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Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ A$ of primes of height $d-1$ such that there is exactly one prime in $\textrm{Spec}\ B$ over a prime of $U$. Is it then true that $\textrm{Spec}\ B$ is an irreducible topological space?

Under various choices of ancillary hypotheses, results like this one follow from Zariski's connectedness principle; see e.g A useful form of principle of connectednessA useful form of principle of connectedness.

Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ A$ of primes of height $d-1$ such that there is exactly one prime in $\textrm{Spec}\ B$ over a prime of $U$. Is it then true that $\textrm{Spec}\ B$ is an irreducible topological space?

Under various choices of ancillary hypotheses, results like this one follow from Zariski's connectedness principle; see e.g A useful form of principle of connectedness.

Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ A$ of primes of height $d-1$ such that there is exactly one prime in $\textrm{Spec}\ B$ over a prime of $U$. Is it then true that $\textrm{Spec}\ B$ is an irreducible topological space?

Under various choices of ancillary hypotheses, results like this one follow from Zariski's connectedness principle; see e.g A useful form of principle of connectedness.

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Olivier
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A weaker form of Zariski's connectedness principle

Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ A$ of primes of height $d-1$ such that there is exactly one prime in $\textrm{Spec}\ B$ over a prime of $U$. Is it then true that $\textrm{Spec}\ B$ is an irreducible topological space?

Under various choices of ancillary hypotheses, results like this one follow from Zariski's connectedness principle; see e.g A useful form of principle of connectedness.