Formalize intuition about comaximal monoids in a commutative ring One way to understand a ring $A$ is to think of it as the ring of nice global functions on a space $X$ into integral domains or fields. Following this geometric analogy one can think of certain ideals in this ring as ideals of global functions which vanish on some subset of the space. Localization represents zooming in on this subset, for example when one passes from global smooth functions on a manifold to germs at a point.
I have recently come across the concept of comaximal monoids. Given a (commutative) ring $A$, monoids (under multiplication) $S_1,\dots,S_n$ in $A$ are said to be comaximal when for any choice of $s_i\in S_i$ for each $i$, $\langle s_1,\dots,s_n\rangle=\langle1\rangle$. We write $A_S$ for the localization of $A$ at a monoid $S$.
I am trying to understand the idea of comaximal monoids (or even just monoids in general) using this geometric analogy. Classically, I believe the prototypical situation is of a monoid which is the complement of a prime ideal of vanishing functions at a point. However, I am working constructively, where we may not have access to prime ideals.
As an example of this concept in use we have the following theorem: Let $S_1,\dots,S_n$ be comaximal monoids in $A$ and suppose $B$ is an $m\times p$ matrix with entries in $A$. For a column vector $C$ with entries in $A$ the following are equivalent:

*

*$Bx=C$ has a solution in $A^p$.

*$Bx=C$ has a solution in $A_{S_i}^p$ for each $i$.

This is referred to as a local-global principle. What I am hoping for is some geometric motivation behind this name. My current intuition about this is comaximal monoids represent a kind of open covering of the space which has the desired properties for allowing one to glue local solutions into global ones. This property is given by the comaximality, which I am thinking of as a kind of "partition of unity" condition.
Actually this theorem holds even when $B$ and $C$ have entries from an $A$-module $M$. I'm not sure what a geometric intuition for modules over a ring is (anyway I understand them better as algebraic objects), but I know finitely generated projective modules are like (the sections of) vector bundles over a sufficiently nice space like a smooth manifold as in e.g. the Serre-Swan theorems.
My questions (hopefully this is not too broad):

*

*How can I understand monoids and comaximal collections of monoids geometrically? Is there a way to formalize the guesses I have already provided?

*What does solving equations mean in this geometric context, especially when the entries for $B$ and $C$ are in a module over $A$?

 A: Yai0Phah's comment (and your intuition) is right -- $\{\operatorname{Spec}(S_i^{-1}A)\to\operatorname{Spec}A\}$ is a faithfully flat cover of $\operatorname{Spec}A.$ Since this collection is finite, this is the same as saying that $A\to\prod_{i = 1}^n S_i^{-1}A$ is faithfully flat. Let's first prove the algebraic result, and then talk about the geometric implications.
Claim: If $S_1,\dots, S_n$ is a collection of comaximal monoids in $A,$ then the natural map
$$
A\to\prod_{i = 1}^n S_i^{-1}A
$$
is faithfully flat.
Proof: First, notice that map $A\to S_i^{-1}A$ is flat, since any localization is flat (and that each $S_i\subseteq A$ is a multiplicative subset, so we can localize). Then, the direct product is also flat, so all we need to prove is that for any maximal ideal $\mathfrak{m}\subseteq A,$ the tensor product $\left(\prod_{i = 1}^n S_i^{-1}A\right)\otimes_A\kappa(\mathfrak{m})\neq 0$ (see here). Since this is a finite product, it agrees with the direct sum, and so it commutes with the tensor product. Thus, we need to check that for any maximal ideal $\mathfrak{m},$ there exists some $i$ such that $S_i^{-1}A\otimes_A\kappa(\mathfrak{m})$ is nonzero.
Suppose for the sake of contradiction that there is some maximal ideal $\mathfrak{m}\subseteq A$ such that $S_i^{-1}A\otimes_A\kappa(\mathfrak{m}) = 0$ for each $i.$ This means that $S_i^{-1}\kappa(\mathfrak{m}) = 0$ for any $i,$ and the localization of a ring is zero if and only if the multiplicative subset includes $0.$ In this case, this means that for each $i,$ there exists some $s_i\in S_i$ such that the image $s_i + \mathfrak{m}\in\kappa(\mathfrak{m})$ is zero. However, this implies that the ideal generated by these $s_i$ is not the unit ideal, which contradicts comaximality of the $S_i.$ $\square$
In case you're not already aware, to any commutative, unital ring $A$ we may associate a topological space $\operatorname{Spec}A$ whose points are the prime ideals of $A.$ This space naturally comes equipped with a sheaf of rings whose global sections are precisely $A.$ So, the elements of the ring $A$ can be thought of as functions on the space $\operatorname{Spec}A.$
We give $\operatorname{Spec}A$ a topology called the Zariski topology, but the cover above is not a cover in this topology. Instead, this collection is a cover in what's called the fpqc topology, which is a generalization of the classical notion of a topology. We don't talk about open sets in the fpqc topology, but we can still make sense of good covers, and we can talk about sheaves with respect to this topology. The point is, $\{\operatorname{Spec}B\to \operatorname{Spec}A\}$ is an fpqc cover precisely if the ring map $A\to B$ is faithfully flat.
So, the above result is saying that the collection $\{\operatorname{Spec}S_i^{-1}A\to\operatorname{Spec}A\}$ form a nice cover of the space $\operatorname{Spec}A$, although the images of these maps need not be open sets in the Zariski topology on $\operatorname{Spec}A$.
In other words, your theorem is indeed saying that if you can find a solution on each portion of the cover, then those solutions can be glued to give a solution on the entire space. (You might not literally be gluing the local solutions to get a global solution in the proof, but I suspect that is what happens.)
Thinking of the $S_i$ (or any selection of one element from each) as being a partition of unity is good geometric intuition. In fact, I believe this terminology is used in Vakil's notes, among other places (perhaps also Eisenbud's "Commutative Algebra"?).
As for the geometric interpretation of modules, you can indeed think of them as generalizations of vector bundles: see here for a discussion.
Then here is one way to interpret your local-global principle. You are given a certain good (fpqc) cover $\{U_i = \operatorname{Spec}S_i^{-1}A\mid i = 1,\dots, n\}$ of the space $\operatorname{Spec}A,$ and a fixed "vector bundle" $\mathcal{E}$ on $\operatorname{Spec}A.$ Given a system of equations
\begin{align*}
x_1 \sigma_{1,1} + x_2 \sigma_{1,2} + \dots + x_r \sigma_{1,r} &= \tau_1\\
x_1 \sigma_{2,1} + x_2 \sigma_{2,2} + \dots + x_r \sigma_{2,r} &= \tau_2\\
\qquad\vdots\qquad\qquad&\\
x_1 \sigma_{k,1} + x_2 \sigma_{k,2} + \dots + x_r \sigma_{k,r} &= \tau_k\\
\end{align*}
where each of the $\sigma_{i,j},\tau_i$ are global sections of $\mathcal{E},$ can we find functions $f_1,\dots, f_r$ on $\operatorname{Spec}A$ which solve these equations simultaneously? Well, the answer is that if you can find collections of functions locally on each piece of the cover which work, you can glue those functions into functions on the entire space which solve your system.
