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

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