Two definitions of Hilbert series/Hilbert function in algebraic geometry

Question

In classical algebraic geometry, suppose $I$ is a reduced homogeneous ideal in $k[x_0,\cdots,x_n]$, where $k$ is algebraically closed field, then $I$ cuts out a projective variety $X$, whose Hilbert function is defined by \begin{equation} \phi(m)=\text{dim}_k~ (k[x_0,\cdots,x_n]/I)_m \end{equation} where $()_m$ means the $m$-th homogeneous part of the quotient ring. When $m$ is large, there exists a polynomial $p_X$ such that \begin{equation} \phi(m)=p_X(m) \end{equation} $p_X$ is the Hilbert polynomial. However in the scheme world, from section 18.6 of Ravi Vakil's book, the Hilbert function for a projective scheme $X \hookrightarrow \mathbb{P}^n$ is defined by \begin{equation} \phi(m)=h^0(X,\mathcal{O}(m)) \end{equation} When $m$ is large, there exists a polynomial $p_X$ such that \begin{equation} \phi(m)=\chi(X,\mathcal{O}(m))=p_X(m) \end{equation} Are the two definitions of Hilbert functions the same? I guess there are issues with reducedness, could anyone give an overall explanation?

Answer 1

Look at the graded $k$-algebra $A=k[x_0,\ldots,x_n]/I$. The graded $k$-algebra surjection $k[x_0,\ldots,x_n]\to A$ yields a closed $k$-immersion $j:X=\mathrm{Proj}(A)\hookrightarrow\mathbf{P}_k^n$ with image $V_+(I)$. The Hilbert function $p_X$ of $X$ relative to the closed immersion $j$ (i.e. to the very ample invertible sheaf $\mathscr{O}_X(1)=j^*(\mathscr{O}_{\mathbf{P}_k^n}(1))$ is given by $p_X(m)=\dim_k(H^0(X,\mathscr{O}_X(m)))$ for $m\gg 0$, i.e., it is the Hilbert function of the finitely generated $k[x_0,\ldots,x_n]$-module $\bigoplus_{m\geq 0}H^0(X,\mathscr{O}_X(m))=\bigoplus_{m\geq 0}H^0(\mathbf{P}_k^n,(j_*(\mathscr{O}_X))(m))$. There is a canonical graded $k$-algebra map $A/I\to\bigoplus_{m\geq 0}H^0(X,\mathscr{O}_X(m))$ and it is an isomorphism in sufficiently large degree (see http://stacks.math.columbia.edu/tag/0AG7). Thus the Hilbert functions agree for sufficiently large $m$, so the Hilbert polynomials coincide.