Euler characteristic is a Hilbert polynomial

Setup and the zero sheaf. By [F1] the scheme $X$ is proper over $k$, every twist $\mathcal{F}(m)$ is coherent, and $\chi (X,\mathcal{F}(m))$ and $h_{\mathcal{F}}(m)$ are defined for every $m\in \mathbb{Z}$. If $\mathcal{F}=0$ then every $\mathcal{F}(m)=0$, all cohomology groups vanish and $\chi (X,\mathcal{F}(m))=0=h_{\mathcal{F}}(m)$ for every $m$ by [F1], so $P_{\mathcal{F}}=0$ is a Hilbert polynomial and part 3 of the statement holds. Assume from 1.2 through 1.8 that $\mathcal{F}\mathrm{\ne }0$ and that $k$ is infinite.

Finite support dimension. By [F5] the support $\mathrm{Supp}\mathcal{F}$ is a nonempty closed subset of the Noetherian space $X$, and its intersections with the standard charts are closed subsets of $\mathrm{Spec}k[t_1,\dots ,t_n]$. Irreducible closed subsets of a spectrum correspond to prime ideals by A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point, and chains in a closed subset are chains in the ambient space. Each chart intersection therefore has dimension at most $n$ by A polynomial ring in n variables over a field has dimension n. Applying Dimension can be computed on an open cover to the induced finite open cover of the support gives $0\le d:=\dim \mathrm{Supp}\mathcal{F}\le n$. Thus the following induction has a finite integer parameter; Noetherianity alone would not suffice.

The induction proposition. For each integer $e\ge 0$ let $\mathrm{\Phi }(e)$ be the assertion: every coherent ${\mathcal{O}}_X$-module $\mathcal{G}$ with $\dim \mathrm{Supp}\mathcal{G}\le e$ admits $Q\in \mathbb{Q}[t]$ with $Q(m)=\chi (X,\mathcal{G}(m))$ for every $m\in \mathbb{Z}$. The zero module satisfies $\mathrm{\Phi }(e)$ with $Q=0$ by [F1] and $\dim \mathrm{Supp}0=-\mathrm{\infty }$ by [F5]. We prove $\mathrm{\Phi }(e)$ for all $e\ge 0$ by induction on $e$.

Base case $e=0$. Let $\mathcal{G}\mathrm{\ne }0$ be coherent with $\dim \mathrm{Supp}\mathcal{G}=0$. By [F4] applied to $\mathcal{G}$ (which is nonzero, with $d=0$) there is $\mathrm{\ell }$ such that for every $m$ the map $\cdot \mathrm{\ell }:\mathcal{G}(m-1)\to \mathcal{G}(m)$ is injective with cokernel ${\mathcal{G}}^{\mathrm{\prime }}(m)$ satisfying ${\mathcal{G}}^{\mathrm{\prime }}(m)=0$. Hence for every $m$ the sequence $0\to \mathcal{G}(m-1)\to \mathcal{G}(m)\to 0\to 0$ is exact, and additivity [F2] gives $\chi (X,\mathcal{G}(m))=\chi (X,\mathcal{G}(m-1))+\chi (X,0)=\chi (X,\mathcal{G}(m-1))$. Thus $m\mapsto \chi (X,\mathcal{G}(m))$ is constant with value $\chi (X,\mathcal{G}(0))$, and the constant polynomial $Q(t)=\chi (X,\mathcal{G}(0))\in \mathbb{Q}[t]$ realizes $\mathrm{\Phi }(0)$.

Induction step, $e\ge 1$. Let $\mathcal{G}\mathrm{\ne }0$ be coherent with $d^{\mathrm{\prime }}:=\dim \mathrm{Supp}\mathcal{G}$ satisfying $1\le d^{\mathrm{\prime }}\le e$. By [F4] applied to $\mathcal{G}$ there is $\mathrm{\ell }$; let ${\mathcal{G}}^{\mathrm{\prime }}$ be the cokernel of $\cdot \mathrm{\ell }:\mathcal{G}(-1)\to \mathcal{G}(0)$, so that ${\mathcal{G}}^{\mathrm{\prime }}$ is a coherent module with $\dim \mathrm{Supp}{\mathcal{G}}^{\mathrm{\prime }}=d^{\mathrm{\prime }}-1\le e-1$ by [F4] (its case $m=0$). By the induction hypothesis $\mathrm{\Phi }(e-1)$ there is $Q\in \mathbb{Q}[t]$ with $Q(m)=\chi (X,{\mathcal{G}}^{\mathrm{\prime }}(m))$ for every $m$.

The difference equation. Fix $m\in \mathbb{Z}$ and tensor the exact sequence $0\to \mathcal{G}(-1)\stackrel{\cdot \mathrm{\ell }}{\to }\mathcal{G}(0)\to {\mathcal{G}}^{\mathrm{\prime }}\to 0$ of 1.5 with the invertible sheaf ${\mathcal{O}}_X(1{)}^{\otimes m}$; by [F3] the result is exact, and by the coherence of the twist conventions of Twists of a quasi-coherent sheaf it is canonically the sequence $0\to \mathcal{G}(m-1)\stackrel{\cdot \mathrm{\ell }}{\to }\mathcal{G}(m)\to {\mathcal{G}}^{\mathrm{\prime }}\otimes {\mathcal{O}}_X(1{)}^{\otimes m}\to 0$. Since the cokernel of the middle map is by [F4] the module ${\mathcal{G}}^{\mathrm{\prime }}(m)$, uniqueness of cokernels gives a canonical isomorphism ${\mathcal{G}}^{\mathrm{\prime }}(m)\cong {\mathcal{G}}^{\mathrm{\prime }}{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(1{)}^{\otimes m}$; consequently additivity [F2] gives $\chi (X,\mathcal{G}(m))-\chi (X,\mathcal{G}(m-1))=\chi (X,{\mathcal{G}}^{\mathrm{\prime }}(m))=Q(m)$ by 1.5.

Antiderivative and the constant. By [F6] choose $R\in \mathbb{Q}[t]$ with $R(m)-R(m-1)=Q(m)$ for every $m\in \mathbb{Z}$, and put $C:=\chi (X,\mathcal{G}(0))-R(0)\in \mathbb{Q}$. Then $D(m):=\chi (X,\mathcal{G}(m))-R(m)$ satisfies $D(m)-D(m-1)=(\chi (X,\mathcal{G}(m))-\chi (X,\mathcal{G}(m-1)))-(R(m)-R(m-1))=Q(m)-Q(m)=0$ for every $m$, so $D$ is constant by [F6] and $D(0)=C$. Hence the polynomial $Q_{\mathcal{G}}(t):=R(t)+C\in \mathbb{Q}[t]$ satisfies $Q_{\mathcal{G}}(m)=\chi (X,\mathcal{G}(m))$ for every $m\in \mathbb{Z}$, which proves $\mathrm{\Phi }(e)$ and completes the induction.

Existence and uniqueness for infinite $k$. Applying 1.3-1.7 with $e=d$ to $\mathcal{F}$ (and the zero-sheaf case 1.1) produces $P_{\mathcal{F}}\in \mathbb{Q}[t]$ with $P_{\mathcal{F}}(m)=\chi (X,\mathcal{F}(m))$ for every $m$. If $P,P^{\mathrm{\prime }}\in \mathbb{Q}[t]$ agree on all integers, then $P-P^{\mathrm{\prime }}$ has infinitely many roots, so $P-P^{\mathrm{\prime }}=0$ because a nonzero polynomial over the field $\mathbb{Q}$ has at most $\mathrm{deg}(P-P^{\mathrm{\prime }})$ roots; hence $P_{\mathcal{F}}$ is unique and parts 1 and 3 of the statement hold when $k$ is infinite.

Arbitrary base field. Now let $k$ be arbitrary and let $K=k(t)$ be the rational function field, an infinite field extension of $k$ by [F8]. Apply steps 1.1-1.8 to the projective pair $X_K\mathrm{/}K$ with its induced closed immersion $i_K:X_K\hookrightarrow {\mathbb{P}}_K^n$ over the infinite field $K$ and to the coherent module ${\mathcal{F}}_K$; all suppliers used there are stated for an arbitrary field in place of $k$, and $X_K$ is proper over $K$ by [F8]. This gives $P\in \mathbb{Q}[t]$ with $P(m)=\chi (X_K,{\mathcal{F}}_K(m))$ for every $m\in \mathbb{Z}$; by [F8] one has $\chi (X_K,{\mathcal{F}}_K(m))=\chi (X_K,(\mathcal{F}(m){)}_K)=\chi (X,\mathcal{F}(m))$ for every $m$, because $g^{\ast }(\mathcal{F}(m))\cong {\mathcal{F}}_K(m)$ and the Euler characteristics are preserved by base change. Hence $P(m)=\chi (X,\mathcal{F}(m))$ for every $m\in \mathbb{Z}$, and $P$ is unique as in 1.8.

Large twists. By [F7] there is $m_1$ with $H^q(X,\mathcal{F}(m))=0$ for every $q>0$ and every $m\ge m_1$. For such $m$ the defining alternating sum of $\chi (X,\mathcal{F}(m))$ in [F1] reduces to the $q=0$ term, so $\chi (X,\mathcal{F}(m))=h_{\mathcal{F}}(m)$; hence $P_{\mathcal{F}}(m)=h_{\mathcal{F}}(m)$ for every $m\ge m_1$, which is part 2 of the statement with $m_0:=m_1$.

Boundaries and choice. If $X=\mathrm{\varnothing }$ then every ${\mathcal{O}}_X$-module is the zero module, so $\mathcal{F}=0$, all groups vanish and $P_{\mathcal{F}}=0$ by 1.1; the zero sheaf is treated there as well and satisfies part 3. The case $n=0$ is included: ${\mathbb{P}}_k^0=\mathrm{Spec}k$ by [F7]'s conventions, $X$ is closed in it, and all steps apply with the single chart. Both an infinite field $k$ (steps 1.1-1.8) and a finite field $k$ (step 1.9) are covered, and the polynomial identity holds at $m=0$ and at negative $m$ as well, since 1.7 and 1.9 produce polynomials agreeing with $\chi (X,\mathcal{F}(m))$ for every $m\in \mathbb{Z}$ and not merely at large twists; the large-twist hypothesis is needed only for the comparison with $h_{\mathcal{F}}$ in 1.10. The Axiom of Choice is consumed exactly through the suppliers invoked: the finiteness corollary and the definition of the Euler characteristic [F1], the hyperplane lemma [F4], the closed-immersion pushforward and Serre vanishing [F7], and the flat base-change isomorphism [F8]; the extension field $k(t)$ is constructed as a fraction field and not selected, and no further family is chosen.