Noetherian scheme

Imagine a big playground with some toys scattered around. This playground is called a scheme. Now, let's say that we don't want to be there forever and we want to stop playing and go home. But how do we know when we can stop playing and leave the playground? This is where the concept of noetherian scheme comes in.

A noetherian scheme is like a well-behaved playground where we can easily count all the toys and know exactly when we've played with them all. This means that if we start playing with the toys and put them back, we can always pick up a new toy to play with until we've played with them all.

In more technical terms, a scheme is said to be noetherian if every decreasing chain of closed subsets (like the number of toys we need to play with) stops after finitely many steps (when we've played with all the toys). This property is important because it allows us to apply certain mathematical tools and simplifies the analysis of the scheme.

So, in summary, a noetherian scheme is a well-behaved playground where we can easily count all the toys and know exactly when we've played with them all. This makes it easy for us to leave and go home once we've played with everything.