The exact sequence for adding one point to a divisor

Setup and local structure at $p$. Write $D={\sum }_x{n}_x[x]$, put $a:=n_p(D)$, and recall $d=[\kappa (p):k]={\dim }_k\kappa (p)$ by [F1]; the divisor $D+p$ has coefficient $a+1$ at $p$ and the same coefficient as $D$ at every other closed point. By [F3], ${\mathcal{O}}_{C,p}$ is a discrete valuation ring with maximal ideal $(t)$ for a uniformizer $t$, the residue field is $\kappa (p)={\mathcal{O}}_{C,p}\mathrm{/}(t)$, ${\mathrm{ord}}_p$ is additive with ${\mathrm{ord}}_p(f)\ge 0$ if and only if $f\in {\mathcal{O}}_{C,p}$, and $t^{m}f\in {\mathcal{O}}_{C,p}$ if and only if ${\mathrm{ord}}_p(f)\ge -m$ for $m\in \mathbb{Z}$.

The curve is locally Noetherian. By [F9] the structure morphism $C\to \mathrm{Spec}k$ is of finite type, so every point of $C$ has an affine open neighbourhood $\mathrm{Spec}B$ with $B$ a finitely generated $k$-algebra; $k$ is Noetherian and a finitely generated algebra over a Noetherian ring is Noetherian, so each such $B$ is Noetherian, and $C$ is locally Noetherian by [F9].

The two subsheaves and their order conditions. By [F2] the divisors $D$ and $D+p$ carry inclusions of invertible subsheaves ${\mathcal{O}}_C(D)\subseteq {\mathcal{O}}_C(D+p)\subseteq K_C$ of the constant sheaf of rational functions such that, for every nonempty open $U$, ${\mathcal{O}}_C(D)(U)$ consists of the $f\in k(C)$ with ${\mathrm{ord}}_x(f)+n_x\ge 0$ for every closed point $x\in U$, and likewise for ${\mathcal{O}}_C(D+p)$ with the coefficient at $p$ raised by one; their section groups on the empty open are zero. In particular, if $p\in U$ and $f\in {\mathcal{O}}_C(D+p)(U)$, then ${\mathrm{ord}}_p(f)\ge -a-1$. By [F4] the natural morphism ${\mathcal{O}}_C(D)\to {\mathcal{O}}_C(D+p)$ of ${\mathcal{O}}_C$-modules is injective, given by the inclusion of subsheaves of $K_C$.

Definition of the evaluation morphism $\psi$. Define for every open $U\subseteq C$ a map ${\psi }_U:{\mathcal{O}}_C(D+p)(U)\to (i_{p,\ast }\kappa (p))(U)$ by ${\psi }_U:=0$ when $p\mathrm{\notin }U$, and by ${\psi }_U(f):=t^{a+1}f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>(t)\in \kappa (p)$ when $p\in U$, where $t$ is the uniformizer of step 1.1. This is well defined: for $f\in {\mathcal{O}}_C(D+p)(U)$ with $p\in U$, step 1.3 gives ${\mathrm{ord}}_p(f)\ge -a-1$, hence ${\mathrm{ord}}_p(t^{a+1}f)\ge 0$, so $t^{a+1}f\in {\mathcal{O}}_{C,p}$ by step 1.1 and its class in $\kappa (p)={\mathcal{O}}_{C,p}\mathrm{/}(t)$ is defined.

Cohomology of the skyscraper. Let $Z=\{p\}$ be the one-point space with the subspace topology, with inclusion $i:Z\hookrightarrow C$, and let $\mathcal{F}$ be the sheaf of abelian groups on $Z$ with $\mathcal{F}(Z)=\kappa (p)$; by [F7] the direct image is $(i_{\ast }\mathcal{F})(V)=\mathcal{F}(i^{-1}V)$, which equals $\kappa (p)=(i_{p,\ast }\kappa (p))(V)$ for opens $V$ containing $p$ and $0$ otherwise, so $i_{\ast }\mathcal{F}=i_{p,\ast }\kappa (p)$ and $H^q(C,i_{p,\ast }\kappa (p))\cong H^q(Z,\mathcal{F})$ for every $q\ge 0$ by [F7]. By [F7] again $H^0(Z,\mathcal{F})\cong \mathcal{F}(Z)=\kappa (p)$ and $H^q(Z,\mathcal{F})=0$ for $q>0$; hence $H^0(C,i_{p,\ast }\kappa (p))\cong \kappa (p)$, of $k$-dimension ${\dim }_k\kappa (p)=d$ by step 1.1, and $H^q(C,i_{p,\ast }\kappa (p))=0$ for every $q\ge 1$. This is part (2) of the Statement.

$\psi$ is a morphism of sheaves of ${\mathcal{O}}_C$-modules. For every open $U$ the map ${\psi }_U$ is additive, because multiplication by $t^{a+1}$ and reduction modulo $(t)$ are additive on the groups involved; it is ${\mathcal{O}}_C(U)$-linear, because for $g\in {\mathcal{O}}_C(U)$ the germ of $g$ at $p$ lies in ${\mathcal{O}}_{C,p}$ with class $\bar{g}\in \kappa (p)$ and $t^{a+1}(gf)=g\cdot t^{a+1}f$ holds in ${\mathcal{O}}_{C,p}$, so ${\psi }_U(gf)=\bar{g}{\psi }_U(f)$; and the maps are compatible with restrictions: for $V\subseteq U$ with $p\mathrm{\notin }U$ both maps are zero, for $p\in V\subseteq U$ the two subsheaves of $K_C$ have the same element $f$ and $i_{p,\ast }\kappa (p)$ has identity restrictions on opens containing $p$, while for $p\in U$, $p\mathrm{\notin }V$ the target $(i_{p,\ast }\kappa (p))(V)$ is the zero group. Hence $\psi$ is a morphism of sheaves of ${\mathcal{O}}_C$-modules.

The kernel of $\psi$ is ${\mathcal{O}}_C(D)$. At an open $U$ with $p\mathrm{\notin }U$, step 1.3 gives ${\mathcal{O}}_C(D)(U)={\mathcal{O}}_C(D+p)(U)$ and ${\psi }_U=0$, so $\ker {\psi }_U={\mathcal{O}}_C(D)(U)$. At an open $U$ with $p\in U$, step 1.1 shows that $f\in \ker {\psi }_U$ is equivalent to $t^{a+1}f\in t{\mathcal{O}}_{C,p}$, equivalently to ${\mathrm{ord}}_p(f)\ge -a$, and together with the order conditions at the other points of $U$, which are the same for $D$ and $D+p$, this is exactly the condition $f\in {\mathcal{O}}_C(D)(U)$; the converse is immediate. Therefore $\ker \psi ={\mathcal{O}}_C(D)$ as subsheaves of ${\mathcal{O}}_C(D+p)$.

Stalks of the skyscraper and surjectivity of $\psi$. By [F7] the skyscraper $i_{p,\ast }\kappa (p)$ has value $\kappa (p)$ on the opens containing $p$, with identity restrictions, and value $0$ on the opens not containing $p$; hence its stalk at $p$ is $\kappa (p)$, while at a point $x\mathrm{\ne }p$ every section over an open containing $x$ restricts to zero on the open complement of the closed set $\{p\}$, which still contains $x$, so the stalk at $x$ is $0$. The map ${\psi }_x$ is a map into the zero group for $x\mathrm{\ne }p$, and at $p$ the map ${\psi }_p$ is surjective: for $u\in {\mathcal{O}}_{C,p}$ the rational function $t^{-a-1}u$ satisfies ${\mathrm{ord}}_p(t^{-a-1}u)\ge -a-1$, hence lies in ${\mathcal{O}}_C(D+p{)}_p$ by the stalk description of [F2], so it is the germ of a section of ${\mathcal{O}}_C(D+p)$ near $p$, and ${\psi }_p(t^{-a-1}u)=u<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>(t)$ because $\psi$ is defined by $f\mapsto t^{a+1}f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>(t)$ on sections over opens containing $p$ as in step 2.1.

Exactness of the single-point sequence. Consider the sequence of sheaves of ${\mathcal{O}}_C$-modules $0\to {\mathcal{O}}_C(D)\to {\mathcal{O}}_C(D+p)\stackrel{\psi }{\to }{i}_{p,\ast }\kappa (p)\to 0$. At a point $x\mathrm{\ne }p$ the stalk sequence is $0\to A_x\to A_x\to 0\to 0$, exact because the two subsheaves agree away from $p$ by step 1.3 and the target stalk is $0$; at $p$ the stalk sequence is $0\to t^{-a}{\mathcal{O}}_{C,p}\to t^{-a-1}{\mathcal{O}}_{C,p}\to \kappa (p)\to 0$ by steps 1.1 and 3.3, and it is exact: the first map is the inclusion of the subgroup $t^{-a}{\mathcal{O}}_{C,p}$, so its kernel vanishes; the kernel of ${\psi }_p$ is the image of that inclusion, because the kernel sheaf of $\psi$ is ${\mathcal{O}}_C(D)$ by step 3.2 and the stalk of a kernel is the kernel of the stalk map by [F5]; and ${\psi }_p$ is surjective by step 3.3. By the stalkwise criterion for exactness [F5] the sequence is a short exact sequence, so the cokernel of the inclusion ${\mathcal{O}}_C(D)\to {\mathcal{O}}_C(D+p)$ is its third term $i_{p,\ast }\kappa (p)$.

Coherence of the third term. By [F9] the curve $C$ is locally Noetherian, and by [F9] the invertible sheaves ${\mathcal{O}}_C(D)$ and ${\mathcal{O}}_C(D+p)$ are coherent ${\mathcal{O}}_C$-modules; the present sequence is a short exact sequence of ${\mathcal{O}}_C$-modules whose left-hand map is a morphism of coherent modules with cokernel $i_{p,\ast }\kappa (p)$, so the cokernel is a coherent ${\mathcal{O}}_C$-module by [F9]. This and step 4.1 give part (1) of the Statement, with the cokernel identified as the skyscraper at $p$ with value $\kappa (p)$.

The extension step for the iteration. Let $p$ be a closed point with $E\ge p$ and put $E^{\mathrm{\prime }}:=E-p$; set $A:={\mathcal{O}}_C(D)$, $B:={\mathcal{O}}_C(D+E^{\mathrm{\prime }})$ and $C^{\mathrm{\prime }}:={\mathcal{O}}_C(D+E)={\mathcal{O}}_C(D+E^{\mathrm{\prime }}+p)$, so that $A\subseteq B\subseteq C^{\mathrm{\prime }}$ are nested subsheaves of $K_C$ by steps 1.3 with injective inclusion morphisms by [F4]. Define $Q_{E^{\mathrm{\prime }}}:=\mathrm{coker}(A\to B)$ and $Q_E:=\mathrm{coker}(A\to C^{\mathrm{\prime }})$. By the third isomorphism theorem in an abelian category [F6], applied to the subobjects $A\le B\le C^{\mathrm{\prime }}$, the quotient $Q_E\mathrm{/}{Q}_{E^{\mathrm{\prime }}}$ is canonically isomorphic to $C^{\mathrm{\prime }}\mathrm{/}B=\mathrm{coker}(B\to C^{\mathrm{\prime }})$, and by step 4.1 applied to the divisor $D+E^{\mathrm{\prime }}$ and the point $p$ the latter is $i_{p,\ast }\kappa (p)$; hence $0\to Q_{E^{\mathrm{\prime }}}\to Q_E\to i_{p,\ast }\kappa (p)\to 0$ is a short exact sequence of sheaves of ${\mathcal{O}}_C$-modules.

Support of $Q_E$. Let $x$ be a closed point with $x\mathrm{\notin }\mathrm{Supp}(E)$; then the coefficients of $D$ and $D+E$ at $x$ coincide, so over an open neighbourhood of $x$ the order conditions defining the two subsheaves of step 1.3 are the same and the inclusion $A\subseteq C^{\mathrm{\prime }}$ induces an isomorphism of stalks at $x$; since the stalk of a cokernel is the cokernel of the stalk map by [F5], the stalk $(Q_E{)}_x$ is the cokernel of an isomorphism and hence $0$. Therefore $Q_E$ is supported on the finite set $\mathrm{Supp}(E)$.

Coherence of $Q_E$. The sheaves ${\mathcal{O}}_C(D)$ and ${\mathcal{O}}_C(D+E)$ are coherent ${\mathcal{O}}_C$-modules by [F9], the curve is locally Noetherian by step 1.2, and $Q_E$ is their cokernel by step 5.2; hence $Q_E$ is a coherent ${\mathcal{O}}_C$-module by [F9].

The induction on ${\mathrm{deg}}_{k}E$. We prove by induction on the nonnegative integer ${\mathrm{deg}}_{k}E$ that ${\dim }_k{H}^0(C,Q_E)={\mathrm{deg}}_k(E)$ and $H^q(C,Q_E)=0$ for every $q\ge 1$, for every effective divisor $E$. If ${\mathrm{deg}}_{k}E=0$ then $E=0$ by [F1] and $Q_E$ is the cokernel of the identity of ${\mathcal{O}}_C(D)$, hence the zero sheaf, so both assertions hold. If $E\mathrm{\ne }0$, choose a closed point $p\in \mathrm{Supp}(E)$ and put $E^{\mathrm{\prime }}:=E-p$, an effective divisor with ${\mathrm{deg}}_k{E}^{\mathrm{\prime }}={\mathrm{deg}}_{k}E-[\kappa (p):k]<{\mathrm{deg}}_{k}E$ by [F1]; by step 5.2 there is a short exact sequence $0\to Q_{E^{\mathrm{\prime }}}\to Q_E\to i_{p,\ast }\kappa (p)\to 0$, whose long exact sequence [F8] contains the exact segment $H^0(Q_{E^{\mathrm{\prime }}})\to H^0(Q_E)\to \kappa (p)\to H^1(Q_{E^{\mathrm{\prime }}})\to H^1(Q_E)\to H^1(i_{p,\ast }\kappa (p))=0$, where $H^q(i_{p,\ast }\kappa (p))=0$ for $q\ge 1$ by step 2.2. Since $H^1(Q_{E^{\mathrm{\prime }}})=0$ by the induction hypothesis, the segment collapses to the exact sequence $0\to H^0(Q_{E^{\mathrm{\prime }}})\to H^0(Q_E)\to \kappa (p)\to 0$, so [F10] gives ${\dim }_k{H}^0(Q_E)={\dim }_k{H}^0(Q_{E^{\mathrm{\prime }}})+{\dim }_k\kappa (p)={\mathrm{deg}}_k{E}^{\mathrm{\prime }}+[\kappa (p):k]={\mathrm{deg}}_{k}E$; and for $q\ge 1$ the exactness of $H^q(Q_{E^{\mathrm{\prime }}})\to H^q(Q_E)\to H^q(i_{p,\ast }\kappa (p))$, with both outer groups zero by the induction hypothesis and step 2.2, gives $H^q(Q_E)=0$.

Conclusion and choice accounting. Step 4.1 gives the short exact sequence of part (1) with its cokernel identified, and step 5.1 the coherence of its terms; step 2.2 gives part (2); and steps 6.1, 6.2 and 6.3 give, for every effective divisor $E$, the short exact sequence with cokernel $Q_E$, its support on $\mathrm{Supp}(E)$, its coherence and the dimension formula ${\dim }_k{H}^0(C,Q_E)={\mathrm{deg}}_k(E)$, hence the promised bounds $0\le {\dim }_k{H}^0(C,Q_E)\le {\mathrm{deg}}_k(E)$. The Axiom of Choice is used only through the suppliers recorded in [F11], namely the sheaf-cohomology results of [F7] and [F8], the coherence supplier of [F9], the flagged Cartier-divisor suppliers of [F2], and the local-DVR supplier of [F3]; the only selections made above are the uniformizer $t$ supplied by [F3] and the point $p$ chosen in the finite set $\mathrm{Supp}(E)$.