The de Rham cohomology of the algebra of polynomial functions on a simplicial complex

Q4 Mathematics
Igor Baskov
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引用次数: 0

Abstract

We consider the algebra $A^0 (X)$ of polynomial functions on a simplicial complex $X$. The algebra $A^0 (X)$ is the $0$th component of Sullivan's dg-algebra $A^\bullet (X)$ of polynomial forms on $X$. Our main interest lies in computing the de Rham cohomology of the algebra $A^0(X)$, that is, the cohomology of the universal dg-algebra $\Omega ^\bullet _{A^0(X)}$. There is a canonical morphism of dg-algebras $P:\Omega ^\bullet _{A^0(X)} \to A^\bullet (X)$. We prove that $P$ is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra $A^0 (X)$ is canonically isomorphic to the cohomology of the simplicial complex $X$ with coefficients in $k$. Moreover, for $k=\mathbb{Q}$ the dg-algebra $\Omega ^\bullet _{A^0 (X)}$ is a model of the simplicial complex $X$ in the sense of rational homotopy theory.
简单复上多项式函数代数的de Rham上同调
我们考虑简单复数X上多项式函数的代数A^0 (X)。代数$A^0 (X)$是在$X$上多项式形式的Sullivan's dg-algebra $A^\bullet (X)$的第0个分量。我们的主要兴趣在于计算代数$A^0(X)$的de Rham上同调,也就是计算泛代数$ Omega ^\bullet _{A^0(X)}$的上同调。g-代数$P:\Omega ^\bullet _{a ^0(X)} \到a ^\bullet (X)$的正则态射。证明$P$是一个拟同构。因此,代数$A^0 (X)$的de Rham上同构与系数在$k$中的单纯复数$X$的上同构。此外,对于$k=\mathbb{Q}$, g-代数$\Omega ^\bullet _{A^0 (X)}$是在有理同伦理论意义上的单纯复$X$的一个模型。
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来源期刊
Chebyshevskii Sbornik
Chebyshevskii Sbornik Mathematics-Mathematics (all)
CiteScore
0.60
自引率
0.00%
发文量
19
期刊介绍: The aim of the journal is to publish and disseminate research results of leading scientists in many areas of modern mathematics, some areas of physics and computer science.
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