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I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of: $$ \mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^2+1)^{-1},(r^2+2)^{-1},\dots] $$ where $r^2=x_1y_1+\dots+x_ny_n$. In other words we are taking the polynomial algebra $k[x_1,\dots,x_n,y_1,\dots,y_n]$ and inverting all functions $(r^{2}+k)$ for $k\in\mathbb{Z}_{>0}$.

Are there any known techniques/tricks for this? I'm hoping for my application that the cohomology is just the constant functions, so if you could at least explain why that is/isn't true I would be very thankful :)

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    $\begingroup$ I would guess that the closed 1-forms $d\log(r^2 + k) = 2rdr/(r^2+k)$ are linearly independent in $H^1_{\rm dR}$, making it infinite-dimensional. $\endgroup$
    – Piotr Achinger
    Commented Sep 13, 2021 at 13:32

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