# Equivariant Sheaf

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?

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 $$f_*$$).
• 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". Mar 21 '19 at 22:12
• 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? Mar 21 '19 at 22:12