I’m taking an introductory course on Scheme theory and I’m looking for a reference for the following construction that was covered in the course.
Let $A$ be a commutative ring, $U$ an open subset of $\text{Spec}(A)$, $f\in A$. We define
$$D(f):=\{ \mathfrak{p}\in \text{Spec}(A) \> | \> f \notin \mathfrak{p} \}$$ $$S(U):=\{f\in A \> | \> U \subset D(f) \}$$
then $S(U)$ is a multiplicative subset of $A$.
Now, the following object is defined in my lecture notes: if $M$ is an $A$-module, we define the $S(U)^{-1}A$-module
$$M_p(U):=S(U)^{-1}M,$$
and since $V \subset U$ implies $S(U) \subset S(V)$ for any open subset $V$, we get a canonical morphism $M_p(U)\rightarrow M_p(V)$ and it can be shown that $V \mapsto M_p(V)$ is an $A$-modules presheaf.
Is there a name for this presheaf/can I find more information about it somewhere/are there any tags I can look for? In particular, I’m interested in the following result:
For all $f\in A$, let $M_f$ be the localization of the $A$-module $M$ at the element $f$, and let $D(f) \mapsto \tilde{M}(D(f))$ be the sheafification of the presheaf $D(f) \mapsto M_p(D(f))$. Then the morphism $M_f \rightarrow \tilde{M}(D(f))$ is an isomorphism of $A$-modules.
We were given a proof of this, but I’d like to find more references on it. I’ve browsed through a couple books (Hartshorne and Görtz & Wedhorn) and didn’t find this particular construction.