Derived tensor product

In algebra, given a <a href="/facts/Differential_graded_algebra/zySud4Kl">differential graded algebra</a> A over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> R, the derived tensor product functor is

−
        
          ⊗
          
            A
          
          
            
              L
            
          
        
        −
        :
        D
        (
        
          
            
              M
            
          
          
            A
          
        
        )
        ×
        D
        (

A
          
        
        
          
            M
          
        
        )
        →
        D
        (

R
          
        
        
          
            M
          
        
        )
      
    
    {\displaystyle -\otimes _{A}^{\textbf {L}}-:D({\mathsf {M}}_{A})\times D({}_{A}{\mathsf {M}})\to D({}_{R}{\mathsf {M}})}

where 
 
 
 
 
 
 
 M
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathsf {M}}_{A}}
 
 and

A
 
 
 
 
 M
 
 
 
 
 {\displaystyle {}_{A}{\mathsf {M}}}
 
 are the <a href="/facts/Category_of_modules/AnnF9JKf">categories of right A-modules</a> and left A-modules and D refers to the homotopy category (i.e., <a href="/facts/Derived_category/pBQHnFYo">derived category</a>). By definition, it is the left derived functor of the <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product functor</a> 
 
 
 
 −
 
 ⊗
 
 A
 
 
 −
 :
 
 
 
 M
 
 
 
 A
 
 
 ×

A
          
        
        
          
            M
          
        
        →

R
 
 
 
 
 M
 
 
 
 
 {\displaystyle -\otimes _{A}-:{\mathsf {M}}_{A}\times {}_{A}{\mathsf {M}}\to {}_{R}{\mathsf {M}}}
 
.

Derived tensor product open-in-new

Derived tensor product