In his 1987 article The geometry of toric varieties Danilov gives a combinatorial presentation of Chow rings of complete smooth toric varietes. Given a complete unimodular fan $\Sigma$ we have $$ A^*(X_\Sigma) \cong \mathbb Z[U]/(I+J), $$ where $U$ is the set of rays in $\Sigma$, $I$ the Stanley-Reisner ideal of the associated simplicial complex (on the rays) and $J$ an ideal depending on the coordinates of the minimal ray generators.

In the recent preprint Hodge theory for combinatorial geometries by Adiprasito, Huh and Katz, the authors state that this isomorphism still holds when $\Sigma$ isn't complete. They cite the following two articles for this:

  • Bifet, De Concini, Procesi, Cohomology of regular embeddings, 1990
  • Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry, 1996

However, I'm unable to extract this result from those articles. Maybe someone can point out why this is implied by the results in the cited articles or possibly knows another reference?


1 Answer 1


If I understand correctly the result you want is the corollary at the end of section 3.1 of the Brion article you mention which states $$R_{\Sigma} / M R_{\Sigma} \cong A^*(X).$$

Notice in section 1.3 of the Brion article at the bottom of page 28 it states $R_{\Sigma}$ is isomorphic to the Stanley-Reisner algebra when $\Sigma$ is a simplicial fan. Also in section 1.4 on page 29 we see that $M R_{\Sigma}$ denotes the ideal generated by linear forms, which is the $J$ in the question.

  • $\begingroup$ I think their $R_\Sigma$ is $\mathbb Q[U]/I$ and the chow ring is with rational coefficients, isn't it? I'm looking at integral coefficients. $\endgroup$
    – Christoph
    Commented Sep 9, 2016 at 17:34
  • $\begingroup$ Yes Brion uses rational coefficients. But perhaps we can still make the argument go through with integral coefficients. The exact sequence in the proof of the theorem before the corollary should still work I think because we know we have the isomorphism for complete fans with integral coefficients. $\endgroup$ Commented Sep 9, 2016 at 23:07

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