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This question concerns the meaning of the data defining a so called equivariant sheaf $F$ on a scheme X. Let denote by $\sigma: G \times_S X \to X$ an action of a group scheme $G$ on $X$ . Then a $O_X$-module $F$ is called equivariant if there exist in isomorphism $\phi: \sigma^* F \simeq p_2^*F$ of $\mathcal{O}_{G \times_S X}$ and additionally the "cocycle" condition $p_{23}^* \phi \circ (1_G \times \sigma)^* \phi = (m \times 1_X)^* \phi$ is satisfied where $p_{23}, 1_G \times \sigma, m \times 1_X$ a maps between $G \times G \times X$ and $G \times X$.

My simple question is what is a pullback of a sheaf morphsim as occuring in the cocycle condition concretely? Namely when $\phi: F \to G$ is a morphism on sheaves on $X$ and $f: Y \to X$ is a morphism of schemes how is the pullback $f^*\phi$ defined?

Is it established by following comutative diagram?

$$ \require{AMScd} \begin{CD} F @>{\phi} >> G \\ @VVV @VVV \\ f^*F @>{f^*\phi}>>f^*G \end{CD} $$

what are the vertical maps? locally $f^*F$ has the shape $O_Y \otimes_{f^{-1}O_X}f^{-1}F$. what bothers me is how can I obtain the arrow $F \to O_Y \otimes_{f^{-1}O_X}f^{-1}F$? as a side note: is the approach by the diagram above correct at all?

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The existence of the morphism $f^*\phi$ follows from functoriality of $f^{-1}$. Once you have a morphism $f^{-1}\phi$, you can tensor with $\mathcal{O}_Y$ to get $f^*\phi$.

Also note that $f^*\mathcal{F}$ is exactly $O_Y \otimes_{f^{-1}O_X}f^{-1}F$ (not only locally).

Thus it doesn't really make sense to speak of a morphism of sheaves $\mathcal{F}\to f^*\mathcal{F}$ as the former is a sheaf on $X$ while the latter is a sheaf on $ Y$. But there is a natural morphism $\mathcal{F}\to f_*f^*\mathcal{F}$ (which is the unit of the adjunction of $f^*$ and $f_*$).

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  • $\begingroup$ yes your answer definitely answers my question but besides I have another question concerning the unit map $u: \mathcal{F}\to f_*f^*\mathcal{F}$ (resp the corresponding counit map $c: \mathcal{G} \to f_*f^*\mathcal{G}$) you have mentioned in the answer. Do you know a nice reference where is rigorously treated the question when $u$ or $c$ are isomorphisms? I know some well known sufficient criterions like for $c$ (when $f$ being an immersion) or for $u$ (when $f$ being projective with connected fibers). But these two exampled criterions seem to be a bit "scattered". $\endgroup$
    – KarlPeter
    Mar 21 '19 at 22:12
  • $\begingroup$ Is there a more deeper/ more intuitive approach to understand when $c$ or $u$ are isomorphisms? Is there maybe a geometric meaning in some simple cases? For example when we consider $F,G$ as quasi coherent over affine schemes? Then this reduces to statements about modules. Do we in these cases have a deeper understanding when $c,u$ are isomorphisms? $\endgroup$
    – KarlPeter
    Mar 21 '19 at 22:12

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