Hilbert function and Euler characteristic on a projective scheme

Assume the Axiom of Choice, inherited from the finiteness corollary cited below (The Axiom of Choice). Let $k$ be a field (Field), let $n\ge 0$, and let $i:X\hookrightarrow {\mathbb{P}}_k^n$ be a closed immersion of schemes (Closed immersions of schemes, Relative projective space from standard charts). Thus the structure morphism $X\to \mathrm{Spec}k$ (The underlying space of an affine spectrum) is projective over $k$ in the finite-dimensional H-projective convention (Projective morphisms before Proj), and it is proper (Projective morphisms are proper). Put ${\mathcal{O}}_X(1)=i^{\ast }{\mathcal{O}}_{{\mathbb{P}}_k^n}(1),$ a pullback of the twisting sheaf along the closed immersion (Pullback of a module along a morphism of ringed spaces); it is an invertible ${\mathcal{O}}_X$-module (Invertible sheaves). The embedding $i$ and the sheaf ${\mathcal{O}}_X(1)$ are fixed once and for all, and below "projective $X\mathrm{/}k$" always means $X$ together with this fixed embedding.

Twists. For $m\in \mathbb{Z}$ and an ${\mathcal{O}}_X$-module $\mathcal{F}$ the $m$-th twist of $\mathcal{F}$ is $\mathcal{F}(m)=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(1{)}^{\otimes m}$ (Tensor product of sheaves of modules, Twists of a quasi-coherent sheaf), where ${\mathcal{O}}_X(1{)}^{\otimes m}$ denotes the $m$-fold tensor power of ${\mathcal{O}}_X(1)$ for $m\ge 0$ and the dual of the $(-m)$-fold tensor power for $m<0$ (Invertible sheaves).

The two functions. Let $\mathcal{F}$ be a coherent ${\mathcal{O}}_X$-module (Coherent module sheaves). Each twist $\mathcal{F}(m)$ is then coherent again: coherence is local on $X$, and on an open set on which the invertible sheaf ${\mathcal{O}}_X(1)$ is trivial the twist is isomorphic to $\mathcal{F}$ (Invertible sheaves, Coherent module sheaves, Tensor product of sheaves of modules). Since $X$ is proper over the field $k$, the finiteness corollary Finite-dimensional coherent cohomology over a field shows that every cohomology group $H^q(X,\mathcal{F}(m))$ (Sheaf cohomology as right derived global sections) is a finite-dimensional $k$-vector space (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) and that only finitely many of these groups are nonzero; hence the Euler characteristic $\chi (X,\mathcal{F}(m))={\sum }_{q\ge 0}(-1{)}^q{\dim }_k{H}^q(X,\mathcal{F}(m))$ of Euler characteristic of a coherent sheaf is a well-defined integer. The Hilbert function of $\mathcal{F}$ with respect to the fixed embedding is the function $\mathbb{Z}⟶{\mathbb{Z}}_{\ge 0},m⟼h_{\mathcal{F}}(m):={\dim }_k{H}^0(X,\mathcal{F}(m)),$ and the Euler-characteristic function of $\mathcal{F}$ is the function $\mathbb{Z}⟶\mathbb{Z},m⟼P_{\mathcal{F}}(m):=\chi (X,\mathcal{F}(m))\mathrm{.}$

Relation between the two. Writing $h_{\mathcal{F}}^q(m)$ for ${\dim }_k{H}^q(X,\mathcal{F}(m))$, only finitely many of which are nonzero, the two functions are related by $P_{\mathcal{F}}(m)={\sum }_{q\ge 0}(-1{)}^q{h}_{\mathcal{F}}^q(m)=h_{\mathcal{F}}(m)-{\sum }_{q>0}(-1{)}^{q-1}{h}_{\mathcal{F}}^q(m)\mathrm{.}$ In particular $P_{\mathcal{F}}(m)=h_{\mathcal{F}}(m)$ whenever the higher cohomology groups $H^q(X,\mathcal{F}(m))$ with $q>0$ all vanish; the higher groups contribute to the Euler-characteristic function but not to the Hilbert function, and the definition imposes no vanishing of them at any particular $m$.

Hilbert polynomial. If there is a polynomial $p\in \mathbb{Q}[t]$ with $p(m)=P_{\mathcal{F}}(m)\text{for every }m\in \mathbb{Z},$ then $p$ is called a Hilbert polynomial of $\mathcal{F}$ with respect to the fixed embedding, and the notation $P_{\mathcal{F}}(t)$ is also used for it once it is known to exist. Existence and uniqueness of such a polynomial are not asserted by this definition.

If $X=\mathrm{\varnothing }$ or $\mathcal{F}=0$, then all groups $H^q(X,\mathcal{F}(m))$ vanish, so $h_{\mathcal{F}}\equiv 0$ and $P_{\mathcal{F}}\equiv 0$, and the zero polynomial is a Hilbert polynomial.