Consider the polynomial ring $S = k[x_0,\cdots, x_n]$ of $n+1$ many variables, where $k$ is a field. Let $I$ and $J$ be homogeneous ideals in $S$. Consider the localization at the variable $x_i$: $$I_{x_i} := \bigg\{\frac{G}{x_i^m}: G\in I \text{ and } m \in \mathbb{N} \bigg\}$$
The text I am reading seem to implied the following: If for some $i$, one has the equality of localization: $I_{x_i} = J_{x_i}$, then $x_i^t J \subseteq I$ for some integer $t$.
How does one establish the existence of such a $t$? To me it is not clear that such a $t$ must necessarily exist (and be finite). Any hints given would be greatly appreciated!
Pick a finite set of generators $f_1,\cdots,f_m$ for $I$ and $g_1,\cdots,g_l$ for $J$ (you can do this and have $l,m<\infty$ because $S$ is noetherian and thus every ideal is finitely generated). These $f$ then generate $I_{x_i}$ as an ideal of $S_{x_i}$ (similarly for $g$ and $J_{x_i}$). Now, $I_{x_i}=J_{x_i}$ so we can write each $f$ in terms of the $g$ and maybe some negative powers of $x_i$, and vice versa. Next, clear denominators so the relation holds in $S$ - intelligently pick $t$ (you need it to satisfy finitely many conditions here, so you can do it) and you get the result.