Fun example: Empty colimit does not commute with empty limit

One important property of filtered colimits is that they commute with finite limits in the category of sets.

Theorem: Let \(F \colon \mathcal{C}\times \mathcal{D} \to \mathbf{Sets}\) be a functor, where \(\mathcal{C}\) is a filtered small category and \(\mathcal{D}\) is a finite category. Then the natural mapping

\[\mathrm{colim}_{\mathcal{C}} \lim_{\mathcal{D}} F (c, d) \to \lim_{\mathcal{D}} \mathrm{colim}_{\mathcal{C}} F(c, d)\]

is an isomorphism.

This statement is used for example to check that …

A non-Noetherian local ring with finitely generated maximal ideal

Some time ago I found the following interesting lemma on the stacksproject:

Theorem: (tag/05GH) Let \(I\) be a finitely generated ideal in a ring \(R\). Then the \(I\)-adic completion \(\widehat{R}\) is Noetherian if \(R/I\) is.

Corollary: Let \(R\) be a complete local ring with a finitely generated maximal ideal \(\mathfrak{m}\). Then \(R\) is …