# 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(?)." – Anton Geraschenko Mar 31 '10 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 – Anton Geraschenko Mar 31 '10 at 2:43
• Thanks for the advice, Anton. I will edit it a bit later. – shenghao Mar 31 '10 at 4:58
• Here is a reference that perhaps addresses the question: stacks.math.columbia.edu/tag/0944 – Kestutis Cesnavicius Jul 15 '16 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.

I'm not sure that analogous results should be expected to hold in arbitrary generality; for example, both references place a restriction on the base scheme, and require quasi-coherence assumptions. (In some sense, one has to reduce to the locally free case, where the statement is obvious. Quasi-coherent sheaves then admit locally free resolutions. The proofs of the cited results apply some form of this argument in rather subtle and sophisticated ways.)

• Thanks, prof. Emerton. What about the etale case (which is the case I'm more interested in)? Do you know if it's in SGA 4? In K. Behrend's 'Derived l-adic categories for algebraic stacks', corollary 6.1.3, in the proof he says "This follows from the general projection formula for morphisms of ringed topoi ...", but doesn't give a precise reference, so I was thinking maybe there are some folklore that I was missing? – shenghao Apr 1 '10 at 4:58
• Corollary 12.11, SGA4, Expose 4 is called "projection formula" but it is not as general as you might like. – S. Carnahan Apr 1 '10 at 5:27
• @Shenghao: In corollary 6.1.3, the morphism is smooth, and so morphism on lisse-etale sites is of the form discussed in SGA4, expose 4, e.g. given via the localization/restriction/comma category construction (so that for example the functor f^* is really easy to describe). – David Zureick-Brown Apr 1 '10 at 6:12