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Tilde Turns Modules into Quasi-Coherent Sheaves by Vicfred

Last updated: Sun 11 January 2026

tags: mathematics, sheaves, quasi-coherent sheaves, differentials

On an affine scheme, localizing a module supplies the sections of its associated sheaf on every basic open. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

The mathematical object

An \(A\)-module \(M\) produces a quasi-coherent sheaf \(\widetilde M\) on \(\operatorname{Spec}A\). Localisation gives its sections on basic opens and its stalks at primes.

$$ X=\operatorname{Spec}A,\qquad M\in A\text{-Mod} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \widetilde M(D(f))=M_f $$

One explicit computation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \begin{aligned}\widetilde M_x&\cong M_{\mathfrak p},\\\Gamma(X,\widetilde M)&\cong M\qquad(x\leftrightarrow\mathfrak p).\end{aligned} $$

Why the identity matters

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad X=\operatorname{Spec}A,\qquad M\in A\text{-Mod},\\[5pt] \mathsf{C}\;&:\quad \widetilde M(D(f))=M_f. \end{aligned} $$

Where it can fail

Sheafification repairs local compatibility, but it does not make every sheaf quasi-coherent. Affine module methods apply only to the quasi-coherent class.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \widetilde M(D(f))=M_f \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Thu 29 December 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.