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For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection formula, and other relatively abstract places. As a somewhat vague question, I'd like to see some non-trivial examples of how to interpret $f^*f_* \mathcal{F}$, or typical examples where this is an important construction. How about the $f^*f_*$ map on sheaves in other sites, such as the etale site?

Some context:

1.) Pullback of coherent sheaves is a fairly natural operation. I'm also perfectly happy with the pushforwards and higher pushforwards. Specifically, Mumford (in "Abelian Varieties") gives some fairly general results about how to interpret stalks of such sheaves (if $f$ is proper flat and $\mathcal{F}$ is locally free or more generally flat over $Y$) in terms of cohomology groups of $\mathcal{F}$ on fibers. More generally the Leray spectral sequence gives you a context to "interpret" higher pushforwards. If the pushforward of the structure sheaf is again the structure sheaf then Stein factorization gives us information about the fibers. More generally we have the theorem of formal functions.

2.) For finite degree $n$ maps of smooth proper curves over $\bar{k}$, the related map$f_*f^*$ can be defined at the level of the Picard group, where it is the multiplication by $n$ map.

$\textbf{Question}$ How should I think about Thm III.8.8 of Hartshorne (below)? Is this just a useful technical result (and where?), or is it more than that? Are there good examples? Affine locally one can certainly describe the natural map $f^*f_*(\mathcal{F}) \rightarrow \mathcal{F}$ explicitly. Is this how one "works" with it in practice, or are there better ways (I hope)?

$\textbf{Thm}$ III.8.8: If $f:X \rightarrow Y$ is a projective map of noetherian schemes, $\mathcal{F}$ coherent on $X$ and $\mathcal{O}_X(1)$ a very ample sheaf on $X$ over $Y$, then $\forall n>> 0$, the natural map $f^*f_*(\mathcal{F}(n)) \rightarrow \mathcal{F(n)}$ is surjective.

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Dear LMN, Why don't you first think about the case when $X$ is a $k$-scheme for some field $k$, and $f:X \to $ Spec $k$ is the natural map. Then $f^* f_* \mathcal F$ is the $\mathcal O_X$-module generated by the global sections of $\mathcal F$, and so the Hartshorne thm. that you ask about is giving a condition for $\mathcal F(n)$ to be generated by global sections, and is much more than a mere technical result. The general case is a relative version of the case of mapping to a point. Regards, Matthew –  Emerton Oct 28 '12 at 3:47
Dear Matt, That's great, Thanks! So one interprets it as a relative version of the case we already know. In hindsight this is much more clear. –  LMN Oct 28 '12 at 4:30

2 Answers 2

To me, pushforword is like taking sections along the fibers, and higher pushforwards are like cohomologies along the fiber.

Think about that $f^*f_*F\to F$ being surjective as globally generated along the fibers, then Hartshorn 8.8 is just a relative version of that twisted by ample you will eventually get globally generated sheaf.

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Let both $f_*$ and $f^*$ be derived pushforard and pullback. Assume that $f_*O_X = O_Y$. Then there is a semiorthogonal decomposition of the derived category $$ D(X) = \langle Ker f^*, f^*(D(Y)) \rangle $$ and $f^*f_*$ is the projection onto the second component.

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