Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a localization of an $A$-module $M$ at $S$ is an $A$-module morphism $M\to M[S^{-1}]$ initial with respect to $f\in A$ acting invertibly.
For modules, principal localization at an element $f\in A$ is a special case of universally inverting an endomorphism $f$ in a category. (This fails for rings because the action of $f$ on $A$ is not a ring morphism.) As always there are two universal constructions - initial and terminal. Localization of a module is the initial variant.
What about the terminal way to invert the action of a ring element on a module? The "colocalization" at $f\in A$ is an $A$-module morphism $R_f(M)\to M$ which is terminal in the category of $A$-module maps to $M$ on whose domain $f$ acts invertibly.
The colocalization may be constructed as the following sequential limit $$R_f(M)\cong\varprojlim(\cdots \overset{f}{\to}M\overset{f}{\to}M),$$so an element is a string $(m_0,m_1,\dots )$ satisfying $m_n=f(m_{n+1})$.
Viewing a $C^\infty(X)$-module as a $C^\infty$ vector bundle over $X$, an element of the colocalization at some $f\in C^\infty(X)$ is like a "section obtained by arbitrarily many multiplications by $f$". I don't understand how to think of this. Maybe as a section "coming from" a certain completion?
Question. What is the geometric meaning of colocalization of a module at an element of the ring? Some kind of "cosupport" (whatever that is)?
