# Proving $R[Y_1, \ldots, Y_r]_{P}$ is integrally closed (trying to prove $\mathbb{P}^r_R$ is normal)

Let $$R$$ be an integral domain with algebraically closed fraction field. Let $$P$$ be a prime ideal of $$R[Y_1, \ldots, Y_r]$$. Then it follows that $$R[Y_1, \ldots, Y_r]_{P}$$ is an integral domain. I would like to prove that $$R[Y_1, \ldots, Y_r]_{P}$$ is integrally closed in its fraction field. Any explanation would be appreciated. Thank you.

Edit. I am asking this question because I wanted to prove that $$\mathbb{P}^r_R$$ is a normal scheme when $$R$$ is a valuation ring with an algebraically closed fraction field.

• If $R$ is a valuation ring then it is integrally closed, then so is any polynomial ring over $R$, and its localizations as well. Mar 24, 2020 at 19:40
• @user26857 that looks like an answer to me - would you care to record it as such? Mar 24, 2020 at 19:48

If $$R$$ is a valuation ring then it is integrally closed, then so is any polynomial ring over $$R$$, and its localizations as well. – user26857