Monotonicity of L(D) in the divisor

Set-up and coefficients. By [F1] write $D={\sum }_x{n}_x[x]$ and $E={\sum }_x{m}_x[x]$ with $n_x\le m_x$ for every closed point $x$, both sums having finite support, and write $c_x:=m_x-n_x\ge 0$ for the coefficient of $E-D$; the degree is ${\mathrm{deg}}_k(E-D)={\sum }_x{c}_x[\kappa (x):k]$.

Order description of the Riemann-Roch space. By [F2], for $f\in k(C{)}^{\times }$ one has $f\in L(D)$ if and only if $\mathrm{div}(f)+D\ge 0$, and by [F3] this is equivalent to the coefficientwise condition ${\mathrm{ord}}_x(f)+n_x\ge 0$ for every closed point $x$; moreover $L(D)$ is a $k$-subspace of $k(C)$ and $0\in L(D)$.

The sheaf picture. By [F2] the attached invertible sheaves are subsheaves ${\mathcal{O}}_C(D)\subseteq {\mathcal{O}}_C(E)\subseteq K_C$ of the constant sheaf of rational functions, with stalks cut out by the very order conditions of step 1.2 and with $H^0(C,{\mathcal{O}}_C(D))=L(D)$ and $H^0(C,{\mathcal{O}}_C(E))=L(E)$.

Monotonicity of the spaces. Let $f\in L(D)$. By step 1.2, ${\mathrm{ord}}_x(f)+n_x\ge 0$ for every closed point $x$; since $n_x\le m_x$ by step 1.1, also ${\mathrm{ord}}_x(f)+m_x\ge 0$ for every $x$, so $f\in L(E)$ by step 1.2 again. Hence $L(D)\subseteq L(E)$ as subsets of $k(C)$, and both are $k$-subspaces by [F2].

The one-point case: the evaluation map. Now let $E=D+p$ for a single closed point $p$, and let $a:=n_p$ be the coefficient of $D$ at $p$. Choose a uniformizer $t$ of the discrete valuation ring ${\mathcal{O}}_{C,p}$. The canonical evaluation of sections of ${\mathcal{O}}_C(D+p)$ at $p$ has target fiber ${\mathcal{O}}_C(D+p){\mathrm{\mid }}_p$; in the local frame $t^{-a-1}$, its coordinate is ${\phi }_t(f)=t^{a+1}f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>(t)\in \kappa (p)$. This coordinate description depends on $t$, but is well defined: $f\in L(D+p)$ gives ${\mathrm{ord}}_p(f)\ge -a-1$, hence ${\mathrm{ord}}_p(t^{a+1}f)\ge 0$ and $t^{a+1}f\in {\mathcal{O}}_{C,p}$ by [F3] and [F4]. The map ${\phi }_t:L(D+p)\to \kappa (p)$ is $k$-linear, since multiplication by $t^{a+1}$ and reduction modulo $(t)$ are $k$-linear on ${\mathcal{O}}_{C,p}$.

Its kernel is $L(D)$. If $f\in L(D)$ then ${\mathrm{ord}}_p(f)\ge -a$, so $t^{a+1}f\in (t)$, and $\phi (f)=0$. Conversely, if $\phi (f)=0$ then $t^{a+1}f\in (t)$, that is, ${\mathrm{ord}}_p(t^{a+1}f)\ge 1$, so ${\mathrm{ord}}_p(f)\ge -a$ by [F3]; for points $x\mathrm{\ne }p$ the conditions $f\in L(D)$ and $f\in L(D+p)$ coincide because $D$ and $D+p$ have the same coefficients away from $p$; hence $f\in L(D)$ by step 1.2. Therefore $\ker \phi =L(D)$.

The morphism of invertible subsheaves. Both ${\mathcal{O}}_C(D)$ and ${\mathcal{O}}_C(E)$ are invertible subsheaves of the constant sheaf $K_C$ by [F2]; the stalkwise inclusion of step 2.1, given by the coefficientwise comparison of step 2.2, defines the natural morphism ${\mathcal{O}}_C(D)\to {\mathcal{O}}_C(E)$, which is injective on every stalk and hence on sections; the induced map on global sections is the inclusion $L(D)\subseteq L(E)$, and the dimension formula $l(D)\le l(E)$ follows from [F5] and [F6].

The quotient embeds in the residue field. By step 2.4 and [F6] the first isomorphism theorem gives a $k$-linear isomorphism $L(D+p)\mathrm{/}L(D)\to \mathrm{im}\phi$, so $L(D+p)\mathrm{/}L(D)$ embeds $k$-linearly into $\kappa (p)$; since $L(D+p)$ is finite-dimensional by [F5], rank-nullity together with the quotient formula gives ${\dim }_{k}L(D+p)\mathrm{/}L(D)={\dim }_{k}L(D+p)-{\dim }_{k}L(D)={\dim }_k\mathrm{im}\phi \le {\dim }_k\kappa (p)=[\kappa (p):k],$ the inequality by subspace monotonicity and the last equality by [F4]. In particular $l(D+p)={\dim }_{k}L(D+p)\le l(D)+[\kappa (p):k]$.

Iteration over the support. Enumerate the finite support of $E-D$ as points $x_1,\dots ,x_r$ and let $c_i=c_{x_i}\ge 1$ be the multiplicities of step 1.1. Consider the finite chain of divisors starting at $D$ and adding one copy of $[x_i]$ at a time, $c_i$ times for each $i$, ending at $E$; every successive difference is a single closed point $q$, so step 3.2 applied to the pair of consecutive divisors gives an increase of $l$ by at most $[\kappa (q):k]$. Summing the $r$ chains of inequalities gives $l(E)\le l(D)+{\sum }_i{c}_i[\kappa (x_i):k]=l(D)+{\mathrm{deg}}_k(E-D)$ by step 1.1.

Conclusion and choice accounting. Step 2.2 gives $L(D)\subseteq L(E)$ and step 3.1 the injective morphism of invertible subsheaves together with $l(D)\le l(E)$; steps 2.3 and 2.4 identify $L(D)$ as the kernel of the evaluation $L(D+p)\to \kappa (p)$, step 3.2 embeds the quotient in $\kappa (p)$ with the bound ${\dim }_{k}L(D+p)\mathrm{/}L(D)\le [\kappa (p):k]$, and step 4.1 gives $l(E)\le l(D)+{\mathrm{deg}}_k(E-D)$ in general. The Axiom of Choice is used only through the suppliers recorded in [F7], namely the DVR structure of [F4], the finiteness results of [F5], and the Cartier-to-Weil route of [F2]; no further selection is made above.