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Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$.

For a simple polytope $P$ let us denote by $\Pi(P)$ the subalgebra of $\Pi$ generated by Minkowski summands of $P$. ($Q$ is a Minkowski summand of $P$ if $P=Q+Q'$ for another convex polytope $Q'$.

As far as I heard the algebra $\Pi(P)$ is isomorphic to the cohomology algebra of the toric variety corresponding to $P$.

Question. I would like to have a reference to this statement and a construction of the isomorphism. (The case of a smooth toric variety would be sufficient for me for the moment.)

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    $\begingroup$ Brion has an article called "THE STRUCTURE OF THE POLYTOPE ALGEBRA" in which he cites Fulton and Sturmfels's "Intersection theory on toric varieties" for the relation between the rational polytope algebra and cohomology of toric varieties. Perhaps there you will find something useful (I know almost nothing about toric varities so I can't tell if this answers your question). $\endgroup$
    – efs
    Commented Nov 13, 2021 at 14:22

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