What's the general projection formula in algebraic geometry, for instance on the level of derived categories of ringed topoi? And what's the reference? I guess it might be in SGA 4, but couldn't find it.

Two examples:

Zariski site, $D^b_{qcoh}$ on schemes. Let $f:X\to Y$ be a proper map of noetherian schemes (maybe there are some other mild conditions), and let $F\in D^b_{qcoh}(X)$ and $G\in D^b_{qcoh}(Y).$ Then $(Rf_*F)\otimes^L G=Rf_*(F\otimes^L Lf^*G)$.

Etale site, say ringed by a torsion ring like $Z/n.$ Let $f:X\to Y$ be a (seperated; but this condition can be removed. See for instancde Laszlo and Olsson, The six operations on Artin stacks...) map of schemes of finite type over some base $S$ ($S$ may need to satisfy some assumptions, in order for the classical results in SGA 4/4.5 or Gabber's new results on finiteness of $f_*$ and dualizing complexes to work; but let's be sloppy). Let $F\in D^-_c(X,Z/n)$ and $G\in D^-_c(Y,Z/n).$ Then $Rf_!F\otimes^L G=Rf_!(F\otimes^L f^*G).$

We used $f_!$ in example 2 in order to allow $F$ and $G$ to be in $D^-_c$ rather than $D^b_c.$ If one restricts to $D^b_c,$ is it also true for $f_*?$

"For an arbitrary morphism $f$ of ringed topoi, is it true that the natural map$\endgroup$`$Rf_*(F)\otimes^L E\to Rf_*(F\otimes^L Lf^*(E))$`

is an isomorphism, where $F$ and $E$ are in the derived categories of coherent sheaves? I know this is true when $f$ is a morphism of schemes(?).""Does the projection formula hold for derived categories of ringed topoi?"and add the tags [derived-category] and [reference-request]. See also mathoverflow.net/howtoask $\endgroup$