An 'away sheaf' on the scheme of a PID: Localize at powers of the generator of the prime ideal?

Question

$\newcommand{\Spec}{\operatorname{Spec}}$ $\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\O}{\mathscr{O}}$ $\newcommand{\mmod}{\operatorname{mod}}$

Background:

Consider a principal ideal domain $R$. The usual structure sheaf on the scheme $\operatorname{Spec}(R)$ defines the ring of functions at each point $\p \in \Spec R$ to be $R$ localized at $S := R\setminus \p$. This localization turns $R$ into a local ring, with $\p$ as a maximal ideal.

On the other hand, since $R$ is a PID, let $\p$ be generated by an element $\p_0 \in R$. So we have $\p = (\p_0)$. We can build a different sheaf on $\Spec(R)$ (call it the away sheaf) which assigns to a point $\p \in \Spec(R)$, $R$ localized at $S := \{ 1, \p_0, \p_0^2, \dots \}$. This localization annihilates all information "at $\p$", and carries information only about what happens "away from $\p$".

For example, let $R = \Z/30\Z$. The primes in this ring are $(2), (3), (5)$. If we consider the "function" $21$ on the usual structure sheaf, we get the local components $21(\p) = 21_{R\setminus \p}$: the residue of $21$ when the ring is localized at the ring $\p$. This gives $21(2) = 1, 21(3) = 0, 21(5) = 1$. By chinese remainder, we can find the original number $21$.

On the other hand, let us consider the away sheaf on the same ring $\Z/30\Z$. It assigns the values $21((\p_0)) = 21_{R \setminus S}$ where $S = \{1, \p_0, \p_0^2, \dots \}$: the residue of $21$ on ignoring the prime $\p_0$. This gives values $21(2) = 6, 21(3) = 1, 21(5) = 3$. That is, we consider:

• $21(2) = 21 ~(\mmod 30/2) = 21 ~(\mmod 15) = 6$
• $21(3) = 21 ~(\mmod 30/3) = 21 ~(\mmod 10) = 1$
• $21(5) = 21 ~(\mmod 30/5) = 21~(\mmod 6)= 3$

To glue together the data, we solve the equations using CRT for non pairwise coprime moduli. Our residues will be consistent because they came from a single number. Alternatively, the "full GCD" of the numbers $\gcd(15, 10, 6) = 1$ thus we can always find a unique solution.

In general, this technique works because for a given number $n = \prod_i p_i$, we are finding the "away factors" $a_i \equiv n/p_i = \prod_{j \neq i} p_j$. Then we will have $\gcd(a_1, \dots, a_k) = 1$ because each $a_i$ is missing one of the primes $p_i$. This will allow us to recover solutions and glue solutions together using CRT.

It's a very different kind of sheaf from the usual, beacuse we localize using the multiples of the generator of the prime ideal, rather than localizing using the complement of the prime ideal.

Questions:

1. This has surely been studied. What is this sort of phenomena called?
2. Are there more uses for this than just a curiosity?

Verification:

• Here is the SAGE code I used to verify the localization computations. It prints the output of what happens when we localize at a multiplicative set.

In this case, it provides the output:

Z/30Z localized at {2^i}:
   0 ->    0 | 1 ->    1
   2 ->    2 | 3 ->    3
   4 ->    4 | 5 ->    5
   6 ->    6 | 7 ->    7
   8 ->    8 | 9 ->    9
  10 ->   10 | 11 ->   11
  12 ->   12 | 13 ->   13
  14 ->   14 | 15 ->    0
  16 ->    1 | 17 ->    2
  18 ->    3 | 19 ->    4
  20 ->    5 | 21 ->    6
  ...
Z/30Z localized at complement of (2):
   0 ->    0 | 1 ->    1
   2 ->    0 | 3 ->    1
   4 ->    0 | 5 ->    1
   6 ->    0 | 7 ->    1
   ...

So the behaviour I claimed on localization is the behaviour that occurs.