Let $A\subset B$ be an inclusion of $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domains. Is the integral closure of $A$ in $B$ also $\mathbb{Z}/n\mathbb{Z}$-graded?


This is true for the $G$-graded case with $G$ a nice torsion-free integral commutative monoid, see Theorem 2.3.2 of [Huneke-Swanson, Integral Closure of Ideals, Rings, and Modules] for $G=\mathbb{Z}^r\times\mathbb{N}^s$ (even without noetherian condition).



Is the integral closure of a $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domain in a bigger $\mathbb{Z}/n\mathbb{Z}$-graded domain also graded?
Example 2.3.3 in Huneke and Swanson answers this in the negative.


