Riemann Roch for Curves via Euler Characteristics

Given: a field $k$, a smooth proper geometrically integral curve $C$ over $k$, and a divisor $D$ on $C$.

A curve over $k$ is geometrically integral, separated and of finite type over $k$, and its underlying topological space has chain dimension one; being geometrically integral, $C$ is an integral $k$-scheme, so it is nonempty, reduced and irreducible, and the adjectives smooth and proper mean that the structure morphism $C\to \mathrm{Spec}k$ is smooth and proper (Curves over a field, Integral schemes).

For an integral $k$-scheme of finite type with underlying space of chain dimension one, the underlying space is Noetherian, and every proper closed subset is a finite set of closed points; the chain dimension is the Krull dimension of Chain dimension and the empty-space convention, so a curve has dimension at most one in the sense required for vanishing theorems (Proper closed subsets of a curve are finite, Chain dimension and the empty-space convention).

If $X$ is a Noetherian topological space with $\dim X\le d$ for an integer $d\ge 0$, then $H^q(X,\mathcal{F})=0$ for every sheaf of abelian groups $\mathcal{F}$ on $X$ and every integer $q>d$ (Grothendieck vanishing on a Noetherian space).

If $X$ is a scheme proper over a field $k$ and $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module, then $H^q(X,\mathcal{F})$ is a finite-dimensional $k$-vector space for every $q\ge 0$, and only finitely many of the groups are nonzero: for a finite affine open cover of $X$ with $n$ members, $H^q(X,\mathcal{F})=0$ for every $q\ge n$ (Finite-dimensional coherent cohomology over a field).

An invertible ${\mathcal{O}}_X$-module is locally free of rank one, and a locally free module is quasi-coherent; a locally free module of rank $r$ is of finite type, since on a chart $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r=\widetilde{A^r}$ with $A^r$ a finitely generated module; on a locally Noetherian scheme a quasi-coherent module is coherent if and only if it is of finite type (Invertible sheaves, Locally free sheaves of finite rank, Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Coherent sheaves on a locally Noetherian scheme).

A scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings; a morphism locally of finite type provides, around every point, an affine chart $\mathrm{Spec}B$ with $B$ a finitely generated algebra over the coordinate ring of an affine open of the target; a field is a Noetherian ring, and a finitely generated algebra over a Noetherian ring is Noetherian (Locally finite type and finite type morphisms, Locally Noetherian and Noetherian schemes, 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).

For a coherent module $\mathcal{F}$ on a scheme proper over $k$ whose cohomology is finite-dimensional with only finitely many nonzero groups, the Euler characteristic $\chi (X,\mathcal{F})={\sum }_{q\ge 0}(-1{)}^q{\dim }_k{H}^q(X,\mathcal{F})$ is an integer; the dimension ${\dim }_k$ of a finite-dimensional $k$-vector space is a nonnegative integer, and $H^q$ denotes sheaf cohomology of the underlying sheaf of abelian groups (Euler characteristic of a coherent sheaf, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Sheaf cohomology as right derived global sections).

Weil-to-Cartier, invertibility, and the Riemann-Roch space. The divisor $D$ is a Weil divisor on the smooth curve, so Cartier and Weil divisors agree on a smooth curve identifies it with a Cartier divisor. The local-equation construction of Invertible sheaf of cartier divisor defines ${\mathcal{O}}_C(D)$, and The sheaf of a Cartier divisor is invertible proves it is invertible. The actual definition The space L(D) identifies $L(D)=\{f\in k(C{)}^{\times }:\mathrm{div}(f)+D\ge 0\}\cup \{0\}$ with the image of $H^0(C,{\mathcal{O}}_C(D))$ in $k(C)$, using Rational sections of line bundles are Cartier divisors. These interfaces supply the uses at steps 1.3 and 5.1.

The Axiom of Choice enters through the proper finiteness theorem [F4], the coherence and Noetherian suppliers [F5] and [F6], and the choice premises of the curve Cartier-to-Weil route [F8]. In ZF, AC implies DC by AC implies DC implies countable choice, so the DC premise of [F8] is available from the stated assumption (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain); no further selection is made below.

Given: the Axiom of Choice inherited from the local-DVR, divisor and finite-dimensionality suppliers; a field $k$, a smooth proper geometrically integral curve $C$ over $k$, and divisors $D\le E$ on $C$.

A divisor on $C$ is a finite formal sum $D={\sum }_x{n}_x[x]$ over the closed points of $C$ with integer coefficients; $D\le E$ means that the coefficients satisfy $n_x\le m_x$ for all $x$, equivalently that $E-D$ is effective; the degree is additive, ${\mathrm{deg}}_k(E-D)={\sum }_x(m_x-n_x)[\kappa (x):k]$, the sum over the finite support, and each residue field $\kappa (x)$ is a finite extension of $k$ with $[\kappa (x):k]={\dim }_k\kappa (x)\ge 1$ (Divisors on a smooth proper curve, Degree divisor proper curve, Divisor support positive negative parts).

The current The space L(D) identifies $L(D)=\{f\in k(C{)}^{\times }:\mathrm{div}(f)+D\ge 0\}\cup \{0\}$ as a $k$-subspace of $k(C)$ with the image of $H^0(C,{\mathcal{O}}_C(D))$. It uses the principal Weil divisor interface Principal weil divisor and class group, the local-equation construction Invertible sheaf of cartier divisor, and the curve Cartier-to-Weil identification Cartier and Weil divisors agree on a smooth curve. The latter's Dependent Choice premise is supplied by AC through AC implies DC implies countable choice. The rational-section dictionary Rational sections of line bundles are Cartier divisors identifies the global sections with the stated rational functions. These interfaces give the section and stalk descriptions used below.

For a normal locally Noetherian integral scheme the order along a prime divisor is a group homomorphism ${\mathrm{ord}}_Z:K(X{)}^{\times }\to \mathbb{Z}$ with ${\mathrm{ord}}_Z(fg)={\mathrm{ord}}_Z(f)+{\mathrm{ord}}_Z(g)$ and ${\mathrm{ord}}_Z(f^{-1})=-{\mathrm{ord}}_Z(f)$, and on an integral scheme ${\mathrm{ord}}_Z(f)\ge 0$ if and only if $f$ lies in the local ring, with equality to zero exactly for units (Order codimension one rational function). The closed points of the smooth curve $C$ are its codimension-one points (Divisors on a smooth proper curve). For the order inequalities below only, put ${\mathrm{ord}}_x(0)=+\mathrm{\infty }$; zero belongs to every $L(D)$ by [F2].

The local ring ${\mathcal{O}}_{C,p}$ of a closed point of the smooth curve $C$ is a discrete valuation ring with maximal ideal generated by a uniformizer $t$; every nonzero $f\in k(C{)}^{\times }$ is $f=t^{m}u$ with $m={\mathrm{ord}}_p(f)\in \mathbb{Z}$ and $u$ a unit, and $\kappa (p)={\mathcal{O}}_{C,p}\mathrm{/}(t)$ is a field, the residue field, of $k$-dimension $[\kappa (p):k]$ (Local rings at closed points of smooth curves are discrete valuation rings, The residue field at a point of an affine scheme, Degree divisor proper curve).

$L(D)=H^0(C,{\mathcal{O}}_C(D))$ is a finite-dimensional $k$-vector space and $l(D)={\dim }_{k}L(D)=h^0(D)$ is a nonnegative integer; the same holds with $D$ replaced by $D+p$ (Finite-dimensionality of the Riemann-Roch space, The Riemann-Roch dimension l(D), Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis).

Linear algebra over $k$: for a linear map $T:V\to W$ with $V$ finite-dimensional, ${\dim }_{k}V={\dim }_k\ker T+{\dim }_k\mathrm{im}T$ (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$); the formula $\tilde{T}(v+\ker T)=T(v)$ defines a linear isomorphism $V\mathrm{/}\ker T\to \mathrm{im}T$ (First isomorphism theorem for vector spaces: $V\mathrm{/}\ker T$ is isomorphic to $\mathrm{im}T$); a subspace of a finite-dimensional space has dimension at most that of the ambient space (If ${\dim }_{F}V=n$ and $U$ is a linear subspace of $V$, then $U$ is finite-dimensional, ${\dim }_{F}U\le n$, and ${\dim }_{F}U=n$ if and only if $U=V$); and for $W\le V$ with $V$ finite-dimensional, ${\dim }_k(V\mathrm{/}W)={\dim }_{k}V-{\dim }_{k}W$ (A quotient basis lifts to a basis adapted to $W$).

The Axiom of Choice enters through the DVR supplier of [F4], the finiteness suppliers of [F5] and the Cartier-to-Weil route of [F2]. In ZF, AC implies DC by AC implies DC implies countable choice, supplying the DC premise of Cartier and Weil divisors agree on a smooth curve. The chosen uniformizer in step 2.3 only specifies a coordinate on the fiber; the kernel is independent of it, and no additional choice principle is used (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain).

Given: a field $k$, a smooth proper geometrically integral curve $C$ over $k$, a divisor $D$ on $C$, a closed point $p\in C$, and, for part (3), an effective divisor $E$ on $C$.

Divisors and degrees. A divisor on $C$ is a finite formal sum $D={\sum }_x{n}_x[x]$ over the closed points, $\mathrm{Supp}(E)$ is the finite set of closed points with nonzero coefficient, $D$ is effective when all coefficients are $\ge 0$, the residue field $\kappa (x)$ of a closed point is a finite extension of $k$ with $[\kappa (x):k]={\dim }_k\kappa (x)\ge 1$, and ${\mathrm{deg}}_k(E)={\sum }_x{n}_x[\kappa (x):k]$; $C$ is geometrically integral, separated and of finite type over $k$, and it is nonempty (Divisors on a smooth proper curve, Divisor support positive negative parts, Degree divisor proper curve, The residue field at a point of an affine scheme, Curves over a field). Moreover ${\mathrm{deg}}_k(E)\ge 0$ for effective $E$, with ${\mathrm{deg}}_k(E)=0$ exactly for $E=0$ (Effective divisors have nonnegative degree).

Weil divisors, their Cartier sheaves, and the order description. In order inequalities below, use the convention ${\mathrm{ord}}_x(0)=+\mathrm{\infty }$; this is notation for the zero section, not an extension of the valuation homomorphism domain. The divisor $D^{\mathrm{\prime }}$ on the smooth curve is a Weil divisor. By Cartier and Weil divisors agree on a smooth curve it is represented by a Cartier divisor whose cycle is $D^{\mathrm{\prime }}$. The local-equation construction of Invertible sheaf of cartier divisor gives the subsheaf ${\mathcal{O}}_C(D^{\mathrm{\prime }})\subseteq {\mathcal{K}}_C$, and The sheaf of a Cartier divisor is invertible proves it invertible. The generic sheaf ${\mathcal{K}}_C$ is constant with value $k(C)$ (Sheaf total quotient rings). For a nonempty open $U\subseteq C$, its sections identify with $k(C)$; a rational function $f$ is a section of ${\mathcal{O}}_C(D^{\mathrm{\prime }})$ on $U$ exactly when it belongs to the stalk at every point of $U$. At the generic point the stalk is $k(C)$ and imposes no condition. At each closed point $x$, the local equation has order $n_x(D^{\mathrm{\prime }})$ by the cycle identification, so the DVR stalk condition is ${\mathrm{ord}}_x(f)+n_x(D^{\mathrm{\prime }})\ge 0$ (Local rings at closed points of smooth curves are discrete valuation rings, Order codimension one rational function). Locality of the subsheaf then gives ${\mathcal{O}}_C(D^{\mathrm{\prime }})(U)=\{f\in k(C):{\mathrm{ord}}_x(f)+n_x(D^{\mathrm{\prime }})\ge 0\text{ for every closed point }x\in U\}\mathrm{.}$ For $U=\mathrm{\varnothing }$, ${\mathcal{O}}_C(D^{\mathrm{\prime }})(U)=0$. Thus $D^{\mathrm{\prime }}\le D^{\mathrm{\prime }\mathrm{\prime }}$ gives the inclusion of these subsheaves and the stated closed-point stalk descriptions. The global-section instance is also the identification of The space L(D) using the rational-section dictionary Rational sections of line bundles are Cartier divisors.

Local structure at $p$. The local ring ${\mathcal{O}}_{C,p}$ is a discrete valuation ring with maximal ideal generated by a uniformizer $t$, so that $\kappa (p)={\mathcal{O}}_{C,p}\mathrm{/}(t)$ and every nonzero element of ${\mathcal{O}}_{C,p}$ is a unit times a power of $t$ (Local rings at closed points of smooth curves are discrete valuation rings). For $f\in k(C{)}^{\times }$ the order ${\mathrm{ord}}_p(f)$ is additive, ${\mathrm{ord}}_p(f)\ge 0$ if and only if $f\in {\mathcal{O}}_{C,p}$, and ${\mathrm{ord}}_p(f)=0$ if and only if $f$ is a unit of ${\mathcal{O}}_{C,p}$ (Order codimension one rational function); consequently $t^{m}f\in {\mathcal{O}}_{C,p}$ if and only if ${\mathrm{ord}}_p(f)\ge -m$, and $t^{m}f\in t{\mathcal{O}}_{C,p}$ if and only if ${\mathrm{ord}}_p(f)\ge -m+1$, for every $m\in \mathbb{Z}$.

Monotonicity supplier. For divisors $D\le E$ the natural morphism of invertible subsheaves ${\mathcal{O}}_C(D)\to {\mathcal{O}}_C(E)$ is injective, and $L(D)\subseteq L(E)$; for $E=D+p$ the quotient $L(D+p)\mathrm{/}L(D)$ embeds in $\kappa (p)$ with dimension at most $[\kappa (p):k]$ (Monotonicity of L(D) in the divisor). Only the injectivity assertion is used below.

Exactness and stalks of sheaves. A sequence of sheaves of ${\mathcal{O}}_C$-modules is exact exactly when it is exact in the abelian category of sheaves of abelian groups, in the sense of Exact sequences of sheaves; the kernel sheaf of a morphism is computed objectwise while the cokernel sheaf is the sheafification of the objectwise cokernel, with the same formulas taken in the module categories on each open set, so cokernels and exactness of ${\mathcal{O}}_C$-module morphisms are computed on underlying sheaves of abelian groups (Kernel sheaves are objectwise, while cokernels and images are sheafified). Since kernels are objectwise, stalks are filtered colimits of section groups (The stalk of a presheaf at a point) and filtered colimits of abelian groups are exact (Filtered colimits of abelian groups are exact), the stalk of the kernel of a morphism is the kernel of the stalk map, and by the same exactness and Sheafification preserves stalks the stalk of the cokernel is the cokernel of the stalk map. Consequently a sequence of sheaves of abelian groups is exact if and only if it is exact on every stalk (A sequence of abelian sheaves is exact exactly when it is exact on every stalk), and the category of sheaves of abelian groups is abelian (Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories).

Abelian-category algebra. For a morphism $f:A\to B$ of an abelian category there is a canonical isomorphism $A\mathrm{/}\ker (f)\cong \mathrm{im}(f)$ (First isomorphism theorem in an abelian category), and for subobjects $C\le B\le A$ there is a canonical isomorphism $(A\mathrm{/}C)\mathrm{/}(B\mathrm{/}C)\cong A\mathrm{/}B$ (Third isomorphism theorem in an abelian category); the quotient $A\mathrm{/}B$ is the cokernel of the representing monomorphism $B\rightarrowtail A$ (The quotient of an object by a subobject).

Skyscraper sheaves. For a point $x\in X$ of a topological space and an abelian group $A$, the skyscraper sheaf $i_{x,\ast }A$ has $(i_{x,\ast }A)(V)=A$ for $x\in V$ with identity restrictions and value $0$ for $x\mathrm{\notin }V$ (A skyscraper sheaf of abelian groups at a point). For a closed subset $Z\subseteq X$ with the subspace topology and inclusion $i:Z\hookrightarrow X$, the direct image $i_{\ast }\mathcal{F}$ of a sheaf $\mathcal{F}$ of abelian groups on $Z$ is given by $(i_{\ast }\mathcal{F})(V)=\mathcal{F}(i^{-1}V)$ (Direct image of a sheaf along a continuous map, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace), and $H^q(Z,\mathcal{F})\cong H^q(X,i_{\ast }\mathcal{F})$ for every $q\ge 0$ (Pushforward along a closed immersion preserves sheaf cohomology). On the one-point space $Z=\{\ast \}$ one has $H^0(Z,\mathcal{F})\cong \mathcal{F}(Z)$ and $H^q(Z,\mathcal{F})=0$ for every $q>0$ (A point has no higher sheaf cohomology).

Long exact sequence. For every short exact sequence of abelian sheaves on $C$ there is a natural long exact sequence of sheaf cohomology groups $\cdots \to H^q(C,{\mathcal{F}}^{\mathrm{\prime }})\to H^q(C,\mathcal{F})\to H^q(C,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})\to H^{q+1}(C,{\mathcal{F}}^{\mathrm{\prime }})\to \cdots$ (Long exact sequence of sheaf cohomology, Sheaf cohomology as right derived global sections).

Coherence. A curve is of finite type over the field $k$ (Curves over a field); a morphism of finite type provides affine charts $\mathrm{Spec}B$ with $B$ a finitely generated algebra over the coordinate ring of an affine open of the target (Locally finite type and finite type morphisms), a field is Noetherian (A field has only the zero ideal and itself, hence is Noetherian) and a finitely generated algebra over a Noetherian ring is Noetherian (Every algebra of finite type over a Noetherian ring is a Noetherian ring), so $C$ is locally Noetherian: it has an affine open cover by spectra of Noetherian rings (Locally Noetherian and Noetherian schemes). For every divisor $D^{\mathrm{\prime }}$ the sheaf ${\mathcal{O}}_C(D^{\mathrm{\prime }})$ is a coherent ${\mathcal{O}}_C$-module (Finite-dimensionality of the Riemann-Roch space), and on a locally Noetherian scheme the kernel, image and cokernel of a morphism of coherent ${\mathcal{O}}_C$-modules are coherent (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves).

Linear algebra over $k$. For a linear map $T:V\to W$ with $V$ finite-dimensional, ${\dim }_{k}V={\dim }_k\ker T+{\dim }_k\mathrm{im}T$ (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$); the formula $\tilde{T}(v+\ker T)=T(v)$ defines an isomorphism $V\mathrm{/}\ker T\to \mathrm{im}T$ (First isomorphism theorem for vector spaces: $V\mathrm{/}\ker T$ is isomorphic to $\mathrm{im}T$); for $W\le V$ with $V$ finite-dimensional, ${\dim }_k(V\mathrm{/}W)={\dim }_{k}V-{\dim }_{k}W$ (A quotient basis lifts to a basis adapted to $W$); and ${\dim }_k$ of a finite-dimensional $k$-vector space is a nonnegative integer (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). Hence for an exact sequence of $k$-vector spaces $0\to A\to B\to C\to 0$ with $A$ and $C$ finite-dimensional one has ${\dim }_{k}B={\dim }_{k}A+{\dim }_{k}C$.

The Axiom of Choice enters through the sheaf-cohomology suppliers of [F7] and [F8], the coherence supplier [F9], the curve Cartier-to-Weil supplier in [F2], and the local-DVR supplier [F3]. The only additional premise needed there is Dependent Choice, which follows from the stated Axiom of Choice by [F12]; the argument below makes no further selection (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain).

In ZF, the Axiom of Choice implies Dependent Choice (AC implies DC implies countable choice). Hence the stated Choice assumption supplies the Dependent Choice premise of Cartier and Weil divisors agree on a smooth curve.