Suppose $\pi : Y \to X$ is a finite 'etale map of degree d. I want a formula for $\pi_* \mathcal O_Y$. I'm happy with a formula in $K$ theory.
There is a $S_d$-torsor $P \to X$ of local isomorphisms $Y \simeq \bigsqcup X$ of $Y$ with the trivial cover over $X$. The quotient of $P$ by $S_{d-1}$ is $Y$. So sheaves on $X$ are $S_d$-equivariant sheaves on $P$, while sheaves on $Y$ are $S_{d-1}$-equivariant ones.
The pushforward of a sheaf $F$ along a $G$-torsor ($G$ finite) is $\bigoplus_{g \in G} g^*F$ I believe.
Write $\sigma = (123 \cdots d)$. I can't shake the feeling that the pushforward $\pi_*F$ should be $\bigoplus_{0 \leq k \leq d-1} (\sigma^k)^*F$. I'm not sure how to endow this with $S_d$ action because each $(\sigma^k)^*F$ should be invariant under a different copy of $S_{d-1} \subseteq S_d$.
Is the above correct?
Are there hypotheses under which the formula $\pi_*F = \bigoplus_{0 \leq k \leq d-1} (\sigma^k)^*F$ makes sense/is true? Note $\mathcal O_Y = \pi^*\mathcal O_X$ actually corresponds to an $S_d$ equivariant sheaf; I really only need a formula for $\pi_* \pi^*G$ or I think even $\pi^* \pi_*G$. In chow, $\pi_* \pi^* = d \cdot$ is multiplication by $d$, but $K$ theory is not so lucky.
Are there any references for pushforward formula in $K$ theory I should know of?
