On the Hilbert function of projective schemes

Question

Let $X \subset \mathbb{P}^n$ be a projective subscheme (not necessarily reduced or irreducible). Denote by $I_X$ the ideal of $X$ i.e., $\Gamma_*(\mathcal{I}_X)$. There are two definitions of Hilbert functions that is used in the literature. One is the function $f(t)=\mbox{dim}(S/I_X)_t$, the $t$-graded part of the module where $S$ is the polynomial ring $k[X_0,...,X_n]$ and $t \in \mathbb{Z}$. The other definition is $f(t)=h^0(\mathcal{O}_X(t))$ for any integer $t$. Why are these two definitions compatible? I understand why they are the same for $t$ large enough.

Answer 1

Note that $I_X(t)=\Gamma_*\mathcal{I}_X(t)$ which is equal to $H^0(\mathcal{I}_X(t))$ by definition. Using the short exact sequence mentioned above, you get the equivalence of the two definitions.