Filtered algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a filtered algebra is a generalization of the notion of a <a href="/facts/Graded_algebra/AIUjkSaP">graded algebra</a>. Examples appear in many branches of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially in <a href="/facts/Homological_algebra/Jfm8w52l">homological algebra</a> and <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>. 
A filtered algebra over the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> 
 
 
 
 (
 A
 ,
 ⋅
 )
 
 
 {\displaystyle (A,\cdot )}
 
 over 
 
 
 
 k
 
 
 {\displaystyle k}
 
 that has an increasing sequence 
 
 
 
 {
 0
 }
 ⊆
 
 F
 
 0
 
 
 ⊆
 
 F
 
 1
 
 
 ⊆
 ⋯
 ⊆
 
 F
 
 i
 
 
 ⊆
 ⋯
 ⊆
 A
 
 
 {\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A}
 
 of subspaces of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 such that

A
        =
        
          ⋃
          
            i
            ∈
            
              N
            
          
        
        
          F
          
            i
          
        
      
    
    {\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i}}

and that is compatible with the multiplication in the following sense:

∀
        m
        ,
        n
        ∈
        
          N
        
        ,
        
        
          F
          
            m
          
        
        ⋅
        
          F
          
            n
          
        
        ⊆
        
          F
          
            n
            +
            m
          
        
        .
      
    
    {\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.}

Filtered algebra open-in-new

Filtered algebra