Lagmental Vicfred

Line Bundles Form the Picard Group by Vicfred

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

Tensor product turns isomorphism classes of invertible sheaves into an abelian group. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Definitions first

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.

$$ \operatorname{Pic}(X)=\{\text{line bundles on }X\}/\cong $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ [\mathcal L]+[\mathcal M]=[\mathcal L\otimes\mathcal M],\qquad-[\mathcal L]=[\mathcal L^\vee] $$

A small case in full

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \operatorname{Pic}(\mathbf P_k^n)\cong\mathbf Z,\qquad d\longmapsto\mathcal O_{\mathbf P^n}(d) $$

The reusable statement

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{Pic}(X)=\{\text{line bundles on }X\}/\cong,\\[5pt] \mathsf{C}\;&:\quad [\mathcal L]+[\mathcal M]=[\mathcal L\otimes\mathcal M],\qquad-[\mathcal L]=[\mathcal L^\vee]. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [\mathcal L]+[\mathcal M]=[\mathcal L\otimes\mathcal M],\qquad-[\mathcal L]=[\mathcal L^\vee] \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sun 03 February 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.