Divisors of rational differentials form one linear equivalence class

By [F1], $\omega$ and ${\omega }^{\mathrm{\prime }}$ are nonzero vectors in the same one-dimensional vector space over $k(C)$, so there is a unique $f\in k(C{)}^{\times }$ with ${\omega }^{\mathrm{\prime }}=f\omega$.

Put $D_{\omega }={\mathrm{div}}_C(\omega )$ and $D_{{\omega }^{\mathrm{\prime }}}={\mathrm{div}}_C({\omega }^{\mathrm{\prime }})$, the Cartier divisors of [F3]. Their Weil cycles are the finite divisors $\mathrm{cyc}(D_{\omega })=\mathrm{div}(\omega )$ and $\mathrm{cyc}(D_{{\omega }^{\mathrm{\prime }}})=\mathrm{div}({\omega }^{\mathrm{\prime }})$, because the cycle coefficient at $x$ is the order of the local equation $g_x$ and finite support follows from [F4].

For each ${\omega }^{\mathrm{\prime }}$, the rational-section theorem [F3] gives ${\mathcal{O}}_C(D_{{\omega }^{\mathrm{\prime }}})\cong {\omega }_C$ carrying the canonical rational section to ${\omega }^{\mathrm{\prime }}$. The curve Cartier-to-Weil isomorphism identifies $D_{{\omega }^{\mathrm{\prime }}}$ with $K_C=\mathrm{div}({\omega }^{\mathrm{\prime }})$, so ${\mathcal{O}}_C(K_C)\cong {\omega }_C$ for every canonical divisor.

Fix a closed point $x$ and any local frame ${\eta }_x$ of ${\omega }_C$, and write $\omega =g_x{\eta }_x$ and ${\omega }^{\mathrm{\prime }}=g_x^{\mathrm{\prime }}{\eta }_x$. The equality ${\omega }^{\mathrm{\prime }}=f\omega$ from step 1.1 gives $g_x^{\mathrm{\prime }}=f{g}_x$, so additivity of the normalized DVR order gives ${\mathrm{ord}}_x({\omega }^{\mathrm{\prime }})={\mathrm{ord}}_x(f)+{\mathrm{ord}}_x(\omega )$. This frame calculation is valid also for inseparable residue extensions.

The pointwise identity of step 2.1 holds at every closed point, and each divisor has finite support by step 1.2, so coefficientwise equality gives $\mathrm{div}({\omega }^{\mathrm{\prime }})=\mathrm{div}(\omega )+{\mathrm{div}}_W(f)$. By [F4], the Cartier cycles satisfy $\mathrm{cyc}(D_{{\omega }^{\mathrm{\prime }}})=\mathrm{cyc}(D_{\omega })+\mathrm{cyc}({\mathrm{div}}_C(f))$; since the cycle map is an isomorphism, $D_{{\omega }^{\mathrm{\prime }}}-D_{\omega }={\mathrm{div}}_C(f)$.

The equality in step 3.1 says the Weil divisors differ by a principal Weil divisor, hence are linearly equivalent by [F5]; its Cartier form $D_{{\omega }^{\mathrm{\prime }}}-D_{\omega }={\mathrm{div}}_C(f)$ makes the corresponding Cartier divisors linearly equivalent by [F5]. Thus every nonzero rational differential gives the same canonical divisor class.

The ratio is unique by step 1.1, the divisor formula is step 3.1, and steps 4.1 and 1.3 prove the asserted class and sheaf conclusions; the choice assumptions are AC and its consequence DC for the cited structural suppliers.