Canonical bundle and canonical divisors

Let $k$ be a field and let $C$ be a smooth curve over $k$ (Curves over a field). The canonical sheaf of $C$ is the sheaf of relative Kähler differentials ${\omega }_C:={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ (Sheaf of relative Kähler differentials). This sheaf is the raw canonical-sheaf object without any choice assumption. When AC is assumed, the smooth differentials theorem gives that ${\omega }_C$ is locally free of rank one, so it is an invertible ${\mathcal{O}}_C$-module, also called the canonical bundle (The Axiom of Choice, Differentials of a smooth morphism, Invertible sheaves).

Assume AC for the divisor and frame descriptions below. Let $\omega$ be a nonzero rational differential, meaning a nonzero rational section of ${\omega }_C$ (Rational section line bundle). For a closed point $x$, the local ring ${\mathcal{O}}_{C,x}$ is a discrete valuation ring (Local rings at closed points of smooth curves are discrete valuation rings). Choose any local frame ${\eta }_x$ of ${\omega }_C$ near $x$ and write $\omega =g_x{\eta }_x$, where $g_x\in k(C{)}^{\times }$. Define ${\mathrm{ord}}_x(\omega ):={\mathrm{ord}}_x(g_x),$ the normalized DVR order of the coefficient (Order codimension one rational function). If another frame is ${\eta }_x^{\mathrm{\prime }}=u{\eta }_x$, then $u\in {\mathcal{O}}_{C,x}^{\times }$ and the new coefficient is $u^{-1}{g}_x$, so its order is unchanged. This frame definition also applies when $\kappa (x)\mathrm{/}k$ is inseparable; it does not assume that the differential of a uniformizer is a frame.

The divisor of $\omega$ is the Weil divisor $\mathrm{div}(\omega )={\sum }_{x\in C}{\mathrm{ord}}_x(\omega )[x]\mathrm{.}$ Here the displayed sum has finite support. To see this, the coefficients $g_x$ are the local equations of the Cartier divisor $D_{\omega }={\mathrm{div}}_C(\omega )$ supplied by the rational-section theorem (Rational sections of line bundles are Cartier divisors, Cartier divisor). Under AC the closed-point local rings are DVRs, the generic local ring is a field, and $C$ is a normal Noetherian scheme: its affine rings are Noetherian because $C$ is finite type over the field, and these local rings are integrally closed (Curves over a field, Local rings at closed points of smooth curves are discrete valuation rings, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Locally Noetherian and Noetherian schemes, normal noetherian ring). AC implies DC by AC implies DC implies countable choice. Thus Cartier divisors on a normal Noetherian scheme give Weil divisors sends $D_{\omega }$ to a locally finite Weil divisor; at $x$ its coefficient is the order of its local equation $g_x$, namely ${\mathrm{ord}}_x(\omega )$. Since a finite-type scheme is quasi-compact, a finite subcover of neighborhoods meeting only finitely many points of this support shows that the support is finite. The resulting Weil divisor is the sum displayed above (Weil divisor normal noetherian scheme). For a smooth proper curve it is also the divisor under the finite-sum convention of Divisors on a smooth proper curve.

The divisor is effective exactly when $\omega$ is a regular differential. Indeed, at each closed point, nonnegative order is equivalent to $g_x\in {\mathcal{O}}_{C,x}$. Such stalk membership gives a regular coefficient on a neighborhood of each point; these local sections agree as rational sections on overlaps and glue. Conversely a regular differential has regular local coefficients and hence nonnegative orders.

When $C$ is smooth proper and geometrically integral, a canonical divisor $K_C$ is $\mathrm{div}(\omega )$ for any nonzero rational differential $\omega$. For two such differentials there is a unique $f\in k(C{)}^{\times }$ with ${\omega }^{\mathrm{\prime }}=f\omega$, since the generic fibre of the invertible sheaf ${\omega }_C$ is one-dimensional. Frame orders give $\mathrm{div}({\omega }^{\mathrm{\prime }})=\mathrm{div}(\omega )+{\mathrm{div}}_W(f),$ where ${\mathrm{div}}_W(f)$ is the principal Weil divisor (Principal weil divisor and class group). Thus all canonical divisors are linearly equivalent as Weil divisors. The Cartier-to-Weil comparison on this curve identifies each canonical divisor with its Cartier divisor $D_{\omega }$ and identifies principal Weil divisors with principal Cartier divisors, so the same relation is linear equivalence of Cartier divisors (Cartier and Weil divisors agree on a smooth curve, Linear equivalence cartier divisors, Principal cartier divisor). Under this comparison, ${\mathcal{O}}_C(K_C)$ means the invertible sheaf of the corresponding Cartier divisor. The rational-section theorem gives ${\omega }_C\cong {\mathcal{O}}_C(D_{\omega })={\mathcal{O}}_C(K_C)$ for every canonical divisor (Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors).