Question regarding the definition of linearization of line bundles

Question

I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$ over an algebraically closed field $k$ via $\sigma: G\times X\rightarrow X$. He then defines the notion of a $G$-linearized line bundle $L$ on $X$, and comments that for each $g\in G$ and $x\in X$, the induced map on the fibers $L_x\rightarrow L_{gx}$ is a linear isomorphism. We can view the set of such isomorphisms as of line bundles $$\overline{\sigma}(g): L\rightarrow g^*L\ ,$$ which satisfy certain cocycle conditions.

Upto here it's fine. Next, he makes the following remark: the collection of isomorphisms $\overline{\sigma}(g)$ can also be viewed as an isomorphism of line bundles $$\Phi : p_2^*L\rightarrow \sigma^*L\ ,$$ where $p_2:G\times X\rightarrow X$ is the second projection.

This last part is not clear to me. Surely, given such a $\Phi$, we can find the $\overline{\sigma}(g)$'s by restricting. But how to go the other way? Meaning, why do the collection of isomorphisms $\overline{\sigma}(g)$ given 'fiberwise' glue to give a global isomorphism? Is there a general theme like this, i.e. to define a morphism of sheaves, it is enough to define it on fibers in certain cases?

Thanks in advance!

Answer 1

I think that you should regard the first definition as an imprecise version of the second definition. For example, suppose that $ X $ is a point and so $ L $ is simply a 1-dimensional vector space. Then the first definition is simply a map $ g : L \rightarrow L $ for each group element $ g $ satisfying multiplicativity; in other words a homorphism of groups $ G \rightarrow GL(L) $ (i.e. a representation of $ G $ as an abstract group). On the other hand, the second definition is equivalent to a homomorphism $ G \rightarrow GL(L) $ as algebraic groups (i.e. a representation of $ G $ as an algebraic group).