## 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 be a functor, where is a filtered small category and is a finite category. Then the natural mapping

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 be a finitely generated ideal in a ring . Then the -adic completion is Noetherian if is.

Corollary: Let be a complete local ring with a finitely generated maximal ideal . Then is …