I'm trying to solve Exercise II.5.18 from Hartshorne. I quote part (b). Here, $X$ is a vector bundle over $Y$, with projection $f$.
(b) For any morphism $f : X \to Y$, a section of $f$ over an open set $U \subseteq Y$ is a morphism $s: U \to X$ such that $f \circ s = Id_U$. It is clear how to restrict sections to smaller open sets, or how to glue them together, so we see that the presheaf $U \to$ {set of sections of $f$ over $U$} is a sheaf of sets of $Y$, which we denote by $S(X,Y)$. Show that $f:X \to Y$ is a vector bundle of rank $n$, then the sheaf of sections $S(X,Y)$ has a natural structure of $O_Y$-module, which makes it a locally free $O_Y$-module of rank $n$. [Hint: it is enough to define the module locally, so we can assume $Y$ = Spec $A$ is affine, and $X = \mathbb{A}^n_Y$. Then a section $s : Y \to X$ comes from an $A$-algebra homomorphism $\theta : A[x_1, ..., x_n] \to A$, which in turn determines an ordered $n$-tuple $<\theta(x_1), ..., \theta(x_n)>$ of elements of $A$. Use this correspondence between sections of $s$ and ordered $n$-tuples of elements of $A$ to define the module structure.]
I've just started Algebraic Geometry; this will probably be a basic question, so please bear with me.
Now, I think I "get" the global picture. For example, I understand the sections of a vector bundles on a differentiable manifold determine a module structure over $C^{\infty}$ functions, where the multiplication is pointwise multiplication in the fibers. I just have trouble translating this general picture in algebraic geometry terms. For example, though I understood the correpondence between morphisms of rings and morphism of their spectra, I don't intuitively "see" the correspondence between sections of $Y$ and the $n$-uple mentioned in the exercise, so I don't see how to define the module structure mentioned in the exercise.