# Cohomology of inverse image sheaf

It seems to me that in Travaux de Griffiths (equation 3.2.1), Deligne casually mentions a theorem along the following lines:

Let $f: X \to B$ be a proper map of topological spaces. Suppose $\mathcal{A}$ denotes one of the constant sheaves $\mathbb{Z}, \mathbb{C}$ (on any space). Let $\mathcal{B}$ be an $\mathcal{A}$-module on $B$. Then there exists a canonical isomorphism $$\mathcal{B} \otimes_{\mathcal{A}_B} R^n f_* (\mathcal{A}_X) \to R^n f_* (f^{-1} \mathcal{B}).$$

Where does this morphism come from in the first place (presumably adjointness of $f_*$, $f^{-1}$?), and why is it an isomorphism? It feels to me like this should be a standard fact (but I haven't come across it so far), any reference would be appreciated.

Note: in the context of this article, $X$, $B$ are complex manifolds and $f$ is smooth and submersive. My gut feeling is that this should not matter here.

Thanks in advance, Tom

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## 1 Answer

Not an answer, just a long comment: This is probably related to what I know as the projection formula. The following is given in Qing Liu's book Algebraic Geometry and Arithmetic Curves on page 190 as Proposition 5.2.32:



Edit In fact, I don't have any other reference, but this post on mathoverflow quotes the formula for ringed spaces.

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 This question references Ravil Vakil's notes: math.stackexchange.com/questions/175807/… I also sketch where the map comes from. – Matt Sep 19 '12 at 13:03 This doesn't use properness at all, whereas the article would make me believe that assumption is crucial ... – Tom Bachmann Sep 20 '12 at 8:15