Let $S=k[x_0,\dots,x_n]$ be the "homogeneous polynomial ring" of $\mathbb P^n$ and let $S(Y)_{x_i}$ denote the localization at the image of $x_i\in S(Y)$ of $S(Y)=S/I(Y)$. In Hartshorne, he asks us to identify a certain ring with the subring of $S(Y)_{x_i}$ consisting of degree-0 elements.
My question is - what grading should these elements be degree zero with respect to? I know that the quotient inherits a natural grading $$S/I=\bigoplus_{d\geq 0}(S_d+I)/I,$$ but I don't know how the localization inherits its grading - in particular, how can we make sense of what has degree zero when we can divide by powers of $x_i$?