Degree of the coherent Hilbert polynomial

Given: The Axiom of Choice as inherited, a field $k$, a closed immersion $i:X\hookrightarrow {\mathbb{P}}_k^n$, the invertible sheaf ${\mathcal{O}}_X(1)=i^{\ast }\mathcal{O}(1)$ with its twists, and a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$.

The Hilbert polynomial: for every coherent $\mathcal{G}$ on $X$ there is a unique $P_{\mathcal{G}}\in \mathbb{Q}[t]$ with $P_{\mathcal{G}}(m)=\chi (X,\mathcal{G}(m))$ for every $m\in \mathbb{Z}$; there is $m_0$ with $P_{\mathcal{G}}(m)=h_{\mathcal{G}}(m)={\dim }_k{H}^0(X,\mathcal{G}(m))$ for all $m\ge m_0$; and $P_{\mathcal{G}}=0$ when $\mathcal{G}=0$. (Euler characteristic is a Hilbert polynomial, Hilbert function and Euler characteristic on a projective scheme, Euler characteristic of a coherent sheaf, Sheaf cohomology as right derived global sections)

Support and dimension: for a coherent $\mathcal{G}$ on $X$ the support $\mathrm{Supp}\mathcal{G}$ is closed in $X$ (Support of a module sheaf), and $X$ is a Noetherian topological space whose closed subsets have a well-defined chain dimension with $\dim \mathrm{\varnothing }=-\mathrm{\infty }$, so $\mathrm{Supp}\mathcal{G}$ has a nonnegative dimension when $\mathcal{G}\mathrm{\ne }0$. The charts of ${\mathbb{P}}_k^n$ are spectra of the Noetherian rings $k[x_{\mathrm{\ell }}^{(i)}]$ (Relative projective space from standard charts, A field has only the zero ideal and itself, hence is Noetherian, If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$, Locally Noetherian and Noetherian schemes, Noetherian topological spaces via ACC on opens or DCC on closed subsets, Subspaces of a Noetherian space and its compact open subsets, Chain dimension and the empty-space convention)

Additivity of $\chi$ in short exact sequences of coherent modules on the proper $k$-scheme $X$ (Euler characteristic is additive in short exact sequences, Exact sequences of sheaves).

The hyperplane lemma: if $k$ is infinite and $\mathcal{G}\mathrm{\ne }0$ is coherent with $d:=\dim \mathrm{Supp}\mathcal{G}$, there is a linear form $\mathrm{\ell }$ such that for every $m$ the map $\cdot \mathrm{\ell }:\mathcal{G}(m-1)\to \mathcal{G}(m)$ is injective with coherent cokernel ${\mathcal{G}}^{\mathrm{\prime }}(m)$ satisfying $\mathrm{Supp}{\mathcal{G}}^{\mathrm{\prime }}(m)=\mathrm{Supp}\mathcal{G}\cap V(\mathrm{\ell })$; if $d\ge 1$ then $\dim \mathrm{Supp}{\mathcal{G}}^{\mathrm{\prime }}(m)=d-1$ for every $m$, and if $d=0$ then ${\mathcal{G}}^{\mathrm{\prime }}(m)=0$ for every $m$. (Regular hyperplane step for coherent support induction)

Exactness and the fixed cokernel: tensoring by the invertible sheaf ${\mathcal{O}}_X(1)$ is exact, twists of coherent modules are coherent, and for the sheaves of [F4] the twist of the exact sequence $0\to \mathcal{G}(-1)\to \mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}(0)\to 0$ by ${\mathcal{O}}_X(1{)}^{\otimes m}$ is canonically $0\to \mathcal{G}(m-1)\to \mathcal{G}(m)\to {\mathcal{G}}^{\mathrm{\prime }}(m)\to 0$, so that ${\mathcal{G}}^{\mathrm{\prime }}(m)\cong {\mathcal{G}}^{\mathrm{\prime }}(0)\otimes {\mathcal{O}}_X(1{)}^{\otimes m}$ and $\chi (X,\mathcal{G}(m))-\chi (X,\mathcal{G}(m-1))=\chi (X,{\mathcal{G}}^{\mathrm{\prime }}(0)(m))$ by [F3]. (Invertible sheaves, Tensor product of sheaves of modules, The stalk of a tensor product sheaf is the tensor product of the stalks, Under the stated choice boundary, free modules are projective and hence flat, A sequence of abelian sheaves is exact exactly when it is exact on every stalk, Twists of a quasi-coherent sheaf)

Positive sections at large twists: let $k$ be any field and $\mathcal{G}\mathrm{\ne }0$ coherent on $X$. The pushforward $i_{\ast }\mathcal{G}$ is a nonzero coherent module on the locally Noetherian ${\mathbb{P}}_k^n$, and $\mathcal{O}(1)$ is ample on ${\mathbb{P}}_k^n$ because the identity is a closed immersion over the affine base $\mathrm{Spec}k$ pulling $\mathcal{O}(1)$ back to itself. Hence Eventual generation of coherent projective twists gives $m_1$ with $(i_{\ast }\mathcal{G})(m)$ globally generated (Global generation by the evaluation map) for every $m\ge m_1$; such a nonzero globally generated module has a nonzero global section, because it is the image of a direct sum of copies of $\mathcal{O}$ indexed by its global sections and hence is zero if all of them vanish. Finally $H^0({\mathbb{P}}_k^n,(i_{\ast }\mathcal{G})(m))\cong H^0({\mathbb{P}}_k^n,i_{\ast }(\mathcal{G}(m)))\cong H^0(X,\mathcal{G}(m))$ for every $m\ge 0$, using the projection identity $i_{\ast }(\mathcal{G}\otimes i^{\ast }\mathcal{H})\cong (i_{\ast }\mathcal{G})\otimes \mathcal{H}$ of a closed immersion (checked on affine charts) and the invariance of cohomology under $i_{\ast }$. Consequently there exist arbitrarily large $m$ with $h_{\mathcal{G}}(m)>0$. (Closed immersion preserves cohomology and coherent pushforward, Closed immersions are affine quotients and survive base change, Direct image of a sheaf along a continuous map, Absolute ampleness by affine section opens, Relative very ampleness in the finite projective-space convention, Relative very ampleness implies relative ampleness, Relative projective space from standard charts, Coherent module sheaves, Sheaf cohomology as right derived global sections)

Base change to an infinite field: for a field extension $K\mathrm{/}k$ with base change $g:X_K\to X$ and ${\mathcal{G}}_K=g^{\ast }\mathcal{G}$, one has $\dim \mathrm{Supp}{\mathcal{G}}_K=\dim \mathrm{Supp}\mathcal{G}$ for every coherent $\mathcal{G}$ (Support dimension under field extension) and $\chi (X_K,{\mathcal{H}}_K)=\chi (X,\mathcal{H})$ for every coherent $\mathcal{H}$ (Flat field extension commutes with coherent cohomology); moreover pullback of quasi-coherent modules is monoidal and $g^{\ast }{\mathcal{O}}_X(1)\cong {\mathcal{O}}_{X_K}(1)$, so $g^{\ast }(\mathcal{H}(m))\cong {\mathcal{H}}_K(m)$ for every $m\in \mathbb{Z}$, exactly as in the base-change step of Euler characteristic is a Hilbert polynomial (where the compatibility of the twisting sheaf with base change and the monoidality of pullback are likewise recorded as proof obligations). The rational function field $K=k(t)$ is an infinite field extension of $k$ (For a field $F$, $F(t)=\mathrm{Frac}(F[t])$ is its rational function field; in particular $\mathbb{R}(t)=\mathrm{Frac}(\mathbb{R}[t])$, The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain, Scheme pullback preserves quasi-coherence, Associativity of tensor products for compatible bimodules, Pullback of a module along a morphism of ringed spaces)

Finite differences of polynomials: for $Q\in \mathbb{Q}[t]$ of degree $e\ge 0$ with leading coefficient $a\mathrm{\ne }0$ and $R\in \mathbb{Q}[t]$ with $R(m)-R(m-1)=Q(m)$ for all $m$, the polynomial $R$ has degree $e+1$ with leading coefficient $a\mathrm{/}(e+1)$; if $R\mathrm{\ne }0$ has degree $f\ge 1$ then $R(m)-R(m-1)$ has degree $f-1$ and leading coefficient $f$ times the leading coefficient of $R$, while if $R$ is constant then $R(m)-R(m-1)=0$. Consequently a polynomial identity $R(m)-R(m-1)=Q(m)$ with $Q$ of exact degree $e\ge 0$ forces $\mathrm{deg}R=e+1$. [algebra]

The Axiom of Choice is the choice principle named in the statement, inherited from the suppliers cited in [F1], [F4], [F6] and [F7]. (The Axiom of Choice)