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Assume $X$ and $Y$ are noetherian schemes over $\mathbb{C}$ and there is a proper and faithfully flat morphism $f: X\rightarrow Y$.

Assume the canonical morphism $F\xrightarrow{\sim} f_{*}f^{*}F$ is an isomorphism for all $F\in Coh(X)$.

Can we say anything about the canonical morphism $f^{*}f_{*}f^{*}F\rightarrow f^{*}F$ $(*)$? Is this also an isomorphism?

We have an isomorphism $f^{*}F\xrightarrow{\sim} f^{*}f_{*}f^{*}F$ if we just pullback the canonical isomorphism. Does this isomorphism have anything to do with $(*)$?

Are there any conditions under which we can deduce that $F\xrightarrow{\sim} f_{*}f^{*}F => f^{*}f_{*}f^{*}F \xrightarrow{\sim} f^{*}F$?

Is there maybe a categorical argument here, since we speak about adjoint functors?

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    $\begingroup$ The definition of adjoint functor implies that the composition $f^\ast F\to f^\ast f_\ast f^\ast F\to f^\ast F$ is the identity. Since the first map is an isomorphism in your situation, then so is the second, which is the canonical map ($\ast$). $\endgroup$
    – Matthias Wendt
    Commented Oct 12, 2015 at 18:03
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    $\begingroup$ Take a step back. All these sorts of adjoint arguments just spit out "the only map you can think of", and so of course (*) will just be the pullback of the canonical morphism and will hence be an isomorphism (because the pullback of an iso is an iso). You should try to turn this into a proof but this should surely be how you are thinking. $\endgroup$
    – eric
    Commented Oct 12, 2015 at 19:29

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In short: always.

Indeed, given a functor $F : \mathcal{C} \to \mathcal{D}$ left adjoint to $G : \mathcal{D} \to \mathcal{C}$, the triangle identities say that the composites of the canonical morphisms $$F X \to F G F X \to F X$$ $$G Y \to G F G Y \to G Y$$ are identities for all $X$ in $\mathcal{C}$ and all $Y$ in $\mathcal{D}$. In particular, if $X \to G F X$ is an isomorphism, then so is $F G F X \to F X$.

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  • $\begingroup$ Thanks, I wasn't aware of these trinagle identities. Should refresh my categroy theory. $\endgroup$
    – Bernie
    Commented Oct 13, 2015 at 11:18

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