Adams resolution

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, there is a resolution analogous to <a href="/facts/Free_resolution/xEFm83U7">free resolutions</a> of <a href="/facts/Spectrum_(topology)/mWho3roy">spectra</a> yielding a tool for constructing the <a href="/facts/Adams_spectral_sequence/XPqI80V4">Adams spectral sequence</a>. Essentially, the idea is to take a <a href="/facts/Connective_spectrum/VvKBoEMT">connective spectrum</a> of finite type 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in 
  
    
      
        
          H
          
            ∗
          
        
        (
        X
        ;
        
          Z
        
        
          /
        
        p
        )
      
    
    {\displaystyle H^{*}(X;\mathbb {Z} /p)}
  
 using <a href="/facts/Eilenberg%E2%80%93Maclane_spectrum/F3yELuIr">Eilenberg–MacLane spectra</a>.
</p><p>This construction can be generalized using a spectrum 
  
    
      
        E
      
    
    {\displaystyle E}
  
, such as the <a href="/facts/Brown%E2%80%93Peterson_cohomology/ASNxFGc6">Brown–Peterson spectrum</a> 
  
    
      
        B
        P
      
    
    {\displaystyle BP}
  
, or the <a href="/facts/Complex_cobordism/tAkFKIL1">complex cobordism</a> spectrum 
  
    
      
        M
        U
      
    
    {\displaystyle MU}
  
, and is used in the construction of the <a href="/facts/Adams%E2%80%93Novikov_spectral_sequence/XPqI80V4">Adams–Novikov spectral sequence</a>pg 49.
</p>

Adams resolution open-in-new

Adams resolution