Say I have a closed embedding $i : S \hookrightarrow Y$ where $Y$ is a degree $d$ smooth projective threefold, and $S$ is a hyperplane section of $Y$. Let $H_Y$ be the polarization of $Y$, and let $H_S$ be the polarization of $S$ (I guess $H_S = S \cap H_Y$), so $H_S^2$ is a degree $2$ class in the Chow ring of $S$.
Q1: Am I correct in thinking that $i_*(H_S^2) = H_Y^3$?
In $S$, $H_S^2 = dP_S$ where $P_S$ is the class of a point in $S$ (but really, I think $H_S$ here can be replaced by any hyperplane of $S$, since we're working modulo rational equivalence), and $i$ is just an inclusion, so a point in $S$ should be taken to that same point, now considered as a point in $Y$. Since a point $P_Y$ in $Y$ satisfies $H_Y^3=dP_Y$, I think we should have $i_*(H_S^2) = H_Y^3$ - is this correct?
Secondly, a Grothendieck-Riemann-Roch calculation involving $i : S \hookrightarrow Y$ and a sheaf $\mathcal{F}$ on $S$ would involve computing $i_*(\mathrm{ch}(\mathcal{F}) . \mathrm{td}(S))$ as the right-hand side of the GRR formula.
Q2: Would it be correct to be using $H_S$ when explicitly writing out $\mathrm{ch}(\mathcal{F})$ and $\mathrm{td}(S)$ (and $H_Y$ when working out the left-hand side) in my situation of $i : S \hookrightarrow Y$ a hyperplane section?
Thanks.