Lagmental Vicfred

Exactness of Sheaves Can Be Checked on Stalks by Vicfred

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

A sequence of sheaves is exact exactly when every induced sequence of stalks is exact. The point is to make the formal expression readable enough to audit line by line.

The data

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.

$$ \mathcal F'\xrightarrow{u}\mathcal F\xrightarrow{v}\mathcal F'' $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \text{exact}\Longleftrightarrow\forall x,\quad\mathcal F'_x\xrightarrow{u_x}\mathcal F_x\xrightarrow{v_x}\mathcal F''_x\ \text{exact} $$

Derivation

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \left(\ker v\right)_x\cong\ker(v_x),\qquad\left(\operatorname{im}u\right)_x\cong\operatorname{im}(u_x) $$

Invariant content

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathcal F'\xrightarrow{u}\mathcal F\xrightarrow{v}\mathcal F'',\\[5pt] \mathsf{C}\;&:\quad \text{exact}\Longleftrightarrow\forall x,\quad\mathcal F'_x\xrightarrow{u_x}\mathcal F_x\xrightarrow{v_x}\mathcal F''_x\ \text{exact}. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \text{exact}\Longleftrightarrow\forall x,\quad\mathcal F'_x\xrightarrow{u_x}\mathcal F_x\xrightarrow{v_x}\mathcal F''_x\ \text{exact} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Thu 23 July 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.