I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging MaBloWriMo.

Let’s summarize the story so far. is a commutative ring, and is the set of maximal ideals of endowed with the Zariski topology, where the sets are a basis for the closed sets. Sometimes we will refer to the closed sets as **varieties**, although this is mildly misleading. Here denotes an element of , while denotes the corresponding ideal as a subset of ; the difference is more obvious when we’re working with polynomial rings, but it’s good to observe it in general.

We think of elements of as functions on as follows: the “value” of at is just the image of in the **residue field** , and we say that vanishes at if this image is zero, i.e. if . (As we have seen, in nice cases the residue fields are all the same.)

For any subset the set is an intersection of closed sets and is therefore itself closed, and it is called the **variety** defined by (although note that we can suppose WLOG that is an ideal). In the other direction, for any subset the set is the **ideal** of “functions vanishing on ” (again, note that we can suppose WLOG that is closed).

A natural question presents itself.

**Question:** What is ? What is ?

In other words, how close are to being inverses?