# Aggressive geometry – Equivariant sheaf: explanations on stems

I have a question about the explanation of the data defining a so-called equivariant sheaf $$F$$ on an X wiki schema: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $$sigma: G times_S X to X$$ an action of a group $$G$$ sure $$X$$ . Then a $$O_X$$-module $$F$$ is called equivariant if it exists in isomorphism $$phi: sigma ^ * F simeq p_2 ^ * F$$ of $$mathcal {O} _ {G times_S X}$$-modules and in addition the condition "cocycle" $$p_ {23} ^ * phi circ (1_G times sigma) ^ * phi = (m times 1_X) ^ * phi$$ is satisfied where $$p_ {23}, 1_G times sigma, m times 1_X$$ a map between $$G times G times X$$ and $$G times X$$.

FOLLOWING EXPLANATION THAT I DO NOT UNDERSTAND: Then, we say that the condition of cocycle indicates that at the level of the stems, the isomorphism $$F_ {gh cdot x} simeq F_x$$ is the same as $$F_ {g cdot h cdot x} simeq F_ {h cdot x} simeq F_x$$; that is, it reflects associativity. (*)

What I do not understand is why induced isomorphism $$(m times 1_X) ^ * phi$$ provides a map $$F_ {gh cdot x} to F_x$$ at the level of the stems?

To know I do not see why $$F_ {gh cdot x}$$ and $$F_x$$ are the domain and the correct codomaine of this map $$(m times 1_X) ^ * phi$$.

Indeed, the induced isomorphism $$(m times 1_X) ^ * phi$$ is an iso of sheaf $$(m times 1_X) ^ * sigma ^ * F = ( sigma circ (m times 1_X)) ^ * F to (m times 1_X) ^ * p_2 ^ * F = (p_2 circ (m times 1_X)) ^ * F$$.

Fix a point $$(g, h, x) in G times G times X$$.

then $$sigma circ (m times 1_X) (g, h, x) = gh cdot x$$ and $$p_2 circ (m times 1_X) (g, h, x) = x$$

I do not understand why the stems of $$(g, h, x)$$ domain and codomaine are given by $$(( sigma circ (m times 1_X)) ^ * F) _ {(g, h, x)} = F_ {gh cdot x}$$ and $$((p_2 circ (m times 1_X)) ^ * F) _ {(g, h, x)} = F_x$$ as shown in (*)?

I've learned that generally for a morphism $$f: X to Y$$ and a sheaf $$F$$ sure $$Y$$ we have the following formula for the stem in $$z in X$$ of the sheaf of draw:

$$(f ^ * F) _z = O_ {X, z} otimes_ {O_ {Y, f (z)}} F_ {f (z)}$$.

Now, we apply this formula to our situation, so $$f = sigma circ (m times 1_X)$$ and $$z = (g, h, z)$$ so we get

$$(( sigma circ (m times 1_X)) ^ * F) _ {(g, h, x)} = O_ {G times G times X, (g, h, x)} otimes_ {O_ {X, gh cdot x}} F_ {gh cdot x}$$

But (*) says that it should be equal to $$F_ {gh cdot x}$$. Why? Where is the error in my reasoning? What's going on with the left "factor"?