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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 $f_*?$

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  • $\begingroup$ 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(?)." $\endgroup$
    – Anton Geraschenko
    Commented Mar 31, 2010 at 2:43
  • $\begingroup$ 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 $\endgroup$
    – Anton Geraschenko
    Commented Mar 31, 2010 at 2:43
  • $\begingroup$ Thanks for the advice, Anton. I will edit it a bit later. $\endgroup$
    – shenghao
    Commented Mar 31, 2010 at 4:58
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    $\begingroup$ Here is a reference that perhaps addresses the question: stacks.math.columbia.edu/tag/0944 $\endgroup$
    – Kestutis Cesnavicius
    Commented Jul 15, 2016 at 20:18

1 Answer 1

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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.)

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  • $\begingroup$ 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? $\endgroup$
    – shenghao
    Commented Apr 1, 2010 at 4:58
  • $\begingroup$ Corollary 12.11, SGA4, Expose 4 is called "projection formula" but it is not as general as you might like. $\endgroup$
    – S. Carnahan ♦
    Commented Apr 1, 2010 at 5:27
  • $\begingroup$ @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). $\endgroup$
    – David Zureick-Brown
    Commented Apr 1, 2010 at 6:12

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