In the web page <http://www.encyclopediaofmath.org/index.php/Moore_space> it can be found the following statement:



> 
> If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers $\mathbb Z$ and $M\_k(G)$ is the Moore space with $\tilde{H}\_k(M\_k(G))=G$, then
> 
> 
> $$lim\_{N\rightarrow\infty}[\Sigma^{N+k}X,K(\mathbb Z, N+n)\bigwedge M\_k(G)]\cong
>  H^n(X,G)$$,
> 
> 
> that is, $\{K(\mathbb Z,n)\wedge M\_k(G)\}$ is the spectrum of the cohomology theory $H^\*(\ ,G)$.
> 
> 
> 


The reference of the webpage is the articule of Moore ´´On homotopy groups of spaces with single non-vanishing homotopy group´´ but I don´t find anything like that on the article. 


It would be very helpful if you could give a reference where I can see the explanation or you can explain me that.



Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?
Let $H\mathbb{Z}$ denote the spectrum for cohomology with coefficients in $\mathbb{Z}$, so your spectrum is $H\mathbb{Z}\wedge M\_k(G)$. Let $$0\to\bigoplus\_I\mathbb{Z}\stackrel{f}{\to}\bigoplus\_J\mathbb{Z}\to G\to 0$$ be a presentation of $G$. Then we can explicitly construct a Moore space $M\_k(G)$ as the cofiber of a map $\bigvee\_I S^k\to \bigvee\_J S^k$ whose induced map on homology is $f$. In spectra, we then have a cofiber sequence $$\bigvee\_I \Sigma^k H\mathbb{Z}\to\bigvee\_J\Sigma^k H\mathbb{Z}\to H\mathbb{Z}\wedge M\_k(G).$$


For any pointed finite CW-complex $X$, we then have a long exact sequence in cohomology of the form $$\dots\to\bigoplus\_IH^{n+k}(X,\mathbb{Z})\to \bigoplus\_J H^{n+k}(X,\mathbb{Z})\to H\mathbb{Z}\wedge M\_k(G)^n(X)\to\dots$$


Now consider $X=S^0$. The long exact sequence above computes that $H\mathbb{Z}\wedge M\_k(G)^{-k}(S^0)=G$ and $H\mathbb{Z}\wedge M\_k(G)^n(S^0)=0$ for $n\neq -k$. Thus up to a degree shift, the cohomology theory $H\mathbb{Z}\wedge M\_k(G)$ satisfies the Eilenberg-Steenrod dimension axiom, so it must coincide with $H^{\*+k}(X,G)$. 


