A remark on the product by the hyperplane class in the Chow ring of some complete intersections

By classical calculation, for a smooth hypersurface $Y\subset {\mathbb{P}}_{\mathbb{C}}^{n+1}$, the product by the hyperplane class is zero on homologically trivial rational cycles i.e. $H_{\mathrm{\mid }Y}\cdot :{\mathrm{C}\mathrm{H}}_i(Y{)}_{hom,\mathbb{Q}}\to {\mathrm{C}\mathrm{H}}_{i-1}(Y{)}_{hom,\mathbb{Q}}$ is $0$ for any $i$. This note extends that result to some complete intersections.