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?