2
$\begingroup$

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $(I_{n})_{n\in\mathbb{N}}$ be a decreasing chain of ideals in $R$ such that $\bigcap_{n \in \mathbb{N}} I_{n} = \{ 0 \}$. Then there is a unique separated topology $\tau$ on $R$ which has the family $(I_{n})_{n\in \mathbb{N}}$ as a zero neighborhood and which turns $R$ into a topological ring. If $I_{n} = \mathfrak{m}^{n}$ then it is known that the completion of $R$ has the same dimension as $R$ and in fact this holds also if $I_{n} = J^{n}$ for any proper ideal $J$ of $R$ (see Krull dimension of a completion ).

Does it hold in general for any chain of ideals $(I_{n})_{n \in \mathbb{N}}$ that the completion of $R$ wrt to the uniformity defined by $(I_{n})_{n \in \mathbb{N}}$ has the same dimension as $R$?

Note that if $R$ is $\mathfrak{m}$-adically complete then it is complete in any separated linear topology (any topology obtained in the above way).

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.