On the integral cohomology of toric varieties (Q2788572)

The main result of this paper claims the following:NEWLINENEWLINELet \(\Sigma\) be a polytopal, simplicial fan in the lattice \(N\) and let \(X_\Sigma\) be the associated projective toric variety. Assume that \(H^*(X_\Sigma;\mathbb{Z})\) is concentrated in even degrees. Then the integral cohomology algebra \(H^*(X_\Sigma;\mathbb{Z})\) of \(X_\Sigma\) is isomorphic to \(\mathbb{Z}[\Sigma]/J_\Sigma\), where \(\mathbb{Z}[\Sigma]\) denotes the Stanley-Reisner algebra of \(\Sigma\) and \(J_\Sigma\) denotes the ideal generated by linear relations determined by \(\Sigma\).NEWLINENEWLINEUnfortunately this claim is false as the following example (provided by Soumen Sarkar) shows:NEWLINENEWLINELet \(\Sigma\) be the complete fan in \(\mathbb{R}^2\) generated by the rays NEWLINE\[NEWLINEu=(1,0);\;v=(0,1);\;w=(-1,-2).NEWLINE\]NEWLINE Then \(X_{\Sigma}\) is the weighted projective space \(\mathbb{P}(1,1,2)\). Moreover, by the above claim NEWLINE\[NEWLINEH^*(X_\Sigma;\mathbb{Z})=\mathbb{Z}[u,v,w]/(uvw,\;u-w,\;v-2w)=\mathbb{Z}[w]/(2w^3),NEWLINE\]NEWLINE which is non-trivial in infinitely many degrees contradicting the fact that \(\mathbb{P}(1,1,2)\) is finite dimensional.