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Under what conditions is the tensor product of two dvrs semilocal?

The same question about being reduced.

Tensor product is taken over another dvr or over a field to make things simpler.

For example, $\mathbb Z_p \otimes_{\mathbb Z_p} \mathbb Z_p$ is clearly reduced. What is (a highbrow) reason for this?

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    $\begingroup$ The tensor products of two fields over a common subfield $k$ is in general not reduced, but it is if one of the two is separable over $k$. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Mar 3, 2011 at 23:28
  • $\begingroup$ Thank you. What if at least one of the rings is not a field? $\endgroup$
    – unknown
    Commented Mar 4, 2011 at 0:33

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If the two dvrs $S$ and $T$ are torsion-free over the base dvr $R$, then $S$ and $T$ are flat over $R$. So the tensor product is flat over $S$ (and over $T$). Therefore if the tensor products of the quotient fields $Q(S)\otimes_{Q(R)}Q(T)$ is reduced, then so is $S \otimes_R T$.

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