# Open subscheme of Noetherian integral scheme is integral?

Let $$(X, \mathcal O_X)$$ be a Noetherian scheme. If $$X$$ is an integral scheme (i.e. $$X$$ is irreducible and reduced scheme), then is it true that for every open subset $$U$$ of $$X$$, the open subscheme $$(U, \mathcal O_U)$$ is integral ? (Here $$\mathcal O_U=\mathcal O_X|_U$$)

I think this is true and here are my thoughts: $$U$$ being an open subset of $$X$$, is irreducible. Moreover, for every $$u\in U$$, we have $$\mathcal O_{U,u}\cong \mathcal O_{X,u}$$ is reduced , hence $$U$$ is reduced. Thus $$U$$ is integral.

Am I correct ?

• That proof is correct, and its a good exercise to also prove this in the affine case for a distinguished open affine $D_f$ in an affine integral scheme $X$, to see the corresponding commutative algebra fact. – user277182 Jun 30 at 3:54