The de Rham cohomology of the algebra of polynomial functions on a simplicial complex
https://doi.org/10.22405/2226-8383-2023-24-1-203-212
Abstract
We consider the algebra 𝐴0(𝑋) of polynomial functions on a simplicial complex 𝑋. The algebra 𝐴0(𝑋) is the 0th component of Sullivan’s dg-algebra 𝐴∙(𝑋) of polynomial forms on 𝑋.
All algebras are over an arbitrary field 𝑘 of characteristic 0.
Our main interest lies in computing the de Rham cohomology of the algebra 𝐴0(𝑋), that is, the cohomology of the universal dg-algebra Ω∙
𝐴0(𝑋). There is a canonical morphism of dgalgebras 𝑃 : Ω∙
𝐴0(𝑋) → 𝐴∙(𝑋). We prove that 𝑃 is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra 𝐴0(𝑋) is canonically isomorphic to the cohomology of the simplicial complex 𝑋 with coefficients in 𝑘. Moreover, for 𝑘 = Q the dg-algebra Ω∙
𝐴0(𝑋) is a model of the simplicial complex 𝑋 in the sense of rational homotopy theory. Our result shows that for the algebra 𝐴0(𝑋) the statement of Grothendieck’s comparison theorem holds (proved
by him for smooth algebras).
In order to prove the statement we consider ˇCech resolution associated to the cover of the simplicial complex by the stars of the vertices.
Earlier, Kan–Miller proved that the morphism 𝑃 is surjective and gave a description of its kernel. Another description of the kernel was given by Sullivan and F´elix–Jessup–Parent.
About the Author
Igor Sergeevich Baskov
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
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Review
For citations:
Baskov I.S. The de Rham cohomology of the algebra of polynomial functions on a simplicial complex. Chebyshevskii Sbornik. 2023;24(1):203-212. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-1-203-212
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