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What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ \mathcal{ O }_Y $ - modules, then $ ( f^* , f_*) $ is a pair of adjoint functors, i.e that $ \mathrm{ Hom } ( f^* \mathcal{G }, \mathcal{F} ) \simeq \mathrm{ Hom } ( \mathcal{G} , f_* \mathcal{ O }_Y ) $ ?

What is meant concretly by saying that $ f^* $ and $ f_* $ are adjoints ?

Thank you in advance for you help.

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  • $\begingroup$ One good property shared by all adjoint functors is that they commute with limits (right adjoints) or colimits (left ones). So in this case, pushforward preserves limits and pullback preserves colimits. $\endgroup$ Apr 28, 2015 at 16:50
  • $\begingroup$ Thank you. Sometimes, i hear that in general, adjunction prepares an equivalence of categories. Is it true ? Which equivalence of categories is concearned here in our case ? :) $\endgroup$
    – Bryan261
    Apr 28, 2015 at 17:12
  • $\begingroup$ I am not sure about the meaning of the statement - I think of adjunction as something like "mutual pseudo-inverses" (see e.g. the triangle identities) and sometimes, well, as pulling some object back and pushing it forward. : ) Great exercise for some intuition is to try to prove that e.g. left adjoints preserve colimits, There one can see what is going on. But this is completely general, I am interested in some insightful answer to your particular question as well... $\endgroup$ Apr 28, 2015 at 17:26
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    $\begingroup$ An adjunction defines an equivalence of categories between the respective subcategories where the adjunction morphisms are isomorphisms. In particular, if the unit and counit are natural isomorphisms, then the two adjoint functors define mutually quasi-inverse equivalences. $\endgroup$ Apr 28, 2015 at 18:20

1 Answer 1

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$\newcommand{\sheaf}[1]{{\mathcal #1}}$ $\DeclareMathOperator{\Hom}{Hom}$

Aside from category-theoretic observations the isomorphism

$$\Hom(f^*\sheaf{G},\sheaf{F}) = \Hom(\sheaf{G},f_*\sheaf{F})$$

has the interpreation that both sides describe a set of morphisms

$$\psi_{U,V}: \sheaf{G}(V) \otimes_{\sheaf{O}_Y(V)} \sheaf{O}_X(U) \to \sheaf{F}(U)$$

for all $U \subseteq X$ and $V \subseteq Y$ open and $f(U) \subseteq V$, that are compatible in the sense that for $U' \subseteq U$ and $V' \subseteq V$ and $f(U') \subseteq V'$ we have

$$\rho^{\sheaf{F}}_{U,U'} \circ \psi_{U,V} = \psi_{U',V'} \circ \rho^{\sheaf{G}}_{V,V'} \otimes \rho^{\sheaf{O}_X}_{U,U'}$$

where $\rho^{\sheaf{F}}, \rho^{\sheaf{G}}$ are the restrictions in the respective sheafs.

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