Polynomial differential form

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the ring of polynomial differential forms on the standard <a href="/facts/Simplex/0FCfNRN8">n-simplex</a> is the <a href="/facts/Differential_graded_algebra/zySud4Kl">differential graded algebra</a>:

Ω
          
            poly
          
          
            ∗
          
        
        (
        [
        n
        ]
        )
        =
        
          Q
        
        [
        
          t
          
            0
          
        
        ,
        .
        .
        .
        ,
        
          t
          
            n
          
        
        ,
        d
        
          t
          
            0
          
        
        ,
        .
        .
        .
        ,
        d
        
          t
          
            n
          
        
        ]
        
          /
        
        (
        ∑
        
          t
          
            i
          
        
        −
        1
        ,
        ∑
        d
        
          t
          
            i
          
        
        )
        .
      
    
    {\displaystyle \Omega _{\text{poly}}^{*}([n])=\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\sum t_{i}-1,\sum dt_{i}).}

Varying n, it determines the <a href="/facts/Simplicial_commutative_ring/6LktuDH8">simplicial commutative dg algebra</a>:

Ω
          
            poly
          
          
            ∗
          
        
      
    
    {\displaystyle \Omega _{\text{poly}}^{*}}

(each 
 
 
 
 u
 :
 [
 n
 ]
 →
 [
 m
 ]
 
 
 {\displaystyle u:[n]\to [m]}
 
 induces the map 
 
 
 
 
 Ω
 
 poly
 
 
 ∗
 
 
 (
 [
 m
 ]
 )
 →
 
 Ω
 
 poly
 
 
 ∗
 
 
 (
 [
 n
 ]
 )
 ,
 
 t
 
 i
 
 
 ↦
 
 ∑
 
 u
 (
 j
 )
 =
 i
 
 
 
 t
 
 j
 
 
 
 
 {\displaystyle \Omega _{\text{poly}}^{*}([m])\to \Omega _{\text{poly}}^{*}([n]),t_{i}\mapsto \sum _{u(j)=i}t_{j}}
 
).

<ul><li>Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2 of: On PL <a href="/facts/De_Rham_theorem/4qGKeJq0">De Rham Theory</a> and <a href="/facts/Rational_homotopy_type/b91f4L7z">Rational Homotopy Type</a>, Memoirs of the A. M. S., vol. 179, 1976.</li>
<li>Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/q-alg/9702015">q-alg/9702015</a>.</li></ul>

Polynomial differential form open-in-new

Polynomial differential form