightarrow H^{\star,\star} _{C _2}(X\times K,\mathds{Z}/n)=colim_U H^{\star,\star} _{C _2}(X\times U,\mathds{Z}/n)$$ so if $\pi^*(x)=0$ then there exists a map $\phi: U\rightarrow Spec (k)$ such that $\phi^*(x)=0$. Because $U$ has a $k$-rational point, $\phi$ yields a splitting and $\phi^*$ is injective. This implies $x=0$ so $\pi^*$ is injective.
 
 Next we show that $\pi^*$ is surjective. For every $\beta\in H^{a+p\sigma,b+q\sigma} _{C _2}(X\times K)$ there exists a map $\phi: Spec (K)\rightarrow U$ such that 
 $\phi^*(\beta ')=\beta$ with $\beta '\in H^{\star,\star} _{C _2}(X\times U)$. 
 If $\xi: Spec(k)\rightarrow U$ a rational point, 
 the maps $\xi\circ \pi$, $\phi: Spec (K)\rightarrow U$ induce $K$-rational points $\phi',\xi': Spec(K)\rightarrow U _K$. According to Theorem \ref{sec} we have that
 $$\phi^{'*}=\xi^{'*}: H^{\star,\star} _{C _2}(X\times U\times K)\rightarrow H^{\star,\star} _{C _2}(X\times K).$$
 For the base change $\underline{\pi}:U _K\rightarrow U$, we have 
$\beta- \pi^*\circ \xi^*(\beta')=\phi^*(\beta')-\pi^*\circ \xi^*(\beta ')=(\phi^{'*}-\xi^{*'})(\underline{\pi}^*(\beta'))=0$, and thus $\beta\in Im(\pi^*)$. 
 \end{proof}
 
 The next corollary computes Bredon and Borel motivic cohomology for algebraically closed fields of characteristic zero.
 \begin{corollary} Let  $K$ an algebraically closed field of characteristic zero and $n>1$. Then $$H^{a+p\sigma,b+q\sigma} _{C _2}(K,\mathds{Z}/n)\simeq H^{a+p\sigma,b+q\sigma} _{C _2}(\C,\mathds{Z}/n)$$
 and $$H^{a+p\sigma,b+q\sigma} _{C _2}(\mathbf{E} C _{2K},\mathds{Z}/n)\simeq H^{a+p\sigma,b+q\sigma} _{C _2}(\mathbf{E} C _2,\mathds{Z}/n),$$
   for any choice of integers $a,b,p,q$. Moreover, the Bredon and Borel cohomology rings of an algebraically closed field of characteristic zero (with finite coefficients) are isomorphic to the Bredon and Borel cohomology rings of the complex numbers (with finite coefficients) computed in the previous section.
 \end{corollary}
  
 \bibliographystyle{plain}
 