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 ?

  • $\begingroup$ 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. $\endgroup$ – user277182 Jun 30 at 3:54

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