Definition of projectively normal in Harshorne Ex II.5.14 In Hartshorne, Ex II.5.14, he defines that a closed subscheme $X \in \mathbb{P}^r_A$ is called projectively normal if the homogeneous coordinate ring $S(X)$ is integrally closed. This is also given by many notes and in wikipedia. But the denominators of elements of the fraction field of $S(X)$ may not be homogeneous, which makes things funny and the problem difficult to solve.
My question is whether this definition is the right one. If yes, how does it make sense? If no, what is the common way to fix the definition?

This definition is fine and part (d) gives you a definition which is equivalent and doesn't encounter this issue of your. Your concern can be fixed with a little ingenuity.
Suppose $R=\bigoplus_d R_d$ is a graded domain, and let $K$ be its quotient field. Assume there exists a nonzero element $y\in R_1$. Let $K_d$ denote the subgroup of elements which can be represented as $\frac{r_1}{r_2}$ for $r_1,r_2$ homogeneous and $\deg r_1-\deg r_2=d$. Then $K'=\bigoplus_{d\in\Bbb Z} K_d$ is a subring of $K$, it's isomorphic to $K_0[y,1/y]$, and $K=K_0(y)=\operatorname{Frac} K'$. We observe that $K_0[y,1/y]$ is integrally closed in its field of fractions, being the localization of a polynomial ring over a field, so the integral closure of $R$ must be a subring of $K'$, and in particular consist of sums of homogeneous fractions. So we've eliminated your difficulty.
