# Is there a general projection formula for morphisms of ringed topoi?

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:

1. 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 $(Rf_*F)\otimes^L G=Rf_*(F\otimes^L Lf^*G)$.

2. 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 $f_*?$

• Perhaps you could clarify your question a bit. Explain what you already know and what you'd like to know. For example, you might rewrite your question as, "For an arbitrary morphism $f$ of ringed topoi, is it true that the natural map $Rf_*(F)\otimes^L E\to Rf_*(F\otimes^L Lf^*(E))$ 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(?)." Mar 31, 2010 at 2:43
• You may also want to change your title to something more descriptive, like "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 Mar 31, 2010 at 2:43
• Thanks for the advice, Anton. I will edit it a bit later. Mar 31, 2010 at 4:58
• Here is a reference that perhaps addresses the question: stacks.math.columbia.edu/tag/0944 Jul 15, 2016 at 20:18

In the context of sheaves of $\mathcal O_X$-modules, there is the following reference: Prop. 3.9.4 in Lipman's Notes on derived functors and Grothendieck duality. A closely related result is in Neeman's paper The Grothendieck duality theorem ...; see Prop. 5.3.